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Symmetries and conservation laws

Acta Wexionensia No 170/2009 Mathematics

Symmetries and conservation laws

Raisa Khamitova

Växjö University Press

Symmetries and conservation laws. Thesis for the degree of Doctor of Philoso-phy, Växjö University, Sweden 2009.

Series editor: Kerstin Brodén ISSN: 1404-4307 ISBN: 978-91-7636-650-9 Printed by: Intellecta Infolog, Göteborg 2009

AbstractConservation laws play an important role in science. The aim

of this thesis is to provide an overview and develop new methodsfor constructing conservation laws using Lie group theory. Thederivation of conservation laws for invariant variational problemsis based on Noether’s theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the numberof basic conserved quantities for differential equations obtained byNoether’s theorem and construct a basis of conservation laws. Sev-eral examples on constructing a basis for some well-known equa-tions are provided.

Moreover, this approach allows one to obtain new conservationlaws even for equations without Lagrangians. A formal Lagrangiancan be introduced and used for computing nonlocal conservationlaws. For self-adjoint or quasi-self-adjoint equations nonlocal con-servation laws can be transformed into local conservation laws.One of the fields of applications of this approach is electromag-netic theory, namely, nonlocal conservation laws are obtained forthe generalized Maxwell-Dirac equations. The theory is also ap-plied to the nonlinear magma equation and its nonlocal conserva-tion laws are computed.

Keywords: conservation law, Noether’s theorem, Lie group anal-ysis, Lie-Bäcklund transformations, basis of conservation laws, for-mal Lagrangian, self-adjoint equation, quasi-self-adjoint equation,nonlocal conservation law

v

AcknowledgmentsMy most sincere thanks are due to my supervisors Professor

Börje Nilsson and Assistant Professor Claes Jougréus for theirguidance and encouragement during my work on this thesis. Mythanks also go to Professor Andrej Khrennikov for offering valuableadvice during my studies as a PhD student.

I would like to express my greatest gratitude to Eva Pettersson,head of the Department of Mathematics and Science, and Jan-OlofGustavsson, head of School of Engineering at Blekinge Institute ofTechnology, for their support of my study.

I want to express my appreciation to Dr. Robert Nyqvist for hishelp with LATEX. My thanks to my colleagues at the Departmentof Mathematics and Science who showed interest towards my workduring this period. I would like to deeply thank the various peoplewho provided me with useful and helpful assistance.

Finally, my special thanks to my husband and our daughtersfor all the good advice and their help and to my friend NadezhdaBalayan for supporting me all these years.

vi

ContentsAbstract vAcknowledgments viPreface 11 Introduction 32 Conservation laws 4

2.1 Concept of a conservation law . . . . . . . . . . . . . . . . . . 42.2 Hamilton’s principle and the Euler-Lagrange equations . . . 62.3 Lie group transformations and Noether’s theorem . . . . . . . 7

3 A basis of conservation laws 93.1 Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Relativistic mechanics . . . . . . . . . . . . . . . . . . . . . . 123.3 Motion in the de Sitter space . . . . . . . . . . . . . . . . . . 143.4 Nonlinear wave equation . . . . . . . . . . . . . . . . . . . . . 153.5 Lin–Reissner–Tsien equation . . . . . . . . . . . . . . . . . . . 153.6 Transonic three-dimensional gas motion . . . . . . . . . . . . 163.7 Short waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.8 Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Equations without Lagrangians 184.1 Formal Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Maxwell-Dirac equations . . . . . . . . . . . . . . . . . . . . . 194.3 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . 204.4 General magma equation . . . . . . . . . . . . . . . . . . . . 21

5 Summary of thesis 216 Summary of papers 22

6.1 Paper 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Paper 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.3 Paper 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4 Paper 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Bibliography 25Paper 1 29Paper 2 37Paper 3 45Paper 4 59

vii

PrefaceThis thesis consists of two parts. The first part is an introductory sectionwhich gives the reader some necessary background on the subject matter.The thesis is based upon 4 papers included in the second part.

Papers included in the thesis:

1. On a basis of conservation laws of mechanics equationsR. KhamitovaDokl. Akad. Nauk SSSR, 248(4), 1979, pp. 798–802.English trans. Soviet Math. Dokl., 248(5), pp. 1071–1075, 1980.

2. Group structure and a basis of conservation lawsR. KhamitovaTeor. i Matem. Fizika, 52(2), pp. 244–251, 1982.English trans. Theor. Math. Phys., 52(2), pp. 777–781, 1983.

3. Conservation laws for Maxwell-Dirac equations with dual Ohm’s lawN. Ibragimov, R. Khamitova and B. ThidéJ. Math. Phys., 48(5), pp. 053523-1–053523-11, 2007.

4. Symmetries and nonlocal conservation laws of the general magma equa-tionR. KhamitovaCom. Nonl. Sci. Num. Sim., 2008, doi: 10.1016/j.cnsns.2008.08.009

Contribution to the paper 3

In the paper 3 mentioned above, the author of this thesis calculated conser-vation laws.

Related papers not included in the thesis:

• On a basis of conservation laws in mechanicsR. KhamitovaContinuum Dynamics, 38, Novosibirsk, pp. 151–159, 1979 (In Rus-sian).

• Adjoint system and conservation laws for symmetrized electromagneticequations with a dual Ohm’s lawN. Ibragimov, R. Khamitova and B. ThidéArchives of ALGA, 3, ALGA publ., BTH, Karlskrona, Sweden,pp. 81–95, 2006.

• Utilization of photon orbital angular momentum in the low-frequencyradio domainB. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi,Ya. N. Istomin, N. H. Ibragimov, and R. KhamitovaPhys. Rev. Lett., 99, pp. 087701-1–087701-4, 2007.

• Self-adjointness and quasi-self-adjointness of the magma equationR. Khamitova

1

in: J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, L. B. Figueiredo(Eds.), Proc. of the 2nd conference on "Nonlinear Science and Com-plexity", July 28–31, 2008, Porto, Portugal, ISBN:978-972-8688-56-1.

• Self-adjointness and quasi-self-adjointness of an equation modelling meltmigration through the Earth’s mantle. Nonlocal conservation lawsR. KhamitovaPreprint in Archives of ALGA, 5, ALGA publ., BTH, Karlskrona, Swe-den, pp. 143–159, 2008.

2

1 Introduction

1 IntroductionConservation laws and symmetries have always been of considerable interestin science. They are important in the formulation and investigation of manymathematical models. They were used, e.g. for proving global existencetheorems [1]–[3], in problems of stability [4], [5], in elasticity for studyingcracks and dislocations [6], [7], in astrophysics [8]–[10], in designing newradio antennas [11] and so on (see also [12]).

Let us look at the use of symmetries and conservation laws, e.g. in celes-tial mechanics. In 1609, J. Kepler formulated two important laws known asKepler’s first and second laws. His first law states that the orbit of a planetis an ellipse with the Sun as its focus. The second law says that if we jointhe Sun and a planet by a straight line, the line will sweep out equal areasat equal times.

What was important in these discoveries is that Kepler explained how theplanets moved. The next step, the explanation why they moved in such away, was given by I. Newton [13] in 1687. He formulated his law of gravity:

F = Gm1m2

r2, (1.1)

where F is the force of gravity between two particles, G is a gravitationalconstant, m1 and m2 are the masses of the particles and r is the distancebetween them. At the beginning Newton tried to use a formula with r3

instead of r2. However, he found out that it was not fruitful. When Newtonused the force of gravity (1.1) in his second law of motion, he obtained thatplanets moved in ellipses. It proved to him that he was on the right track.Thus, his way of discovery was by trial and error.

P.–S. Laplace showed that planets’ movement along ellipses followed fromthe conservation law calculated by him, i.e. the conservation law for thevector (see [14], Vol.1, Book II, Chap. III, Section 18):

A = v × M+ μxr

, (1.2)

where v is the velocity of a planet, M = m(x×v) is the angular momentum,m is the planet’s mass, x is a position-vector of the planet and r is themagnitude of x. Laplace used the formal definition of a conservation law forcalculation of this conserved vector.

In 1983, N. H. Ibragimov [15] showed that it was possible to calculatethe vector (1.2) by using a certain symmetry of the Newton gravitationalfield, a Lie-Bäcklund symmetry. This symmetry is more complicated than,e.g. rotations, it depends not only on the position vector x but also on thevelocity v. Thus, the idea of symmetry and the corresponding conservationlaw helps to explain the movement of planets in ellipses.

Kepler’s second law, the conservation of areas, follows from the conser-vation of angular momentum. This was established [16] independently byL. Euler and D. Bernoulli. The angular momentum corresponds to thecental symmetry of Newton’s gravitational field.

3

2 Conservation laws

There are several ideas for constructing conservation laws. One of them isto use the direct method, when a conservation law for a differential equationis derived by using its definition. As mentioned earlier, Laplace was the firstone who used this idea in 1798.

Another idea, that certain conservation laws for differential equationsobtained from a variational principle could appear from their symmetries,followed from the works of Jacobi, Klein and Noether. In 1884, Jacobi [17]showed a connection between conserved quantities and symmetries of theequations of a particle’s motion in classical mechanics. Similar result wasobtained by Klein [18] for the equations of the general relativity. Klein pre-dicted that a connection between conservation laws and symmetries could befound for any differential equation obtained from a variational principle. Hesuggested to Emmy Noether to investigate the possibility. She showed [19]in 1918 that the conservation laws were associated with invariance of varia-tional integrals with respect to continuous transformation groups. Noetherobtained the sufficient condition for existence of conservation laws. How-ever, there are no explicit expressions for resulting conservation laws inNoether’s work. In 1921, following Noether’s oral remark, Bessel-Hagen[20] applied Noether’s theorem with the so-called "divergence" condition tothe Maxwell equations and calculated their conservations laws.

In 1951, Hill wrote a remarkable review paper [21] where he discussedNoether’s theorem and presented the explicit formula for conservation lawsin the case of a first-order Lagrangian. The formula is written in terms ofvariations (see [21], Eq. (43)). In 1969, inspired by Hill’s article, Ibragimov[22] proved the generalized version of Noether’s theorem. In this theoremconservations laws are related to the invariance of the extremal values ofvariational integrals. He derived the necessary and sufficient condition forexistence of conservation laws. He also presented the explicit expressionsfor calculating conservation laws in the case of a Lagrangian of any order.On the basis of these theorems many conservations laws for differentialequations having a Lagrangian were calculated (see collected examples in[23]–[25]).

2.1 Concept of a conservation law

Let us consider an ordinary differential equation

F (t, q, q, q) = 0 (2.1)

describing a motion of a dynamical system. Here t is time, q = (q1, ..., qs)are the position coordinates, q = q(t), and v = q ≡ dq

dt is the velocity,q = d2q

dt2 .

4

2 Conservation laws

Definition 2.1. A function C = C(t, q, v) is called a conserved quantityfor Eq. (2.1) if

dC

dt= 0 (2.2)

on every solution of Eq. (2.1).In other words, the conserved quantity C(t, q, v) is constant on each tra-

jectory q = q(t) and therefore is called a constant of motion.In classical mechanics Eq. (2.1) has the form

mx = 0 (2.3)

and describes a free motion of a particle with the mass m and a positionvector x = (x1, x2, x3). The equation has several conserved quantities, e.g.the energy E = 1

2mv2 and the linear momentum p = mv.Let us now consider a partial differential equation of p-th order

F (x, u, u(1), u(2), . . . , u(p)) = 0 (2.4)

where the function F depends on n independent variables x, x = (x1, . . . , xn),m dependent variables u, u = (u1, . . . , um), and the first, second, ..., p-thorder derivatives of u with respect to x denoted as u(1) = {uα

i }, u(2) ={uα

ij}, . . . , u(p) = {uαi1i2...ip

} respectively, α = 1, . . . , m and other indiceschange from 1 to n.Definition 2.2. A vector C = (C1, C2, . . . , Cn) where

Ci = Ci(x, u, u(1), . . .), i = 1, . . . , n,

is called a conserved vector for Eq. (2.4) if

div C = 0 (2.5)

on every solution of Eq. (2.4). We can also say that Eq. (2.5) is a conser-vation law for Eq. (2.4).

A conservation law for a system of partial differential equations can bedefined similarly.

Instead of dealing with functions uα = uα(x) and their derivatives, whichare also functions of x, one can treat all variables, x, u and derivatives of u, asindependent variables, called differential variables. Variables with the sameset of subscripts will be symmetric, for example uij = uji and so on. Usingthe idea of differential variables [26] one can reformulate the definition ofa conservation law by introducing the operator of total differentiation withrespect to xi:

Di =∂

∂xi+ uα

i

∂

∂uα+ uα

ij

∂

∂uαj

+ · · ·+ uαij1...jk

∂

∂uαj1...jk

+ · · · (2.6)

where the usual convention of summation over repeated upper and lowerindices is used. Hence

div C∣∣(2.4)

≡ Di(Ci)∣∣(2.4)

= 0 (2.7)

5

where the notation∣∣(2.4)

means that the relation holds on any solutionof Eq. (2.4). If one of the variables, for example x1, is time t then thecomponent C1 is called the density of the conservation law.

Remark 2.1. In practical calculations the conservation law (2.7) can berewritten to an equivalent form. If

C1∣∣(2.4)

= C1 +D2(h2) + · · ·+Dn(hn).

then one obtains the following conservation law:

Dt(C1) +D2(C2) + . . .+Dn(Cn) = 0

whereC2 = C2 +Dt(h2), . . . , Cn = Cn +Dt(hn)

because DtDi(hi) = DiDt(hi).

I have used this in my calculations of conservation laws.By employing differential variables one can also rewrite Eq. (2.2) in the

following form:dC

dt

∣∣∣(2.1)

≡ Dt(C)∣∣(2.1)

= 0. (2.8)

Thus, conserved quantities and conserved vectors can be computed withthe help of Eq. (2.8) and Eq. (2.7), respectively (see, e.g. [27]–[31]).

2.2 Hamilton’s principle and the Euler-Lagrange equa-tions

Consider again a motion of a dynamical system with a kinetic energy T (t, q, q)and a potential energy U(t, q). The function

L(t, q, v) = T (t, q, q)− U(t, q)

is called the Lagrangian of the system.Hamilton’s principle, or the principle of least action, states that the true

motion of the system between two chosen times t1 and t2 is described bythe fact that the trajectories of the particles provide an extremum of theaction functional ∫ t2

t1

L(t, q, v) dt. (2.9)

This requirement is equivalent to the statement that the Euler-Lagrangeequations:

∂L∂qα

− Dt

( ∂L∂vα

)= 0, α = 1, . . . , s (2.10)

hold. They give a necessary condition for g(t) to provide an extremum ofthe integral (2.9).

6

2 Conservation laws

In the case of several independent variables x = (x1, . . . , xn) and depen-dent variables u = (u1, . . . , um) an action integral has the form∫

V

L(x, u, u(1), . . . , u(p)) dx (2.11)

where V is an arbitrary n-dimensional volume in the space of the variablesx and the Lagrangian L is a function depending on a finite number ofdifferential variables. The corresponding Euler-Lagrange equations havethe form:

δLδuα

= 0, α = 1, ..., m, (2.12)

where

δ

δuα=

∂

∂uα− Di

∂

∂uαi

+ . . .+ (−1)sDi1Di2 . . . Dis

∂

∂uαi1i2...is

+ . . . (2.13)

is the variational derivative.In my first two articles I discuss conservation laws for the Euler-Lagrange

equations.

Definition 2.3. A conservation law is called a trivial conservation law if

Di(Ci) ≡ 0or Ci are smooth functions of δL

δuα , DiδLδuα , . . . . Two conservation laws which

only differ by a trivial conservation law are regarded as equivalent.

2.3 Lie group transformations and Noether’s theoremAssume that the Euler-Lagrange equations (2.12) admit a one-parameterLie transformation group G, i.e. a local group of transformations

x = ϕ(x, u, a), u = ψ(x, u, a),

whereϕ = (ϕ1, ..., ϕn), ψ = (ψ1, ..., ψm),

andϕ(x, u, 0) = x, ψ(x, u, a) = u.

The infinitesimal generator of the group G has the form

X = ξi(x, u)∂

∂xi+ ηα(x, u)

∂

∂uα, (2.14)

where

ξi(x, u) =∂ϕi(x, u, a)

∂a

∣∣∣a=0

, ηα(x, u) =∂ψα(x, u, a)

∂a

∣∣∣a=0

.

7

Definition 2.4. A variational integral (2.10) is invariant under the groupG if ∫

V

L(x, u, u(1), . . . , u(p)) dx =∫

V

L(x, u, u(1), . . . , u(p)) dx.

The invariance condition is given by the following lemma.Lemma 2.1. An integral (2.11) is invariant under the group G if and onlyif [15]

X(L) + LDi(ξi) = 0. (2.15)Here X is a prolonged version of the generator (2.14):

X = ξi ∂

∂xi+ ηα ∂

∂uα+ ζα

i

∂

∂uαi

+ · · ·+ ζαi1...is

∂

∂uαi1...is

+ · · · , (2.16)

whereζαi = Di(ηα − ξjuα

j ) + ξjuαji ,

ζαi1...is = Di1 ...Dis(η

α − ξjuαj ) + ξjuα

ji1...is .

Noether proved her theorem by the application of the variational proce-dure to the integral of action. Using her idea Hill presented the explicit formof conserved quantities in the case of the first-order Lagrangians L(x, u, u(1))(see [21], Eq. (43)). In my articles I have used the following generalizedform of Noether’s theorem proved by Ibragimov [22], [32] on the basis ofthe group-theoretical approach.Theorem 2.1. Let the variational integral (2.11) be invariant with respectto a group G with generators (2.14). Then a vector C with components

Ci = N i(L), i = 1, 2, ..., n, (2.17)

is a conserved vector for the Euler-Lagrange equations (2.12), i.e.

Di(Ci)∣∣2.12)

= 0. (2.18)

Here N i are Ibragimov’s operators [32], [15]:

N i = ξi +Wα{ ∂

∂uαi

+∑s≥1

(−1)sDj1 ...Djs

∂

∂uαij1...js

}(2.19)

+∑r≥1

Dk1 ...Dkr (Wα)

{ ∂

∂uαik1...kr

+∑s≥1

(−1)sDj1 ...Djs

∂

∂uαik1...krj1...js

},

where Wα = ηα − ξjuαj .

Corollary. If for some one-parameter transformation group the invariancecondition (2.15) is not satisfied but the "divergence" condition

X(L) + LDi(ξi) = Di(Bi) (2.20)

holds, then the components of the corresponding conserved vector have theform:

Ci = N i(L)− Bi, i = 1, 2, ..., n. (2.21)

8

3 A basis of conservation laws

3 A basis of conservation lawsBesides the operator X, we shall also use its equivalent canonical Lie-Bäcklund operator [33], [34]:

X = X − ξjDj = ηα ∂

∂uα+ ζα

i

∂

∂uαi

+ · · ·+ ζαi1...is

∂

∂uαi1...is

+ · · · , (3.1)

where

ηα = Wα = ηα − ξjuαj , ζα

i = Di(ηα) , ζαi1...is = Di1 ...Dis(η

α).

Some conservation laws can be obtained more readily by using the com-mutativity of the generators X (or generators (2.14) with ξ1 = const., ...,ξn = const. ) and Di.

Lemma 3.1. A canonical Lie–Bäcklund operator X and an operator oftotal differentiation Di are commutative [15]:

XDi = DiX.

Lemma 3.2. If C = (C1, ..., Cn) satisfies a conservation law for some dif-ferential equation and a generator X is admitted by the equation in questionthen the vector with the components

C′i = X(Ci) (3.2)

also satisfies a conservation law [15].

Hence lemmas 3.1 and 3.2 furnish the basis for another idea for calculat-ing conservation laws. Moreover, conserved vectors can be computed for adifferential equation without any Lagrangian if it has a known conservationlaw (see, e.g. [35] and [36]).

The property (3.2) makes it possible to introduce the concept of a basis(with respect to the group G) of the conservation laws and thus reducethe number of vectors C that must be constructed by means of Noether’stheorem.

Definition 3.1. Let {C} be a set of vectors satisfying the conservation law(2.18). A basis of the set {C} is its minimal subset from which {C} can beobtained by repeated application of (3.2) and by linear combinations. Theconservation laws corresponding to the basis vectors form the basis of theconservation laws.

Using an example of gasdynamics equations, it was conjectured in [32]that the following diagram is commutative:

X1adX−→ X2

Ni1

⏐⏐� ⏐⏐�Ni2

C1X−→ C2

9

and this statement can be used for construction of a basis of conservedvectors. The operators N i

1 and N i2 in the diagram are given by (2.19) and

X, X1, X2 by (2.14), the action adX is defined as follows:

adX(X1) ≡ [X, X1] = XX1 − X1X.

Hence the commutator [X, X1] has the form

[X, X1] =(X(ξi

1)− X1(ξi)) ∂

∂xi+

(X(ηα

1 )− X1(ηα)) ∂

∂uα· (3.3)

Following this idea I verified the validity of the statement by means ofseveral examples [37], [38] and then proved the following general result [39].

Theorem 3.1. Let generators X, X1, X2 of the form (2.14) be admittedby the Euler-Lagrange equations (2.12). Let the conserved vectors C1, C2

correspond (by Noether’s theorem) to the generators X1, X2 and let

[X, X1] = X2.

Then the vectors X(C1) and C2 define equivalent conserved vectors, i.e.

X(C1) = C2.

Remark 3.1. The theorem also holds when instead of the invariance con-dition (2.15) of a variational integral we have the "divergence" condition(2.20.)

The proof of the theorem is given in [39]. Later this theorem was formu-lated in another form in [12]. Specific examples given in [37] were used byTsujishita [40] as applications in modern formal differential geometry.

Let us consider several examples.

3.1 Classical mechanicsLet Eq. (2.3),

mx = 0, x = (x1, x2, x3),

describe the free motion of a particle of mass m. The equation has theLagrangian

L = 12

m|x|2

and admits a 10-parameter point transformation group containing spacetranslations with the generators

Xμ =∂

∂xμ, μ = 1, 2, 3, (3.4)

translation of time with the generator

X4 =∂

∂t, (3.5)

10


rotations of the vector x with the generators

Xμν = xν ∂

∂xμ− xμ ∂

∂xν, μ, ν = 1, 2, 3, (3.6)

and Galilean transformations with the generators

Xμ4 = t∂

∂xμ, (μ = 1, 2, 3). (3.7)

Hence according to Noether’s theorem Eq. (2.3) has 10 conservation lawsof the form

Dt(C)∣∣(2.3)

= 0 (3.8)

defined by the following conserved quantities: the linear momentum

p = mx,

the energy

E =12

m|x|2

the angular momentumM = p × x

and the vectorq = m(x − xt).

From the table of commutators, Table 3.1, it is easy to notice that onlyX4 and one of generators Xμν can not be obtained by using adX. Thus,employing Theorem 3.1 we can conclude the following:A basis of conservation laws consists of two conservation laws defined bythe energy E and one of the components of the angular momentum M.

Indeed, if we choose as a basis of conserved quantities E and, e.g. M1 wecan obtain other conserved quantities by means of the generators Xμ4 andXμν written in the prolonged form:

Xμ4 = t∂

∂xμ+

∂

∂xμ(μ = 1, 2, 3)

and

Xμν = xν ∂

∂xμ− xμ ∂

∂xν+ xν ∂

∂xμ− xμ ∂

∂xν(μ, ν = 1, 2, 3). (3.9)

ThenXμ4E = pμ, X24M

1 = q3,

X12(M1) =M2, X13(M1) =M3.

X34M2 = q1, X14M

3 = q2.

11

Table 3.1: Table of commutators (Classical mechanics)

X1 X2 X3 X4 X12 X23 X13 X14 X24 X34

X1 0 0 0 0 −X2 0 −X3 0 0 0

X2 0 0 0 X1 −X3 0 0 0 0

X3 0 0 0 X2 X1 0 0 0

X4 0 0 0 0 X1 X2 X3

X12 0 −X13 X23 X24 −X14 0

X23 0 −X12 0 X34 −X24

X13 0 X34 0 −X14

X14 0 0 0

X24 0 0

X34 0

3.2 Relativistic mechanicsThe equation of free motion of a relativistic particle in the Minkowski spacewith the metric

ds2 = c2dt2 − dx2 − dy2 − dz2

has the Lagrangian

L = −mc2√1− |v|2

c2, vi =

dxi

dt(i = 1, 2, 3),

where c is a constant equal to the light velocity in vacuum,

x1 = x, x2 = y, x3 = z.

It admits the 10-parameter non-homogeneous Lorentz group with the gen-erators (3.4)–(3.6) and the generators of the Lorentz transformations

Xμ4 = x4∂

∂xμ+1c2

xμ ∂

∂x4, (μ = 1, 2, 3) (3.10)

where x4 = t.The corresponding conservations laws have the form similar to (3.8). Ac-

cording to Noether’s theorem they are defined by the following conservedquantities:

p0 = mc x, E0 = mc3x4, M0 = p0 × x, Q0 = mc(xx4 − xx4) (3.11)

12


Table 3.2: Table of commutators (Relativistic mechanics)

X1 X2 X3 X4 X12 X23 X13 X14 X24 X34

X1 0 0 0 0 −X2 0 −X31c2 X4 0 0

X2 0 0 0 X1 −X3 0 0 1c2 X4 0

X3 0 0 0 X2 X1 0 0 1c2 X4

X4 0 0 0 0 X1 X2 X3

X12 0 −X13 X23 X24 −X14 0

X23 0 −X12 0 X34 −X24

X13 0 X34 0 −X14

X14 0 −1c2 X12

−1c2 X13

X24 0 −1c2 X23

X34 0

where x = (x1, x2, x3) and the dot denotes differentiation with respect tothe length of the arc s in the Minkowski space. Comparing Table 3.2 andTable 3.1, one can see a significant difference. Namely, in the case of rela-tivistic mechanics the time translation generator X4 can be obtained fromother operators by using adX. Therefore, employing Theorem 3.1 we canconclude thatA basis of conserved quantities (3.11) (with respect to group G) is definedby one conserved quantity, e.g. any of the components of the angular mo-mentum M0.

Indeed, if we choose M10 as a basis of conserved quantities, under the

action of the generators of the Lorentz transformations written in the pro-longed form

Xμ4 = x4∂

∂xμ+1c2

xμ ∂

∂x4+ x4

∂

∂xμ+1c2

xμ ∂

∂x4, (μ = 1, 2, 3) (3.12)

and the generators of rotation (3.9) we obtain:

X24(M10 ) = Q3

0, X12(M10 ) =M2

0 , X13(M10 ) =M3

0 .

Then we haveX34(M2

0 ) = Q10, X14(M3

0 ) = Q20,

E0 can be obtained from Q0 with the help of the translation generators Xμ,

13

i.e.c2Xμ(Q

μ0 ) = E0

and the energy E0 transforms into the momentum p0,

Xμ4(E0) = pμ

0 .

Remark 3.2. On the other hand it is also possible to choose any componentof the vector Q0 as a basis of conserved quantities.

3.3 Motion in the de Sitter spaceConsider the space V4 with the metric

ds2 =1Φ2(c2dt2 − dx2 − dy2 − dz2), (3.13)

whereΦ = 1 +

K

4r2, r2 = c2t2 − x2 − y2 − z2 (3.14)

and K = const. denotes the curvature of the Sitter space-time.As well as the equation of free motion of a particle in Minkowski space a

similar equation in the de Sitter space has the Lagrangian

L = −mc2

Φ

√1− |v|2

c2, vμ =

dxi

dt(μ = 1, 2, 3),

also admits 10-parameter group G with the generators of rotations and thegenerators of Lorentz transformations of the form

Xμν = xν ∂

∂xμ− xμ ∂

∂xν(μ < ν, μ = 1, 2, 3; ν = 1, 2, 3, 4), (3.15)

but the generators of space translations (3.4) and translations of time (3.5)are replaced by the generators

Xν =[K

2xνxi + (Φ− 2)δνi

] ∂

∂xi, ν, i = 1, 2, 3, 4. (3.16)

Here the generators are written in the coordinates

x1 = x, x2 = y, x3 = z, x4 = ict

and δki is a Kronecker symbol.According to Noether’s theorem there are 10 conserved quantities similar

to (3.11), the linear momentum pk,, the energy EK , the angular momentumMK and the vector QK .

The structure of Lie algebra with the basis (3.15)–(3.16) is determinedby the commutators

[Xμ, Xν ] = KXμν , [Xμ, Xμν ] = Xν , [Xμν , Xα] = 0 (α �= μ, α �= ν),

[Xμν , Xαβ ] = δμαXνβ + δνβXμα − δμβXνα − δναXμβ .

14


Hence, for the equation of free motion of a particle in the de Sitter space,we arrive at the following assertion:A basis of conserved quantities with respect to the group G is defined by oneconserved quantity, e.g. any of the components of the angular momentumMK =M0/Φ2.Remark 3.3. In this case any conserved quantity, i.e the energy, or anycomponent of the linear momentum or any component of the vector QK

can also be chosen as a basis of the conserved quantities.

3.4 Nonlinear wave equationThe equation

utt −Δu+ λu3 = 0 (3.17)

has the Lagrangian

L = |Δu|2 − u2t +12λu4

where Δu = uxx + uyy + uzz, λ = const.. Eq. (3.17) describes string vi-bration immersed in nonlinear medium. Eq. (3.17) is also used in quantumnonlinear field theory. It admits the 15-dimensional group of conformaltransformations in the Minkowski space and has, correspondingly, 15 con-servation laws of the form

Dt(C1) +Dx(C2) +Dy(C3) +Dz(C4) = 0.

The basis of conserved vectors for the nonlinear wave equation also consistsof one conserved vector [37], [38].

3.5 Lin–Reissner–Tsien equationThe equation [41]

−ϕxϕxx − 2ϕxt + ϕyy = 0

describes the non-steady-state potential gas flow with transonic velocities.It has the Lagrangian

L = |Δu|2 − u2t +12λu4, Δu = uxx + uyy, λ = const.

and admits an infinite transformation group [42]. Accordingly, by Noether’stheorem, the family of conservation laws [43] is infinite. Meanwhile, employ-ing Theorem 3.1 we can conclude the following:

The basis of conservation laws consists of one conservation law [37], namely

Dt(C1) +Dx(C2) +Dy(C3) = 0

where

C1 =12ϕ2

y − 16ϕ3

x, C2 = ϕ2t +

12ϕ2

xϕt, C3 = −ϕtϕy.

15

3.6 Transonic three-dimensional gas motionThe equation

−uxuxx − 2uxt + uyy + uzz = 0

of transonic gas motion has the Lagrangian

L = 16

u3x + uxut − 12

u2y − 12

u2z

and admits an infinite group transformations (the generators [44], [45], [39]depends on arbitrary functions).

The basis of conservation laws

Dt(C1) +Dx(C2) +Dy(C3) +Dz(C4) = 0

is defined by two vectors A1 and A4 where

A11 =

16

u3x−12

u2y−12

u2z, A21 = −ut

(12

u2x+ut

), A3

1 = utuy, A41 = utuz;

A14 = ηux, A2

4 = η(12

u2x + ut

), A3

4 = −ηuy + zL, A44 = −ηuz − yL,

η = yuz − zuy.

3.7 Short wavesDuring first underwater nuclear and thermonuclear explosions near the arc-tic island Novaja Zemlja in the USSR it was discovered that weak waveswere drastically increasing the destructive force of a shock wave [46]. Rizhovand Khristianovich [47] presented the equations describing the behavior ofthese so-called "short waves".

The equations of short waves

uy − 2vt − 2(v − x)vx − 2kv = 0, vy + ux = 0, k = const.,

admit an infinite-dimensional group [48]. They can be reduced by the sub-stitution u = ϕy, v = −ϕx to the equation

ϕyy + 2ϕxt − 2(x+ ϕx)ϕxx + 2kϕx = 0, (3.18)

which has the Lagrangian

L = (ϕtϕx − 13ϕ3

x − xϕ2x +

12ϕ2

y) exp[2(k + 1)t].

In [39] I calculated the following generators for Eq. (3.18):

X1 = 9∂

∂t− 4(k + 1)x ∂

∂x− 2(k + 1)y ∂

∂y− 8(k + 1)ϕ ∂

∂ϕ, (3.19)

X2 = −μ′y

∂

∂x+ μ

∂

∂y−

[xy(μ

′′+ μ

′)− y3

3dkμ

′] ∂

∂ϕ, (3.20)

16


X3 = κ∂

∂x+

[y2dkκ − x(κ

′+ κ)

] ∂

∂ϕ, (3.21)

X4 = λy∂

∂ϕ, X5 = σ

∂

∂ϕ, X0 = y

∂

∂t(3.22)

where μ, κ, λ, σ are arbitrary functions of t, the prime denotes differentiationwith respect to t, dk = d2/dt2 + (k + 1)d/dt+ k.

Although the constant k in Eq. (3.18) takes only the values 0 and 1 inaccordance with the physical content of the problem, it can be regarded asan arbitrary parameter. For k = 2; 1/2 there is an extension of the group,the following generators are added:

X6 = 9a∂

∂t+{[3a′−4(k+1)a]x−[3a′′−(k+1)a′

]y2} ∂

∂x+[6a

′−2(k+1)a]y ∂

∂y

+{−[3a′

+8(k+1)a]ϕ− x2

2[3a

′′+(5−4k)a′

]+[3a′′′+(2−k)a

′′−(k+1)a′]xy2

−y4

6[3aiv + 2(k + 1)a

′′′ − (k2 − k + 1)a′′ − k(k + 1)a

′]} ∂

∂ϕ·

The operator X0 does not satisfy the conditions of Noether’s theorem andthe generators X4 and X5 give only trivial conservation laws.

Among the commutation relations for X1, X2, X3, X6 we can distinguish

ad X2(X1) = −X2 < 9μ′+ 2(k + 1)μ >,

ad X3(X1) = −X3 < 9κ′+ 4(k + 1)κ >, ad X6(X1) = −X6 < 9a

′> .

Here the brackets < ... > mean that instead of the arbitrary function occur-ring in the coordinates of the generator (or conserved vector) it is necessaryto substitute the expression in these brackets.

Therefore, the basis of the conservation laws

Dt(C1) +Dx(C2) +Dy(C3) = 0

is determined by one vector corresponding to X1, i.e. A1 with coordinates

A11 = η1Eϕx + 9L, A2

1 = η1E(ϕt − ϕ2x − 2xϕx)− 4(k + 1)xL,

A31 = η1Eϕy−2(k+1)yL, η1 = −8(k+1)ϕ−9ϕt+4(k+1)xϕx+2(k+1)yϕy.

3.8 Dirac equationsThe Dirac equations

γk ∂ψ

∂xk+mψ = 0,

∂ψ

∂xkγk − mψ = 0, (3.23)

are the relativistic quantum mechanical wave equations used for the descrip-tion of fermions, i.e. elementary particles having half-integer spin number(say, 1

2 , 32 , 52 , ...). Eqs (3.23) have the Lagrangian

L = 12

{ψ

(γk ∂ψ

∂xk+mψ

)−

( ∂ψ

∂xkγk − mψ

)ψ

}, k = 1, 2, 3, 4.

17

Here the independent variables are

x1 = x, x2 = y, x3 = z, x4 = ict,

the dependent variables

ψ = (ψ1, ..., ψ4) ψ = (ψ1, ..., ψ4)

are 4-dimensional complex vectors and and γk are 4× 4 complex matrices.The maximal group admitted by Eqs. (3.23) is obtained in [22]. The

Dirac equations have an infinite number of conservation laws

Dt(C1) +Dx(C2) +Dy(C3) +Dz(C4) = 0.

Using Theorem 3.1 we obtain the following result [39].For m = 0 the basis of conserved vectors is formed by 3 vectors, A12, A5, A8,and for m �= 0 by 2 vectors A12, A5.Their coordinates have the form

Ak12 =

14{ψ(γkγ1γ2 + γ1γ2γk)ψ}+ x2Ak

1 − x1Ak2 ,

Ak5 = −iψγkψ, Ak

8 = iψγkγ5ψ,

where

Akl =

12

{ ∂ψ

∂xlγkψ − ψγk ∂ψ

∂xl

}+ δk

l , k = 1, 2, 3, 4, l = 1, 2,

and δkl is a Kronecker’s symbol.

4 Equations without Lagrangians

4.1 Formal LagrangianMany differential equations cannot be formulated as the Euler–Lagrangeequations since they have no Lagrangians. Therefore, it is impossible toapply Noether’s theorem for calculating conservation laws. However, ac-cording to [49] and [50], it is possible to introduce a formal Lagrangian ifany given system of equations is taken into consideration together with theadjoint system. In his recent paper [50] Ibragimov has proved that the ad-joint system inherits symmetries of the given system and has suggested anew theorem on nonlocal conservation laws.

Consider an arbitrary system of sth-order partial differential equations

Fα

(x, u, u(1), . . . , u(s)

)= 0, α = 1, . . . , m. (4.1)

where the functions Fα(x, u, u(1), . . . , u(s)) depend on n independent vari-ables x = (x1, . . . , xn), m dependent variables u = (u1, . . . , um), u = u(x),and their derivatives up to an arbitrary order s.

18

4 Equations without Lagrangians

Definition 4.1. The adjoint system to Eqs (4.1) is defined by [51]

F ∗α

(x, u, v, u(1), v(1), . . . , u(s), v(s)

) ≡ δ(vβFβ)δuα

= 0, α = 1, . . . , m, (4.2)

where v = (v1, . . . , vm) are new dependent variables, v = v(x), and δδuα is

the variational derivative (2.13).

In the case of linear equations this definition is equivalent to the standardone.

Remark 4.1. The variables v = (v1, . . . , vm) were called in [50] nonlocalvariables in accordance with the general concept of nonlocal symmetries.Therefore, conservation laws involving v were named nonlocal conservationlaws.

Using the new definition of the adjoint system, it can be shown that anysystem of sth-order differential equations (4.1) considered together withits adjoint equation (4.2) has a Lagrangian. Namely, the Euler-Lagrangeequations with the Lagrangian

L = vβFβ

(x, u, u(1), . . . , u(s)

)(4.3)

provide the simultaneous system of equations (4.1), (4.2) with 2m dependentvariables u = (u1, . . . , um) and v = (v1, . . . , vm).

Definition 4.2. The system (4.1) is called self-adjoint if the substitutionv = u gives

F ∗ = λ(x, u, u(1), . . . , u(s))F. (4.4)

The system (4.1) is called quasi-self-adjoint [52] if there exists a functionh(u) such that Eq (4.4) holds upon the substitution v = h(u).

4.2 Maxwell-Dirac equationsWe have the system of equations

∇× E+∂B∂t+ σmB = 0,

∇× B − ∂E∂t

− σeE = 0, (4.5)

∇ · E − ρe = 0,

∇ · B − ρm = 0,

where σm, σe = const. The system (4.5) has eight equations for eight de-pendent variables: six coordinates of the electric and magnetic vector fieldsE = (E1, E2, E3) and B = (B1, B2, B3), respectively, and two scalar quan-tities ρe and ρm, the electric and magnetic monopole charge densities.

19

Using (4.3) we write the Lagrangian (4.3) for Eqs. (4.5) in the followingform [54] :

L = V ·(∇× E+

∂B∂t+ σmB

)+Re

(∇ · E − ρe

)

+W ·(∇× B − ∂E

∂t− σeE

)+Rm

(∇ · B − ρm

), (4.6)

where V, W, Re, Rm are adjoint variables. With this Lagrangian theadjoint equations for the new dependent variables V, W, Re, Rm have theform [55]

∇× V+∂W∂t

− σeW = 0,

∇× W − ∂V∂t+ σmV = 0, (4.7)

Re = 0, Rm = 0.

4.3 Conservation lawsEach generator

X = ξi(x, u)∂

∂xi+ ηα(x, u)

∂

∂uα,

admitted by a first-order system

Fα(x, u, u(1)) = 0, α = 1, ..., m

leads to a conserved vector with the components

Ci = vβ[ξiFβ + (ηα − ξjuα

j

∂Fβ

∂uαi

)]

(4.8)

where i = 1, ..., n and vβ solve the adjoint system

F ∗α(x, u, u(1), v(1)) = 0, α = 1, ..., m.

The conservation law for Eqs (4.5) has the form

Dt(τ) + div χ = 0, (4.9)

which holds on the solutions of Eqs (4.5) and (4.7). Here τ is the densityof the conservation law (4.9), χ = (χ1, χ2, χ3), and

div χ ≡ ∇ · χ = Dx(χ1) +Dy(χ2) +Dz(χ3).

The Maxwell-Dirac equations are neither self-adjoint nor quasi-self-adjoint.Consequently, the conservation laws obtained by using Eqs (4.8) are nonlo-cal.

20

5 Summary of thesis

4.4 General magma equationThe equation

∂f

∂t+

∂

∂z

{fn

[1− ∂

∂z

( 1fm

∂f

∂t

)]}= 0 (4.10)

models the migration of melt through the Earth’s mantle. It follows fromthe equations

uz = −ft, u = fn[1 +

∂

∂z

( 1fm

∂u

∂z

)](4.11)

where u is the vertical barometric flux of melt, f is the volume fractionof melt, z is a vertical space coordinate and t is time. All the variablesare dimensionless. Eqs. (4.11) were proposed by Scott and Stevenson [56].They suggested that 2 ≤ n ≤ 5 or even bigger and supposed that 0 ≤ m ≤ 1.Some authors discussed Eq. (4.10) for any values of n and m.

I denote f by u in order to make Eq. (4.10) compatible with the generalnotation used above. It has the form

F ≡ ut +Dz

{un

[1− Dz

(u−mut

)].}= 0 (4.12)

The general magma equation does not have any Lagrangian and thereforethe formal Lagrangian is introduced. Using the Lagrangian and employinginfinitesimal symmetries of Eq. (4.10) nonlocal conservation laws are ob-tained in my articles [57]– [59]. The central part of these articles is theproof of the remarkable property of Eq. (4.12) to be quasi-self-adjoint forany values of the parameters m and n. This property allows us to obtainlocal conservation laws from nonlocal ones. They include the local conser-vation laws obtained by the direct method by Barcilon and Richter [27] andHarris [28] and later discussed in [36].

5 Summary of thesis

This thesis is a collection of four papers. I present the outline of each of thembelow. My investigations in this thesis concern symmetries and conservationlaws. Two ideas are discussed throughout the text. The first one is aboutconstructing a basis of conservation laws (with respect to admitted groupG) obtained from Noether’s theorem (papers 1 and 2). It is proven thatthere is a connection between a basis of conservation laws and the structureof the Lie algebra of G.

The second idea is the contraction of conservation laws for differentialequations without Lagrangians (papers 3 and 4). Nonlocal conservationlaws are constructed for Dirac’s symmetrized Maxwell-Lorentz equationswith dual Ohm’s law and the equation modelling a melt migration throughthe Earth’s mantle.

21

6 Summary of papers

6.1 Paper 1Lie-Bäcklund groups allow us to reduce a number of independent conservedquantities for differential equations and construct a basis of conservationlaws. This article is devoted to constructing a basis for some equations inmechanics by the method described in [32]. The results are the following:for the equation of free motion of a particle in classical mechanics a basis(with respect to the admitted group G) of the set of 10 conserved quantitiesconsists of two conserved quantities, namely the energy and one of thecomponents of the angular momentum.

For the motion of a relativistic particle a basis of the set of 10 conservedquantities is formed by one conserved quantity. In the case of the Minkowskispace it can be any component of the angular momentum M0 or any com-ponent of the vector Q0, in the case of the de Sitter space it can be anyof the conserved quantities, e.g. any component of the angular momentumMK .

The nonlinear wave equation, describing string vibration immersed innonlinear media, has a set of 15 conserved vectors. The basis is defined byone conserved vector.

The Lin-Reissner-Tsien equation for the non-steady-state potential gasflow with transonic velocities possesses an infinite set of conservation laws.The basis consists of one conservation law.

6.2 Paper 2The derivation of conservation laws for invariant variational problems isbased on fundamental identity connecting Lie-Bäcklund, Euler-Lagrangeand Ibragimov’s operators. It is shown that this identity also makes it pos-sible to establish a connection between a basis of conservation laws (withrespect to the group G admitted by the considered system of differentialequations) and the structure of the Lie algebra of G. This provides a jus-tification for the basis construction scheme proposed by Ibragimov. Thetheorem is applied to the short waves equation, to the transonic three-dimensional gas motion equation and the Dirac equations. As a result abasis of conservation laws for each equation is obtained.

6.3 Paper 3Some differential equations have no Lagrangian. However using a generaltheorem on conservation laws for arbitrary differential equations proven byIbragimov [50] it is possible to introduce a formal Lagrangian. We have de-rived conservation laws for Dirac’s symmetrized Maxwell-Lorentz equationsunder the assumption that both the electric and magnetic charges obey lin-ear conductivity laws (dual Ohm’s law). The conserved quantities obtainedin the article involve solutions v and w of the adjoint equations. It may beuseful for applications to use an alternative representation of the conserved

22

6 Summary of papers

quantities in terms of the electric and magnetic vector fields E and B only.It is shown that nonlocal conservation laws (see Remark 4.1) can be writtenin two-solution or one-solution representations.

6.4 Paper 4A general magma equation models a melt migration through the Earth’smantle. It is a nonlinear third-order differential equation which does nothave any Lagrangian. Therefore, the recent theorem on nonlocal conser-vation laws [50] is applied to this equation. It is shown that the equa-tion has a remarkable property: for any values of the parameters m and nthe general magma equation is quasi-self-adjoint in the terminology of [52].The self-adjoint equations are also singled out. Nonlocal conservation lawsare obtained using the symmetries of the equation. By employing quasi-self-adjointness of the equation it is shown that all local conservation lawscalculated by direct method are consequences of the nonlocal ones.

23

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[37] Khamitova, R. On a basis of conservation laws of mechanics equations,Dokl. Akad. Nauk SSSR, 248(4), 1979, pp. 798–802. English trans.Soviet Math. Dokl., 248(5), pp. 1071–1075, 1980.

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[49] Atherton, R. W. and Homsy, G. M. On the existance and formulation ofvariational principles for nonlinear differential equations, Studies Appl.Math., LIV (1), pp. 31–60, 1975.

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[54] Ibragimov, N. H., Khamitova, R. and Thidé, B. Conservation laws forMaxwell-Dirac equations with dual Ohm’s law, J. Math. Phys., 48(5),pp. 053523-1–053523-11, 2007.

[55] Ibragimov, N. H., Khamitova, R. and Thidé, B. Adjoint system andconservation laws for symmetrized electromagnetic equations with adual Ohm’s law, Archives of ALGA, 3, ALGA publ., BTH, Karlskrona,Sweden, pp. 81–95, 2006.

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[57] Khamitova, R. Self-adjointness and quasi-self-adjointness of the magmaequation, in: J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, L. B.Figueiredo (Eds.), Proc. of the 2nd conference on "Nonlinear Scienceand Complexity", July 28–31, 2008, Porto, Portugal, ISBN:978-972-8688-56-1.

[58] Khamitova, R. Self-adjointness and quasi-self-adjointness of an equa-tion modelling melt migration through the Earth’s mantle. Nonlocalconservation laws, Preprint in Archives of ALGA, 5, ALGA publ.,BTH, Karlskrona, Sweden, pp. 143–159, 2008.

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28

Paper I

29

Paper II

37

Paper III

45

Conservation laws for the Maxwell-Dirac equationswith dual Ohm’s law

Nail H. Ibragimova� and Raisa KhamitovaDepartment of Mathematics and Science, Research Centre ALGA: Advances in Lie GroupAnalysis, Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden

Bo Thidéb�

Ångström Laboratory, Swedish Institute of Space Physics, P.O. Box 537, SE-751 21Uppsala, Sweden

�Received 5 March 2007; accepted 9 April 2007; published online 31 May 2007�

Using a general theorem on conservation laws for arbitrary differential equationsproved by Ibragimov �J. Math. Anal. Appl. 333, 311–320 �2007��, we have derivedconservation laws for Dirac’s symmetrized Maxwell-Lorentz equations under theassumption that both the electric and magnetic charges obey linear conductivitylaws �dual Ohm’s law�. We find that this linear system allows for conservation lawswhich are nonlocal in time. © 2007 American Institute of Physics.�DOI: 10.1063/1.2735822�

I. INTRODUCTION

In all areas of physics, conservation laws are essential since they allow us to draw conclusionsof a physical system under study in an efficient way.

Electrodynamics, in terms of the standard Maxwell electromagnetic equations for fields invacuum, exhibit a rich set of symmetries to which conserved quantities are associated. Recently,there has been a renewed interest in the utilization of such quantities. Here we use a theorem ofIbragimov �2006� to derive conservation laws for Dirac’s symmetric version of the Maxwell-Lorentz microscopic equations, allowing for magnetic charges and magnetic currents, where thelatter, just as electric currents, are assumed to be described by a linear relationship between thefield and the current, i.e., Ohm’s law. The method of Ibragimov �2006� produces two new adjointvector fields which fulfil, Maxwell-like equations. In particular, we obtain conservation laws forthe symmetrized electromagnetic field which are nonlocal in time.

II. PRELIMINARIES

A. Notation

We will use the following notation �see, e.g., Ibragimov, 1999�. Let x= �x1 , . . . ,xn� be inde-pendent variables and u= �u1 , . . . ,um� be dependent variables. The set of the first-order partialderivatives ui

�=�u� /�xi will be denoted by u�1�= �ui��, where �=1, . . . ,m and i , j ,¯ =1, . . . ,n.

The symbol Di denotes the total differentiation with respect to the variable xi:

Di =�

�xi + ui� �

�u� + uij� �

�uj� + uijk

� �

�ujk� + ¯ .

We employ the usual convention of summation in repeated indices.Recall that a necessary condition for extrema of a variational integral

a�Electronic mail: [email protected]�Also at LOIS Space Centre, School of Mathematics and Systems Engineering, Växjö University, SE-351 95 Växjö,Sweden; electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS 48, 053523 �2007�

48, 053523-10022-2488/2007/48�5�/053523/11/$23.00 © 2007 American Institute of Physics

Downloaded 05 Jun 2007 to 130.238.30.29. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

�V

L�x,u,u�1��dx , �1�

with a Lagrangian L�x ,u ,u�1��, depending on first-order derivatives, is given by the Euler-Lagrange equations

�L�u�

�L�u� − Di �L

�ui�� = 0, � = 1, . . . ,m . �2�

We will understand by a symmetry of a certain system of differential equations a generator

X = �i�x,u��

�xi + ��x,u��

�u� �3�

of a continuous transformation group admitted by differential equations under consideration.A vector field C= �C1 , . . . ,Cn� is said to be a conserved vector for the differential equations �2�

if the equation

Di�Ci� = 0 �4�

holds for any solution of Eq. �2�.If one of the independent variables is time, e.g., xn= t, then the conservation law is often

written in the form

dE

dt= 0,

where

E = �Rn−1

Cn�x,u�x�,u�1��x��dx1¯ dxn−1. �5�

Accordingly, Cn is termed the density of the conservation law.

B. Basic conservation theorem

We will employ the recent general theorem �Ibragimov, 2006� on a connection betweensymmetries and conservation laws for arbitrary systems of sth-order partial differential equations

F��x,u,u�1�, . . . ,u�s�� = 0, � = 1, . . . ,m , �6�

where F��x ,u ,u�1� , . . . ,u�s�� involves n independent variables x= �x1 , . . . ,xn� and m dependentvariables u= �u1 , . . . ,um�, u=u�x� together with their derivatives up to an arbitrary order s. For ourpurposes, we formulate the theorem in the case of systems of first-order differential equations.

Theorem 2.1: �See Ibragimov, 2006, Theorem 3.5�. Let an operator �3� be a symmetry of asystem of first-order partial differential equations,

F��x,u,u�1�� = 0, � = 1, . . . ,m , �7�

where v= �v1 , . . . ,vm�. Then the quantities

Ci = v��iF� + �� − � juj��

�F�

�ui� , i = 1, . . . ,n �8�

furnish a conserved vector C= �C1 , . . . ,Cn� for the equations �7� considered together with theadjoint system

053523-2 Ibragimov, Khamitova, and Thidé J. Math. Phys. 48, 053523 �2007�


F�*�x,u,v,u�1�,v�1��

�L�u� = 0, � = 1, . . . ,m , �9�

where

�

�u� =�

�u� − Di�

�ui� , � = 1, . . . ,m ,

and v= �v1 , . . . ,vm� are new dependent variables, i.e., v=v�x�.Remark 2.1: The simultaneous system of Eqs. �7� and �9� with 2m dependent variables u

= �u1 , . . . ,um�, v= �v1 , . . . ,vm� can be obtained as the Euler-Lagrange equations �2� with the La-grangian

L = v�F��x,u,u�1�, . . . ,u�s�� . �10�

Indeed,

�L�v� = F��x,u,u�1�� , �11�

�L�u� = F�

*�x,u,v,u�1�,v�1�� . �12�

Remark 2.2: The conserved quantities �8� can be written in terms of the Lagrangian �10� asfollows:

Ci = L�i + �� − � juj��

�L�ui

� . �13�

Remark 2.3: If Eqs. �7� have r symmetries X1 , . . . ,Xr of the form �3�,

X� = ��i �x,u�

�

�xi + ��x,u�

�

�u� , � = 1, . . . ,r ,

then Eqs. �8� provide r conserved vectors C1 , . . . ,Cr with the components

C�i = L��

i + �� − ��

j uj��

�L�ui

� , � = 1, . . . ,r, i = 1, . . . ,n .

III. ELECTROMAGNETIC EQUATIONS

A. Basic equations and the Lagrangian

Adopting Dirac’s ideas on the existence of magnetic monopoles �Dirac, 1931�, one can for-mulate a symmetrized version of Maxwell’s electromagnetic equations �Schwinger, 1969�. InSysteme International �SI� units and in microscopic �Lorentz� form, these equations are �cf. Thidé,2006, Eqs. �1.50��

� � E +�B

�t+ �0jm = 0 , �14a�

� � B −1

c2�E

�t− �0je = 0 , �14b�

� · E − �0c2e = 0, �14c�

053523-3 Conservation laws for the Maxwell-Dirac equations J. Math. Phys. 48, 053523 �2007�


� · B − �0m = 0, �14d�

together with dual Ohm’s law

je = �eE, jm = �mB , �15�

where m and e are constant scalar �rank zero� quantities. The first equation in Eq. �15� is Ohm’slaw for electric currents. The second equation is dual Ohm’s law for magnetic currents, that, forsymmetry reasons, was introduced by Thidé, 2006, Eqs. �2.60�; see also Eq. �5� by Meyer-Vernet,1982, Eq. �38� by Olesen, 1996, and its generalization by Coceal et al., 1996.

Now we substitute Eqs. �15� into Eqs. �14a� and �14b�. The ensuing equations involve, alongwith the light velocity c, three other constants, e, m, and �0. We eliminate two constants bysetting

t = ct, B = cB, e = c�0e, m =�0

cm,

e = c2�0e, m = c�0m, �16�

and rewrite our basic Maxwell-Dirac equations �14�, discarding tilde, as follows:

� � E +�B

�t+ mB = 0 ,

� � B −�E

�t− eE = 0 ,

� · E − e = 0,

� · B − m = 0. �17�

The system �17� has eight equations for eight dependent variables: six coordinates of the electricand magnetic vector fields E= �E1 ,E2 ,E3� and B= �B1 ,B2 ,B3�, respectively, and two scalar quan-tities, viz., the electric and magnetic monopole charge densities e and m.

Using the method of Ibragimov �2006� we write the Lagrangian �10� for Eqs. �17� in thefollowing form:

L = V · � � E +�B

�t+ mB� + Re�� · E − e� +W · � � B −

�E

�t− eE� + Rm�� · B − m� ,

�18�

where V ,W ,Re ,Rm are adjoint variables �we note in passing that V is a pseudovector and Rm apseudoscalar�. With this Lagrangian we have

�L�V

= � � E +�B

�t+ mB,

�L�Re

= � · E − e,

�L�W

= � � B −�E

�t− eE,

�L�Rm

= � · B − m, �19�

and



�L�E

= � � V +�W

�t− eW − �Re,

�L�e

= − Re,

�L�B

= � � W −�V

�t+ mV − �Rm,

�L�m

= − Rm. �20�

It follows from Eqs. �19� and �20� that the Euler-Lagrange equations �2� for the Lagrangian �18�provide the electromagnetic equations �17� and the following adjoint equations for the new de-pendent variables V ,W ,Re ,Rm:

� � V +�W

�t− eW = 0,

� � W −�V

�t+ mV = 0,

Re = 0, Rm = 0. �21�

Remark 3.1: Let the spatial coordinates x1 ,x2 ,x3 be x ,y ,z. For computing the variationalderivatives �L /�E and �L /�B in Eqs. �20�, it is convenient to use the coordinate representation ofthe Lagrangian �18�, namely,

L = V1�Ey3 − Ez

2 + Bt1 + mB1� + V2�Ez

1 − Ex3 + Bt

2 + mB2� + V3�Ex2 − Ey

1 + Bt3 + mB3� + Re�Ex

1 + Ey2

+ Ez3 − e� + W1�By

3 − Bz2 − Et

1 − eE1� + W2�Bz

1 − Bx3 − Et

2 − eE2� + W3�Bx

2 − By1 − Et

3 − eE3�

+ Rm�Bx1 + By

2 + Bz3 − m� . �22�

B. Symmetries

Equations �17� are invariant under the translations of time t and the position vector x= �x ,y ,z� as well as the simultaneous rotations of the vectors x, E, and B due to the vectorformulation of Eqs. �17�. These geometric transformations provide the following seven infinitesi-mal symmetries:

X0 =�

�t, X1 =

�

�x, X2 =

�

�y, X3 =

�

�z,

X12 = y�

�x− x

�

�y+ E2 �

�E1 − E1 �

�E2 + B2 �

�B1 − B1 �

�B2 ,

X13 = z�

�x− x

�

�z+ E3 �

�E1 − E1 �

�E3 + B3 �

�B1 − B1 �

�B3 ,

X23 = z�

�y− y

�

�z+ E3 �

�E2 − E2 �

�E3 + B3 �

�B2 − B2 �

�B3 . �23�

The infinitesimal symmetries for the adjoint system �21� are obtained from Eq. �23� by replacingthe vectors E and B by V and W, respectively. Moreover, since Eqs. �17� are homogeneous, theyadmit simultaneous dilations of all dependent variables with the generator



T = E ·�

�E+ B ·

�

�B+ e

�

�e+ m

�

�m, �24�

where

E ·�

�E=�

i=1

3

Ei �

�Ei , B ·�

�B=�

i=1

3

Bi �

�Bi .

Recall that the Maxwell equations in vacuum admit also the one-parameter group ofHeaviside-Larmor-Rainich duality transformations

E = E cos � − B sin �, B = E sin � + B cos � �25�

with the generator

X = E ·�

�B− B ·

�

�E �

i=1

3 Ei �

�Bi − Bi �

�Ei� .Also recall that the “mixing angle” � in Eq. �25� is a pseudoscalar.

It was shown �Ibragimov, 2006� that the group �25� provides the conservation of energy forthe Maxwell equations. Let us clarify whether Eqs. �17� admit a similar group. Let therefore

X = E ·�

�B− B ·

�

�E+ e

�

�m− m

�

�e. �26�

The prolongation of the operator �26� is written

X = E ·�

�B− B ·

�

�E+ e

�

�m− m

�

�e+ Et ·

�

�Bt− Bt ·

�

�Et+ Ex ·

�

�Bx− Bx ·

�

�Ex+ Ey ·

�

�By

− By ·�

�Ey+ Ez ·

�

�Bz− Bz ·

�

�Ez. �27�

Reckoning shows that the operator �27� acts on the left-hand sides of Eqs. �17� as follows:

X�� E + Bt + mB� = − �� B − Et − mE� ,

X�� B − Et − eE� = � � E + Bt + eB ,

X�� · E − e� = − �� · B − m� ,

X�� · B − m� = � · E − e.

It follows that the operator �26� is admitted by Eqs. �17� only in the case

m = e. �28�

IV. CONSERVATION LAWS

A. Derivation of conservation laws

We will write the conservation law �4� in the form

Dt�� + div � = 0, �29�

where the pseudoscalar � is the density of the conservation law �29�, the pseudovector current�= ��1 ,�2 ,�3�, and



div � � · � = Dx��1� + Dy��2� + Dz��3� .

Let us find the conservation law furnished by the symmetry �26� when the condition �28� issatisfied, m=e. Applying the formula �8� to the symmetry �26� and to the Lagrangian �18�, weobtain the following density of the conservation law �29�:

� = E ·�L�Bt

− B ·�L�Et

= E · V + B ·W .

Thus,

� = E · V + B ·W . �30�

The pseudovector � is obtained likewise. For example, using the Lagrangian in the form �22�, wehave

�1 = E ·�L�Bx

− B ·�L�Ex

= E2W3 − E3W2 − B2V3 + B3V2.

The other coordinates of � are computed likewise, and the final result is

� = �E � W� − �B � V� . �31�

One can readily verify that Eqs. �30� and �31� provide a conservation law for Eqs. �17� consideredtogether with the adjoint equations �21�. Indeed, using the well-known formula � · �a�b�=b · ��a�−a · ��b� and Eqs. �17� and �21�, we obtain

Dt�� = Et · V + E · Vt + Bt ·W + B ·Wt = V · �� B − eE� + E · �� W + mV�

−W�� E + mB� − B · �� V − eW� ,

� · � = � · �E � W� − � � · B � V� =W · �� E� − E · �� W� − V · �� B� + B · �� V� .

Whence,

Dt�� + div � = �m − e��E · V − B ·W� .

It follows again that the conservation law is valid only if m=e.Remark 4.1: The conservation law given by Eqs. �30� and �31� depends on solutions �V ,W�

of the adjoint system �21�. However, substituting into Eqs. �30� and �31� any particular solution�V ,W� of the adjoint system �21� with m=e, one obtains the conservation law for Eqs. �17� notinvolving V and W. Let us denote m=e= and take, e.g., the following simple solution of theadjoint system �21�:

V1 = et, V2 = V3 = 0, W1 = et, W2 = W3 = 0.

Then Eqs. �30� and �31� yield

� = �E1 + B1�et,

�1 = 0, �2 = �E3 − B3�et, �3 = �B2 − E2�et.

Remark 4.2: The operator �26� generates the one-parameter group

E = E cos � − B sin �, B = E sin � + B cos � ,



e = e cos � − m sin �, m = e sin � + m cos � .

where, again, the “mixing angle” � is a pseudoscalar.Remark 4.3: In the original variables used in Eqs. �14a�, �14b�, and �15�, the operator �26� is

written as

X =1

cE ·

�

�B− cB ·

�

�E+ ce

�

�m−1

cm

�

�e.

Applying similar calculations to the generator �24� of the dilation group provides the conser-vation law with

� = B · V − E ·W, � = �E � V� + �B � W� . �32�

This conservation law is valid for arbitrary m and e. Indeed,

Dt�� = Bt · V + B · Vt − Et ·W − E ·Wt = − V · �� E + mB� + B · �� W + mV�

−W · �� B − eE� + E · �� V − eW� = − V · �� E� + B · �� W�

−W · �� B� + E · �� V� ,

� · � = � · �E � V� + � · �B � W� = V · �� E� − E · �� V� +W · �� B� − B · �� W� .

Hence, Dt��+div �=0.Let us find the conservation law provided by the symmetry X0=� /�t from Eq. �23�. Formula

�8� yields

� = L − Et ·�L�Et

− Bt ·�L�Bt

= L + Et ·W − Bt · V .

Since the Lagrangian L given by Eq. �18� vanishes on the solutions of Eqs. �17�, we can take

� = Et ·W − Bt · V �33�

or

� =W · �� B� − eE� + V · �� E� + mB� . �34�

Let us calculate the pseudovector �. Formula �8� yields

�1 = − Et ·�L�Ex

− Bt ·�L�Bx

.

Using the Lagrangian in the form �22�, we have

�1 = − Et2V3 + Et

3V2 − Bt2W3 + Bt

3W2.

The other coordinates of � are computed similarly, and the final result is

� = �V � Et� + �W � Bt� . �35�

Thus, the time translational invariance of Eqs. �17� leads to the conservation law �29� with � and� given by Eqs. �34� and �35�, respectively.

Remark 4.4: Let us substitute in Eqs. �34� and �35� the following simple solution of theadjoint system �cf. Remark 4.1�:

V1 = emt, V2 = V3 = 0, W1 = eet, W2 = W3 = 0.

Then Eqs. �34� and �35� yield



� = �By3 − Bz

2 − eE1�eet + �Ey

3 − Ez2 + mB1�emt,

�1 = 0, �2 = − Et3emt − Bt

3eet, �3 = Et2emt + Bt

2eet.

The conservation law provided by the symmetry X1=� /�x from Eq. �23� has the followingdensity:

� = − Ex ·�L�Et

− Bx ·�L�Bt

= Ex ·W − Bx · V .

For the pseudovector � the formula �8� yields

�1 = L − Ex ·�L�Ex

− Bx ·�L�Bx

.

Using the Lagrangian in the form �22�, we have

�1 = L − Ex2V3 + Ex

3V2 − Bx2W3 + Bx

3W2.

The other coordinates of � are calculated similarly:

�2 = Ex1V3 − Ex

3V1 + Bx1W3 − Bx

3W1,

�3 = − Ex1V2 + Ex

2V1 − Bx1W2 + Bx

2W1.

We can ignore L in �1 since DxL=0 on solutions of Eqs. �17� and �21�, and the final result is

� = �V � Ex� + �W � Bx� . �36�

Replacing x by y and z we obtain the following conservation laws corresponding to X2=� /�y andX3=� /�z, respectively:

� = Ey ·W − By · V, � = �V � Ey� + �W � By�

and

� = Ez ·W − Bz · V, � = �V � Ez� + �W � Bz� .

Applying formula �8� to the symmetry X12 and to the Lagrangian �22�, we obtain the followingdensity of the conservation law:

� = E2�L�Et

1 − E1�L�Et

2 + �xEy − yEx� ·�L�Et

+ B2�L�Bt

1 − B1�L�Bt

2 + �xBy − yBx� ·�L�Bt

= W2E1 − W1E2

+ �yEx − xEy� ·W − �V2B1 − V1B2� − �yBx − xBy� · V .

The densities of the conservation laws for X13 and X23 are

� = W3E1 − W1E3 + �zEx − xEz� ·W − �V3B1 − V1B3� − �zBx − xBz� · V

and

� = W3E2 − W2E3 + �zEy − yEz� ·W − �V3B2 − V2B3� − �zBy − yBz� · V ,

respectively. Finally, the densities of conservation laws corresponding to the rotation generatorsXij can be written as one vector:

� =W � E +W · �x � ��E − V � B − V · �x � ��B , �37�

where x= �x ,y ,z�.



The operator X12 provides the following pseudovector �:

�1 = − V3E1 − y�Ex2V3 − Ex

3V2� + x�Ey2V3 − Ey

3V2� − W3B1 − y�Bx2W3 − Bx

3W2� + x�By2W3 − By

3W2� ,

�2 = − V3E2 + y�Ex1V3 − Ex

3V1� − x�Ey1V3 − Ey

3V1� − W3B2 + y�Bx1W3 − Bx

3W1� − x�By1W3 − By

3W1� ,

�3 = V1E1 + V2E2 − y�Ex1V2 − Ex

2V1� + x�Ey1V2 − Ey

2V1� + W1B1 + W2B2 − y�Bx1W2 − Bx

2W1� + x�By1W2

− By2W1� .

The pseudovector � for the operator X13 has the following form:

�1 = V2E1 − z�Ex2V3 − Ex

3V2� + x�Ez2V3 − Ez

3V2� + W2B1 − z�Bx2W3 − Bx

3W2� + x�Bz2W3 − Bz

3W2� ,

�2 = − V1E1 − V3E3 + z�Ex1V3 − Ex

3V1� − x�Ez1V3 − Ez

3V1� − W1B1 − W3B3 + z�Bx1W3 − Bx

3W1�

− x�Bz1W3 − Bz

3W1� ,

�3 = V2E3 − z�Ex1V2 − Ex

2V1� + x�Ez1V2 − Ez

2V1� + W2B3 − z�Bx1W2 − Bx

2W1� + x�Bz1W2 − Bz

2W1� .

The operator X23 provides the following pseudovector �:

�1 = V2E2 + V3E3 − z�Ey2V3 − Ey

3V2� + y�Ez2V3 − Ez

3V2� + W2B2 + W3B3 − z�By2W3 − By

3W2� + y�Bz2W3

− Bz3W2� ,

�2 = − V1E2 + z�Ey1V3 − Ey

3V1� − y�Ez1V3 − Ez

3V1� − W1B2 + z�By1W3 − By

3W1� − y�Bz1W3 − Bz

3W1� ,

�3 = − V1E3 − z�Ey1V2 − Ey

2V1� + y�Ez1V2 − Ez

2V1� − W1B3 − z�By1W2 − By

2W1� + y�Bz1W2 − Bz

2W1� .

B. Two-solution representation of conservation laws

The conserved quantities obtained in Sec. IV A involve solutions V ,W of the adjoint equa-tions �21�. It may be useful for applications to give an alternative representation of the conservedquantities in terms of the electric and magnetic vector fields E ,B only.

We suggest here one possibility based on the observation that one can satisfy the adjointsystem �21� by letting

V�x,t� = B�x,− t� ,

W�x,t� = E�x,− t� ,

Re�x,t� = � · E�x,− t� − e�x,− t� ,

Rm�x,t� = � · B�x,− t� − m�x,− t� , �38�

where E�x ,s� ,B�x ,s� solve Eqs. �17� with s=−t. Indeed, employing the substitution �38� and thenotation s=−t we have

� � V +�W

�t− eW = � � B�x,s� +

�E�x,s��s

�s

�t− eE�x,s� ,

� � W −�V

�t+ mV = � � E�x,s� −

�B�x,s��s

�s

�t+ mB�x,s� ,



Re = � · E�x,s� − e�x,s� ,

Rm = � · B�x,s� − m�x,s� . �39�

Hence, the adjoint Eqs. �21� reduce to Eq. �17�:

� � E�x,s� +�B�x,s�

�s+ mB�x,s� = 0,

� · B�x,s� −�E�x,s�

�s− eE�x,s� = 0,

� · E�x,s� − e�x,s� = 0,

� · B�x,s� − m�x,s� = 0. �40�

Let �E�x , t� ,B�x , t�� and �E��x , t� ,B��x , t�� be any two solutions of the electromagnetic equa-tions �17�. Substituting in Eq. �38� the solution �E� ,B��, we obtain the two-solution representa-tions of the conservation laws. For example, the conservation law given by Eqs. �30� and �31� hasin this representation the following coordinates:

� = E�x,t� · B��x,− t� + B�x,t� · E��x,− t� ,

� = �E�x,t� � E��x,− t�� − �B�x,t� � B��x,− t�� . �41�

In particular, if the solutions �E�x , t� ,B�x , t�� are identical, Eq. �41� provides the one-solutionrepresentation:

� = E�x,t� · B�x,− t� + B�x,t� · E�x,− t� ,

� = �E�x,t� � E�x,− t�� − �B�x,t� � B�x,− t�� . �42�

All other conservation laws can be treated likewise, e.g., the conservation law given by Eqs.�33� and �35� has the following two-solution representation:

� = Et�x,t� · E��x,− t� − Bt�x,t� · B��x,− t� ,

� = �B��x,− t� � Et�x,t�� + �E��x,− t� � Bt�x,t�� . �43�

ACKNOWLEDGMENT

One of the authors �B.T.� gratefully acknowledges the financial support from the SwedishGovernmental Agency for Innovation Systems �VINNOVA�.

1N. H. Ibragimov, J. Math. Anal. Appl. 333, 311–320 �2007�.2N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, 2nd ed. �Wiley, Chichester,1999�.

3P. A. M. Dirac, Proc. R. Soc. London, Ser. A 133, 60 �1931�.4 J. Schwinger, Science 165, 757 �1969�.5B. Thidé, Electromagnetic Field Theory �Upsilon Books, Uppsala, Sweden, 2006�; http://www.plasma.uu.se/CED/Book6N. Meyer-Vernet, Am. J. Phys. 50, 846 �1982�.7P. Olesen, Phys. Lett. B 366, 117 �1996�.8O. Coceal, W. A. Sabra, and S. Thomas, Europhys. Lett. 35, 277 �1996�.



Paper IV

59

Symmetries and nonlocal conservation laws of the general magma equation

Raisa Khamitova *

Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden

a r t i c l e i n f o

Article history:Received 22 August 2008Accepted 22 August 2008Available online xxxx

PACS:02.30.Jr02.20.Sv

Keywords:Magma equationSelf-adjointnessQuasi-self-adjointnessNonlocal conservation laws

a b s t r a c t

In this paper the general magma equation modelling a melt flow in the Earth’s mantle isdiscussed. Applying the new theorem on nonlocal conservation laws [Ibragimov NH. Anew conservation theorem. J Math Anal Appl 2007;333(1):311–28] and using the sym-metries of the model equation nonlocal conservation laws are computed. In accordancewith Ibragimov [Ibragimov NH. Quasi-self-adjoint differential equations. Preprint inArchives of ALGA, vol. 4, BTH, Karlskrona, Sweden: Alga Publications; 2007. p. 55–60,ISSN: 1652-4934] it is shown that the general magma equation is quasi-self-adjoint forarbitrary m and n and self-adjoint for n = �m. These important properties are used forderiving local conservation laws.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

The equation

ofot

þ ooz

f n 1� ooz

1f m

ofot

� �� ¼ 0 ð1Þ

models the migration of melt through the Earth’s mantle. In [3] it is called the general magma equation. It follows from theequations

uz ¼ �ft ; u ¼ f n 1þ ooz

1f m

ouoz

� �� ; ð2Þ

where u is the vertical barometric flux of melt, f is the volume fraction of melt, z is a vertical space coordinate and t is time.All the variables are dimensionless. Eqs. (2) were proposed by Scott and Stevenson [4]. They suggested that according to Dul-lien [5] 2 6 n 6 5 or even bigger and supposed that 0 6m 6 1. Some authors discussed Eq. (1) for any values of n and m.

The general magma equation does not have any Lagrangian and therefore it is impossible to apply Noether’s theorem forcalculating conservation laws. However, according to Atherton and Homsy [6] and Ibragimov [1], it is possible to introduce aformal Lagrangian if any given system of equations is taken into consideration together with the adjoint system. In his recentpaper [1], Ibragimov has proved that the adjoint system inherits symmetries of the given system and has suggested a newtheorem on nonlocal conservation laws. The nonlocal conserved vectors presented in my article are obtained by applying thisnew theorem to the infinitesimal symmetries of Eq. (1). Preliminary results were published in the preprint [7] and in theproceedings of the 2nd conference on ‘‘Nonlinear Science and Complexity” [8].

1007-5704/$ - see front matter � 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.cnsns.2008.08.009

* Tel.: +46 455 385 462.E-mail address: [email protected]

Commun Nonlinear Sci Numer Simulat xxx (2008) xxx–xxx

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Please cite this article in press as: Khamitova R, Symmetries and nonlocal conservation laws of the general magma equa-tion, Commun Nonlinear Sci Numer Simulat (2008), doi:10.1016/j.cnsns.2008.08.009

I denote f by u in order to make Eq. (1) compatible with the usual notation in the Lie group analysis. Using the notation ofdifferential algebra it is rewritten as follows:

F � ut þ Dzfun½1� Dzðu�mutÞ�g ¼ 0 ð3Þwhere

Dz ¼ ooz

þ uzoou

þ uzjoouj

þ uzjko

oujkþ � � �

is the operator of total differentiation with respect to z. Here the usual convention of summation over repeated indices isused. Correspondingly, Dt is the operator of total differentiation with respect to t.

The local conservation laws for Eq. (3),

½DtðC1Þ þ DzðC2Þ�ð3Þ ¼ 0;

were obtained by Barcilon and Richter [9] and Harris [10] using the direct method of calculating components of conservedvectors. These conservation laws were also discussed in [11]. It is shown that the local conserved vectors of the form

C1 ¼ C1ðu;uzÞ and C2 ¼ Aðu;uzÞut þ Bðu;uzÞutz þ Sðu;uzÞ;which are presented in [11] as Case B in Table 2 can be calculated by using self-adjointness and quasi-self-adjointness of Eq.(3). The local conserved vector with the components

eC1 ¼ u; eC2 ¼ mun�m�1uzut � u1�2mutz þ n1�m

u1�m ð4Þ

can be derived as a particular case of a nonlocal conservation law (see [11], Case A in Table 2).

2. Self-adjointness of the equation

After differentiations Eq. (3) transforms to

F � ut � un�mutzz þ ð2m� nÞun�m�1uzutz þmun�m�1utuzz þmðn�m� 1Þun�m�2utu2z þ nun�1uz ¼ 0: ð5Þ

The adjoint equation F* = 0 is defined according to [1]

F� � ddu

ðvFÞ ¼ 0 ð6Þ

where v = v(t, z) is a new dependent variable and

ddu

¼ oou

� Diooui

þ DiDjo

ouij� DiDjDk

oouijk

þ � � �

is the variational derivative, D1 = Dt and D2 = Dz, i,j,k = 1,2. Eqs. (5) and (6) yield

F� � �vt þ vtzzun�m þ nvtzun�m�1uz þ nvzzun�m�1ut þ nvz½2un�m�1utz þ ðn�m� 1Þun�m�2utuz � un�1� ¼ 0: ð7ÞLetting v = u in Eq. (7) and comparing it with Eq. (5) we obtain that F* = �F = 0 if m = �n. We have proved the following:

Proposition 2.1. Eq. (5) is self-adjoint if m = �n, i.e. when it has the following form:

F � ut � u2nutzz � 3nu2n�1uzutz � nu2n�1utuzz � nð2n� 1Þu2n�2utu2z þ nun�1uz ¼ 0: ð8Þ

3. Quasi-self-adjointness of the equation

According to Ibragimov [2] an equation is quasi-self-adjoint if there exists a function h(u) such that F* = k(u)F upon thesubstitution v = h(u).

Proposition 3.1. Eq. (5) is quasi-self-adjoint for arbitrary m and n.

Proof. Calculation of the derivatives of v = h(u):

vt ¼ h0ut ; vz ¼ h0uz; vtz ¼ h00utuz þ h0utz; vzz ¼ h00u2z þ h0uzz; vtzz ¼ h000utu2

z þ 2h00uzutz þ h00utuzz þ h0utzz

and substitution of the results in (7) give

F� ¼ �h0ðut � utzzun�mÞ þ ð2uh00 þ 3nh0Þun�m�1uzutz þ ðuh00 þ nh0Þun�m�1utuzz þ ½u2h000 þ 2nuh00

þ nðn�m� 1Þh0�un�m�2utu2z � h0nuuz: ð9Þ

2 R. Khamitova / Commun Nonlinear Sci Numer Simulat xxx (2008) xxx–xxx

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Comparing with Eq. (5) we obtain k(u) = �h0and

2uh00 þ 3nh0 ¼ �ð2m� nÞh0; uh00 þ nh0 ¼ �mh0

; ð10Þu2h000 þ 2nuh00 þ nðn�m� 1Þh0 ¼ �mðn�m� 1Þh0

: ð11ÞEqs. (10) reduce to one equation, namely:

uh00 þ ðnþmÞh0 ¼ 0:

The latter equation has two solutions:

h ¼ � C1

ðnþm� 1Þunþm�1 þ C2; when nþm–1

and

h ¼ C1 ln juj þ C2; when nþm ¼ 1;

C1 and C2 are arbitrary constants. These functions also satisfy Eq. (11). Thus Eq. (5) is quasi-self-adjoint. We can choose

hðuÞ ¼ u1�n�m ð12Þand

hðuÞ ¼ ln juj ð13Þfor n +m– 1 and n +m = 1, respectively. h

4. Nonlocal conservation laws: general form

In accordance with [6,1] we introduce the formal Lagrangian

L ¼ vF:

I will write it in the symmetrized form:

L¼ v ut �13un�mðutzzþuztz þuzztÞþ1

2ð2m�nÞun�m�1uzðutzþuztÞþmun�m�1utuzz þmðn�m�1Þun�m�2utu2

z þnun�1uz

� �:

ð14ÞEq. (5) is said to have a nonlocal conservation law if there exits a vector C = (C1, C2) satisfying the condition

DtðC1Þ þ DzðC2Þ ¼ 0 ð15Þon any solution of Eqs. (5) and (7). Eq. (5) has a local conservation law if (15) is satisfied on any solution of the equation inquestion. We will not take into consideration trivial conservation laws.1 Conservation laws are regarded as equivalent if theydiffer only by a trivial conservation law.

The conserved vector corresponding to an operator

X ¼ n1ðt; z;uÞ oot

þ n2ðt; z;uÞ oo z

þ gðt; z; uÞ oou

;

admitted by Eq. (5), is obtained by the formula [1]

Ci ¼ niLþWoLoui

� DjoLouij

� �þ DjDk

oLouijk

� �� þ DjðWÞ oL

ouij� Dk

oLouijk

� �� þ DjDkðWÞ oL

ouijk; ð16Þ

where

W ¼ g� niui;

i,j,k = 1,2. Since the LagrangianL is equal to zero on solutions of the equation F = 0 we can calculate Ci without the term niL.Furthermore, we can rewrite the density

C1 ¼ WoLout

� DzoLoutz

� �þ D2

zoLoutzz

� �� þ DzðWÞ oL

outz� Dz

oLoutzz

� �� þ D2

z ðWÞ oLoutzz

1 A conservation law is trivial if Dt(C1) + Dz(C2) � 0 or C1 and C2 are smooth functions of F, F*, Di(F), Di(F*),. . .

R. Khamitova / Commun Nonlinear Sci Numer Simulat xxx (2008) xxx–xxx 3

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using the following identities:

�WDzoLoutz

� �¼ �Dz W

oLoutz

� �þ DzðWÞ oL

outz;

WD2z

oLoutzz

� �¼ D2

z WoLoutzz

� �� 2DzðWÞDz

oLoutzz

� �� D2

z ðWÞ oLoutzz

�

Hence we obtain

C1 ¼ WoLout

þ DzðWÞ 2oLoutz

� 3DzoLoutzz

� �� ; ð17Þ

the remaining term

�Dz WoLoutz

� �þ D2

z WoLoutzz

� �ð18Þ

will be included in C2, where it will have the form

�DtðWÞ oLoutz

�WDtoLoutz

� �þ DtDzðWÞ oL

outzzþ DtðWÞDz

oLoutzz

� �þ DzðWÞDt

oLoutzz

� �þWDtDz

oLoutzz

� �: ð19Þ

Calculation of C2 from Eq. (16) gives

C2 ¼ WoLouz

� DtoLouzt

� �� Dz

oLouzz

� �þ DtDz

oLouztz

� �þ DzDt

oLouzzt

� �� þ DtðWÞ oL

ouztþ DzðWÞ oL

ouzz� DtðWÞDz

oLouztz

� �� DzðWÞDt

oLouzzt

� �þ DtDzðWÞ oL

ouztzþ DzDtðWÞ oL

ouzzt:

Using the fact that the Lagrangian (14) is symmetrical with respect to the mixed derivatives and adding the expression (19)we can simplify C2 and obtain the following:

C2 ¼ WoLouz

� 2DtoLoutz

� �� Dz

oLouzz

� �þ 3DtDz

oLoutzz

� �� þ DzðWÞ oL

ouzzþ 3DtDzðWÞ oL

outzz: ð20Þ

Invoking (14) we get the final expressions for the components of a nonlocal conserved vector from (18) and (20):

C1 ¼ vfW½1þmun�m�1uzz þmðn�m� 1Þun�m�2u2z � þ ð2m� nÞun�m�1uzDzðWÞg þ DzðWÞDzðvun�mÞ ð21Þ

and

C2 ¼ Wfv ð2m� nÞun�m�1utz þ 2mðn�m� 1Þun�m�2utuz þ nun�1� � ð2m� nÞDtðvun�m�1uzÞ �mDzðvun�m�1utÞ� DtDzðvun�mÞg þmvun�m�1utDzðWÞ � vun�mDtDzðWÞ: ð22Þ

Thus Eqs. (21) and (22) define components of a nonlocal conservation law for the system of Eqs. (5), (7) corresponding to anyoperator X admitted by Eq. (5).

5. Formulation of the results

Here I present the conservation laws obtained in this article. Their computation is given in Sections 6–8.

5.1. Nonlocal conservation laws, n– 0

The symmetries of Eq. (5) for n– 0 provide the following nonlocal conserved vectors:Translation of time:

C1 ¼ �vzun;

C2 ¼ vtðmun�m�1utuz � un�mutz þ unÞ þ nvzun�m�1u2t þ vtzun�mut ;

Translation of the space coordinate z:

C1 ¼ ½uþmun�m�1u2z � un�muzz�vz;

C2 ¼ vtzun�muz � vtuþ nvn�m�1z utuz;

ð23Þ


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Dilations with the operator X3 ¼ ð2� n�mÞt oot þ ðn�mÞz o

oz þ 2u oou:

C1 ¼ vz½ðn�mÞzðuþmun�m�1u2z � un�muzzÞ þ ð2� n�mÞðun�muz � tunÞ� þ ð2þ n�mÞvu;

C2 ¼ ð2þ n�mÞv½un � un�mutz þmun�m�1utuz� þ vt ½ð2� n�mÞt un þmun�m�1utuz � un�mutz �

� ðn�mÞzu� þ nvz½ð2� n�mÞtun�m�1u2t � 2un�mut þ ðn�mÞzun�m�1utuz� þ vtz½ð2� n�mÞtun�mut

� 2un�mþ1 þ ðn�mÞzun�muz�:

ð24Þ

5.2. Local conservation laws, n– 0

Dilations:

Case 1. Eq. (5) is self-adjoint.n– 0, m = �n, m– 1:

C1 ¼ u�2mu2z þ u2; C2 ¼ 2 mu�2mutuz � u1�2mutz � m

1�mu1�m

� :

n = �1, m = 1:

C1 ¼ u�2u2z þ u2; C2 ¼ 2ðu�2utuz � u�1utz � lnuÞ:

Case 2a. Eq. (5) is quasi-self-adjoint, n +m– 1:Case (i) n +m– 1, n +m– 2, m– 1:

C1 ¼ 12ð1� n�mÞu�2mu2

z þ1

2� n�mu2�n�m;

C2 ¼ mu�2mutuz � u1�2mutz þ n1�m

u1�m:

ð25Þ

Case (ii) n +m = 2, m– 1:

C1 ¼ �12u�2mu2

z þ lnu;

C2 ¼ mu�2mutuz � u1�2mutz þ ð2�mÞ1�m

u1�m:

ð26Þ

Case (iii) n– 1, m = 1:

C1 ¼ �n2u�2mu2

z þ1

1� nu1�n;

C2 ¼ u�2utuz � u�1utz þ n lnu:ð27Þ

Case (iv) n = 1, m = 1:

C1 ¼ �12u�2u2

z þ lnu;

C2 ¼ u�2utuz � u�1utz þ lnu:

Case 2b. Eq. (5) is quasi-self-adjoint, n +m = 1, m– 1:

C1 ¼ 12u�2mu2

z þ u lnu

C2 ¼ lnu mu�2mutuz � u1�2mutz þ u1�m Þ þmu�2mutuz � u1�2mutz � m

1�mu1�m:

5.3. Nonlocal conservation laws, n = 0

When n = 0 Eq. (5) transforms to the form:

Dt u� D2z

u1�m

1�m

� �� ¼ 0; if m–1;

Dt½u� D2z ðlnuÞ� ¼ 0; if m ¼ 1:

Translation of the space coordinate z. The conserved vector has the components (23) with n = 0.Dilations with the operator eX3 ¼ �mz o

oz þ 2u oou

n = 0, m– 0:


ARTICLE IN PRESS


C1 ¼ vz½�mzðuþmu�m�1u2z � u�muzzÞ þ ð2�mÞu�muz� þ ð2�mÞvu;

C2 ¼ v½�ð2�mÞu�mutz þmð2�mÞu�m�1utuz� þmzuvt � ½2u1�m þmzu�muz�vtz:Transformation with the operator X4 ¼ �z2 o

oz þ 3zu oou, n = 0, m ¼ 4

3:

C1 ¼ v�8u�4

3uz þ z 3uþ 4u�43uzz þ 4u�7

3u2z

h iþz2 uz þ 4u�7

3uzuzz � 289

u�103 u3

z

� ��þ 3uþ 5zuz þ z2uzz �

Dz vu�43

� ;

C2 ¼ v u�43ut þ z 3u�4

3utz � 12u�73utuz

� � z2 ut þ 28

9u�10

3 utu2z

� ��83Dt vu�7

3uz

� � 43Dz vu�7

3ut

� � DtDz vu�4

3

� �

5.4. Local conservation laws, n = 0

Dilations with the operator eX3 ¼ �mz ooz þ 2u o

ou

The equation is quasi-self-adjoint.

Case (i): n = 0, m– 2, m– 1, follows from (25):

C1 ¼ 12ð1�mÞu�2mu2

z þ1

2�mu2�m;

C2 ¼ mu�2mutuz � u1�2mutz:

Case (ii): n = 0, m = 2, follows from (26):

C1 ¼ �12u�4u2

z þ lnu;

C2 ¼ 2u�4utuz � u�3utz:

Case (iii): n = 0, m = 1, follows from (27):

C1 ¼ 12u�2u2

z þ u lnu;

C2 ¼ lnu½�u�1utz þ u�2utuz� þ u�2utuz � u�1utz:

6. Computation of conservation laws: n 6¼ 0

6.1. Translation of time

Nonlocal conservation lawIt is obvious that Eq. (5) with arbitrarym and n is invariant under the translation of time t. The operator for the time trans-

lation, X1 ¼ oot, has n1 = 1, n2 = 0, g = 0, therefore W = �ut. Hence from (21) the density C1 has the following form:

C1 ¼ �vfut ½1þmun�m�1uzz þmðn�m� 1Þun�m�2u2z � þ ð2m� nÞun�m�1uzutzg � utzDzðvun�mÞ:

It follows from Eq. (5), F = 0, that

ut þ ð2m� nÞun�m�1uzutz þmun�m�1utuzz þmðn�m� 1Þun�m�2utu2z ¼ un�mutzz � nun�1uz ð28Þ

which gives

C1 ¼ v½�un�mutzz þ nun�1uz� � utzDzðvun�mÞ ¼ Dz½vð�un�mutz þ unÞ� � vzun:

Hence

C1 ¼ �vzun: ð29ÞUsing (22) and adding the term Dt[v(�un�mutz + un)] corresponding to Dz[v(�un�mutz + un)] we compute the component C2:

C2 ¼ �utfv½ð2m� nÞun�m�1utz þ 2mðn�m� 1Þun�m�2utuz þ nun�1� � ð2m� nÞDtðvun�m�1uzÞ �mDzðvun�m�1utÞ� DtDzðvun�mÞg �mvun�m�1ututz þ vun�muttz þ Dt ½vð�un�mutz þ unÞ�: ð30Þ


ARTICLE IN PRESS


The reckoning shows that the expression

v½ð2m� nÞun�m�1utz þ 2mðn�m� 1Þun�m�2utuz þ nun�1� � ð2m� nÞDtðvun�m�1uzÞ �mDzðvun�m�1utÞ � DtDzðvun�mÞis equal to

vð�nun�m�1utz þ nun�1Þ �mvtun�m�1uz � nvzun�m�1ut � vtzun�m:

Hence

C2 ¼ vtðmun�m�1utuz � un�mutz þ unÞ þ nvzun�m�1u2t þ vtzun�mut : ð31Þ

Thus Eqs. (29) and (31) define a nonlocal conservation law corresponding to the translation of the coordinate t. Indeed, cal-culations give that

DtðC1Þ þ DzðC2Þ ¼ vtF þ utF�

which equals to zero on solutions of Eqs. (5) and (7).Local conservation laws:Eq. (5) is self-adjoint when n = �m. Substitution of v = u in (32) yields

C1 ¼ �uzun ¼ �Dz1

nþ 1unþ1

� �; if n–� 1;

C1 ¼ �uzu�1 ¼ �DzðlnuÞ; if n ¼ �1:

In the case of quasi-self-adjointness of the equation F = 0, when n +m– 1 and v = u1�n�m the component C1 has the followingform:

C1 ¼ �ð1� n�mÞu�muz ¼ �Dz1� n�m1�m

u1�m

� �; if m–1;

C1 ¼ nu�1uz ¼ Dzðn lnuÞ; if m ¼ 1;n–0:

For the second case of quasi-self-adjointness, when n +m = 1 and v = lnu, we have

C1 ¼ �un�1uz ¼ �Dz1nun

� �; if n–0;

C1 ¼ �u�1uz ¼ �DzðlnuÞ; if m ¼ 1; n ¼ 0:

Thus for arbitrary m and n only trivial local conservation laws correspond to the operator X1.

6.2. Translation of the space coordinate z

Nonlocal conservation law:Eq. (5) with arbitrary m and n is also invariant under the translation of the coordinate z with the operator X2 ¼ o

oz : In thiscase n1 = 0, n2 = 1,g = 0, therefore W = �uz. From (21) the corresponding component C1 has the form:

C1 ¼ �vfuz½1þmun�m�1uzz þmðn�m� 1Þun�m�2u2z � þ ð2m� nÞun�m�1uzuzzg � uzzDzðvun�mÞ

or

C1 ¼ ½uþmun�m�1u2z � un�muzz�vz þ Dz½�vðuþmun�m�1u2

z Þ�:Hence we can choose

C1 ¼ ½uþmun�m�1u2z � un�muzz�vz ð32Þ

as the density of a nonlocal conservation law corresponding to the translation of z. Adding to C2 in (22) the termDt½�vðuþmun�m�1u2

z Þ� corresponding to Dz½�vðuþmun�m�1u2z Þ� we obtain:

C2 ¼ �uzfv½ð2m� nÞun�m�1utz þ 2mðn�m� 1Þun�m�2utuz þ nun�1� � ð2m� nÞDtðvun�m�1uzÞ �mDzðvun�m�1utÞ� DtDzðvun�mÞg �mvun�m�1utuzz þ vun�mutzz þ Dt ½�vðuþmun�m�1u2

z Þ�:or

C2 ¼ vtzun�muz � vtuþ nvzun�m�1utuz � vF:

Excluding the trivial part vF we finally have

C2 ¼ vtzun�muz � vtuþ nvn�m�1z utuz: ð33Þ


ARTICLE IN PRESS


Thus Eqs. (32) and (33) define a nonlocal conservation law corresponding to the translation of the coordinate z. Indeed, cal-culations give that

DtðC1Þ þ DzðC2Þ ¼ vzF þ uzF�;

which equals to zero on solutions of Eqs. (5) and (7).Local conservation laws:Self-adjointness of Eq. (5). Substitution of v = u and n = �m in (29) gives

C1 ¼ ½uþmu�2m�1u2z � u�2muzz�uz ¼ Dz

12ðu2 � u�2mu2

z Þ� �

:

Quasi-self-adjointness of Eq. (5): When n +m– 1 and v = u1�n�m the component C1 has the following form:

C1 ¼ ½uþmun�m�1u2z � un�muzz�ð1� n�mÞu�n�muz:

Hence

C1 ¼ Dz ðnþm� 1Þ u2�n�m

nþm� 2þ u�2mu2

z

2

� �� ; if nþm–2

or

C1 ¼ Dz � lnuþ u�2mu2z

2

� �; if nþm ¼ 2:

In the second case of quasi-self-adjointness, when n +m = 1 and v = lnu, we obtain

C1 ¼ ½uþmu�2mu2z � u1�2muzz�u�1uz ¼ Dzðu� u�2mu2

z Þ:Thus for arbitrary m and n only trivial local conservation laws correspond to the operator X2.

6.3. Dilations

Nonlocal conservation law:When the parameter n– 0, Eq. (5) admits the following operator of dilation [11]:

X3 ¼ ð2� n�mÞt oot

þ ðn�mÞz ooz

þ 2uoou

:

For this operator

n1 ¼ ð2� n�mÞt; n2 ¼ ðn�mÞz and g ¼ 2u;

therefore

W ¼ 2u� ð2� n�mÞtut � ðn�mÞzuz:

From (21) we have the following component C1:

C1 ¼ vf½2u� ð2� n�mÞtut � ðn�mÞzuz� � ½1þmun�m�1uzz þmðn�m� 1Þun�m�2u2z � þ ð2m� nÞun�m�1uzDz½2u

� ð2� n�mÞtut � ðn�mÞzuz�g þ ½vzun�m þ ðn�mÞvun�m�1uzÞ�Dz½2u� ð2� n�mÞtut � ðn�mÞzuz�:Collecting first all terms containing tut and invoking (28) we obtain:

C11 ¼ ð2� n�mÞtfv½nun�1uz � un�mutzz � ðn�mÞun�m�1uzutz� � vzun�mutzg¼ �ð2� n�mÞtunvz þ Dzfð2� n�mÞtvðun � un�mutzÞg:

For the terms containing 2u � (n �m)zuz we have

C12 ¼ vz½ðn�mÞzðuþmun�m�1u2

z � un�muzzÞ þ ð2� n�mÞun�muz� þ ð2þ n�mÞvuþ Dzfv½2mun�muz

� ðn�mÞzðuþmun�m�1u2z Þ�g: ð34Þ

Combining all nontrivial terms in C11 and C2

1 we obtain the density of a nonlocal conservation law:

C1 ¼ vz ðn�mÞzðuþmun�m�1u2z � un�muzzÞ þ ð2� n�mÞðun�muz � tunÞ� þ ð2þ n�mÞvu: ð35Þ

Using (22) and adding the term corresponding to

Dzfv½2mun�muz � ðn�mÞzðuþmun�m�1u2z Þ þ ð2� n�mÞtðun � un�mutzÞ�g


ARTICLE IN PRESS


we obtain the component C2:

C2 ¼ Wfv½ð2m� nÞun�m�1utz þ 2mðn�m� 1Þun�m�2utuz þ nun�1� � ð2m� nÞDtðvun�m�1uzÞ �mDzðvun�m�1utÞ� DtDzðvun�mÞg þmvun�m�1utDzðWÞ � vun�mDtDzðWÞ þ Dtfv½2mun�muz � ðn�mÞzðuþmun�m�1u2

z Þþ ð2� n�mÞtðun � un�mutzÞ�g:

Collecting together all the terms containing t again, we get:

C21 ¼ ð2� n�mÞfvun þ t½vtðun þmun�m�1utuz � un�mutzÞ þ nvzun�m�1u2

t þ vtzun�mut �g:The remaining terms in C2 will give:

C22 ¼ v½2nun � ð2þ n�mÞun�mutz þmð2þ n�mÞun�m�1utuz� � ðn�mÞzuvt � ½2u� ðn�mÞzuz�ðnvzun�m�1ut þ vtzun�mÞ

ð36Þand the term �(n �m)zvF, which we can ignore. Thus

C2 ¼ C21 þ C2

2 ¼ ð2þ n�mÞv½un � un�mutz þmun�m�1utuz� þ vt ½ð2� n�mÞtðun þmun�m�1utuz � un�mutzÞ � ðn�mÞzu�þ nvz½ð2� n�mÞtun�m�1u2

t � 2un�mut þ ðn�mÞzun�m�1utuz� þ vtz½ð2� n�mÞtun�mut � 2un�mþ1

þ ðn�mÞzun�muz�: ð37ÞThus Eqs. (35) and (37) define the components of a nonlocal conserved vector corresponding to the operator X3.

Local conservation laws

Remark. The adjoint Eq. (7) has a solution v = const. If we choose v ¼ 12þn�m we immediately obtain from (35), (37) the local

conserved vector with the components (4),eC1 ¼ u; eC2 ¼ mun�m�1utuz � un�mutz þ un:

Case 1. Eq. (5) is self-adjoint: n = �m.

Substitution of v = u in (35) yields

C1 ¼ uz½�2mzðuþmu�2m�1u2z � u�2muzzÞ þ 2ðu�2muz � tu�mÞ� þ ð2� 2mÞu2

whence

C1 ¼ ð2�mÞu�2mu2z þ ð2�mÞu2 þ Dz½mzð�u2 þ u�2mu2

z Þ� � 2tu�muz: ð38ÞSince

�2tu�muz ¼ Dz2t

m�1 u1�m

�; if m–1;

Dzð�2t lnuÞ; if m ¼ 1;

(ð39Þ

therefore for n +m = 0, m– 1 the component C1 has the form:

C1 ¼ ð2�mÞðu�2mu2z þ u2Þ þ Dz½mzð�u2 þ u�2mu2

z Þ� þ Dz2t

m� 1u1�m

� �ð40Þ

and for n +m = 0, m = 1

C1 ¼ u�2u2z þ u2 þ Dz½zð�u2 þ u�2u2

z Þ� þ Dzð�2t lnuÞ: ð41ÞUsing (37) and adding

Dt ½mzð�u2 þ u�2mu2z Þ� þ Dt

2tm� 1

u1�m

� �corresponding to the remaining terms in (40) we obtain C2:

C2 ¼ ð2� 2mÞu½u�m � u�2mutz þmu�2m�1utuz� þ ut ½2tðu�m þmu�2m�1utuz � u�2mutzÞ þ 2mzu��muz½2tu�2m�1u2

t � 2u�2mut � 2mzu�2m�1utuz� þ utz½2tu�2mut � 2u1�2m � 2mzu�2muz� þ Dt ½mzð�u2 þ u�2mu2z Þ�

þ Dt2t

m� 1u1�m

� �¼ 2ð2�mÞ mu�2mutuz � u1�2mutz � m

1�mu1�m

h i:

Thus for n +m = 0 and m– 1 the density has the form:

C1 ¼ u�2mu2z þ u2 ð42Þ


ARTICLE IN PRESS


and

C2 ¼ 2 mu�2mutuz � u1�2mutz � m1�m

u1�mh i

ð43Þ

(which also holds for m = 2). Indeed, Dt(C1) + Dz(C2) = 2uF which is equal to zero on solutions of the equation F = 0.For n = �1 and m = 1 we have the density from (41)

C1 ¼ u�2u2z þ u2: ð44Þ

From (37), after addition of the term Dt ½zð�u2 þ u�2u2z Þ� þ Dtð�2t lnuÞ corresponding to the remaining terms in (41), we ob-

tain the component C2:

C2 ¼ ut ½2tðu�1 þ u�3utuz � u�2utzÞ þ 2zu� � uz½2tu�3u2t � 2u�2ut � 2zu�3utuz� þ utz½2tu�2ut � 2u�1 � 2zu�2uz�

þ Dt½zð�u2 þ u�2u2z Þ� þ Dtð�2t lnuÞ

whence

C2 ¼ 2 u�2utuz � u�1utz � lnu �

: ð45ÞThus for n = �1 and m = 1 the conserved vector has the components (44) and (45).

Case 2a. Eq. (5) is quasi-self-adjoint, n +m– 1 and v = u1�n�m.

According to (35) the density has the following form:

C1 ¼ ð1� n�mÞu�n�muz½ðn�mÞzðuþmun�m�1u2z � un�muzzÞ þ ð2� n�mÞðun�muz � tunÞ� þ ð2þ n�mÞu2�n�m

whence

C1 ¼ 12ð1� n�mÞð4� n� 3mÞu�2mu2

z þ ð2þ n�mÞu2�n�m � Dz12ðn�mÞð1� n�mÞzu�2mu2

z

� �þ ðn�mÞð1� n�mÞzu1�n�muz � ð1� n�mÞð2� n�mÞtu�muz: ð46Þ

Transformation of two last terms gives the following result:

ðn�mÞð1� n�mÞzu1�n�muz ¼

Dzðn�mÞð1�n�mÞ

2�n�m zu2�n�mh i

� ðn�mÞð1�n�mÞ2�n�m u2�n�m;

if nþm–2;�Dz½2ð1�mÞz lnuÞ� þ 2ð1�mÞ lnu

if nþm ¼ 2;

8>>>>>>><>>>>>>>:� ð1� n�mÞð2� n�mÞtu�muz ¼ �Dz

ð1�n�mÞð2�n�mÞ1�m tu1�m

h i; if m–1;

Dz½nð1� nÞt lnu�; if m ¼ 1:

(

It shows that we have four possibilities:

ðiÞ nþm–2; m–1; ðiiÞ nþm ¼ 2; m–1; ðiiiÞ n–1; m ¼ 1; ðivÞ n ¼ 1; m ¼ 1:

When n +m– 2, m– 1 the remaining terms in C1 in (46) are

�Dz12ðn�mÞð1� n�mÞzu�2mu2

z

� �þ Dz

ðn�mÞð1� n�mÞ2� n�m

zu2�n�m

� �� Dz

ð1� n�mÞð2� n�mÞ1�m

tu1�m

� �; ð47Þ

for n +m = 2, m– 1

Dz½ð1�mÞzu�2mu2z � � Dz½2ð1�mÞz lnuÞ�; ð48Þ

and for n– 1, m = 1

Dzn2ðn� 1Þzu�2u2

z

h iþ Dz½nzu1�n� þ Dz½nð1� nÞt lnu�: ð49Þ

Case (i) n +m– 2, m– 1.C1 has the form:

C1 ¼ 12ð1� n�mÞð4� n� 3mÞu�2mu2

z þ ð2þ n�mÞu2�n�m � ðn�mÞð1� n�mÞ2� n�m

u2�n�m


ARTICLE IN PRESS


or

C1 ¼ ð4� n� 3mÞ 12ð1� n�mÞu�2mu2

z þ1

2� n�mu2�n�m

� �: ð50Þ

Using (37) and (48) we compute the component C2:

C2 ¼ ð2þ n�mÞu1�n�m½un � un�mutz þmun�m�1utuz� þ ð1� n�mÞu�n�mut ½ð2� n�mÞtðun þmun�m�1utuz � un�mutzÞ� ðn�mÞzu� þ nð1� n�mÞu�n�muz½ð2� n�mÞtun�m�1u2

t � 2un�mut þ ðn�mÞzun�m�1utuz�þ ð1� n�mÞ½�ðnþmÞu�n�m�1utuz þ u�n�mutz� � ½ð2� n�mÞtun�mut � 2un�mþ1 þ ðn�mÞzun�muz�

� Dt12ðn�mÞð1� n�mÞzu�2mu2

z

� �þ Dt

ðn�mÞð1� n�mÞ2� n�m

zu2�n�m

� �� Dt

ð1� n�mÞð2� n�mÞ1�m

tu1�m

� �whence

C2 ¼ ð4� n� 3mÞ mu�2mutuz � u1�2mutz þ n1�m

u1�mh i

: ð51Þ

Thus for n +m– 1, n +m– 2 and m– 1 it follows from (50) and (51) that the components

C1 ¼ 12ð1� n�mÞu�2mu2

z þ1

2� n�mu2�n�m: ð52Þ

and

C2 ¼ mu�2mutuz � u1�2mutz þ n1�m

u1�m ð53Þdefine a nontrivial conserved vector.Case (ii) n +m = 2, m– 1.

For these values of m and n the term

12ð4� n� 3mÞð1� n�mÞu�2mu2

z þ ð2þ n�mÞu2�n�m

in (46) becomes equal to �ð1�mÞu�2mu2z þ 2ð1�mÞ. Therefore we have

C1 ¼ �ð1�mÞu�2mu2z þ 2ð1�mÞ lnu ¼ 2ð1�mÞ �1

2u�2mu2

z þ lnu� �

:

Ignoring the constant 2(1 �m) we can choose

C1 ¼ �12u�2mu2

z þ lnu ð54Þas the density of the conservation law in this case. Invoking (48) we have the corresponding component C2 from (37):

C2 ¼ 2ð2�mÞu�1½u2�m � u2�2mutz þmu1�2mutuz� þ u�2ut½ð2� 2mÞzu� � ð2�mÞu�2uz½�2u2�2mut þ ð2� 2mÞzu1�2mutuz�þ ½2u�3utuz � u�2utz�½�2u3�2m þ ð2� 2mÞzu2�2muz� þ Dt½ð1�mÞzu�2mu2

z � � Dt½2ð1�mÞz lnuÞ�

or

C2 ¼ 2ð1�mÞ mu�2mutuz � u1�2mutz þ ð2�mÞ1�m

u1�m

� �:

Hence, C2 corresponding to (54), has the form:

C2 ¼ mu�2mutuz � u1�2mutz þ ð2�mÞ1�m

u1�m: ð55Þ

Thus, for n +m = 2 and m– 1, the conserved vector has the components (54) and (55).Case (iii) n– 1, m = 1.

Substituting m = 1 in (50) we obtain that

C1 ¼ ð1� nÞ �n2u�2mu2

z þ1

1� nu1�n

� �: ð56Þ

Using (49) we calculate C2 from (37):

C2 ¼ ð1þ nÞu�n½un � un�1utz þ un�2utuz� � nu�n�1ut ½ð1� nÞtðun þ unutuz � un�1utzÞ � ðn� 1Þzu�� n2u�n�1uz½ð1� nÞtun�2u2

t � 2un�1ut þ ðn� 1Þzun�2utuz� � n½�ðnþ 1Þu�n�2utuz þ u�n�1utz�½ð1� nÞtun�1ut � 2un

þ ðn� 1Þzun�1uz� þ Dtn2ðn� 1Þzu�2u2

z

h iþ Dt ½nzu1�n� þ Dt½nð1� nÞt lnu�


ARTICLE IN PRESS


or

C2 ¼ 1þ nþ ð1� nÞðu�2utuz � u�1utz þ n lnuÞ:whence (ignoring the constant 1 + n)

C2 ¼ ð1� nÞðu�2utuz � u�1utz þ n lnuÞ: ð57ÞHence, it follows from (56) and (57), that when n– 1, m = 1, the components of the conserved vector are

C1 ¼ � n2u�2mu2

z þ1

1� nu1�n;

C2 ¼ u�2utuz � u�1utz þ n lnu: ð58ÞCase (iv) m = 1, n = 1.

In (46) we substitute only m = 1 in the terms which can be equal to 0. Then we have

C1 ¼ 12ð1� nÞu�2u2

z þ 2� Dz1� n2

� �zu�2u2

z

� �þ ðn� 1Þzu�1uz þ ð1� nÞtu�1uz:

After ignoring the trivial part, the constant 2, we notice that all the other terms are multiplied by 1 � n. Therefore the densityhas the form:

C1 ¼ �12u�2u2

z þ ln u ð59Þ

and the remaining terms are

Dz1� n2

zu�2u2z

� �� Dz½ð1� nÞz ln u� þ Dz½ð1� nÞt lnu�

Repeating the steps for calculating C2 for Case (iii), and replacing Dt[nzu1�n] by �Dt[(1 � n)zlnu] we obtain:

C2 ¼ ð1� nÞðu�2utuz � u�1utz þ lnuÞ:Hence

C2 ¼ u�2utuz � u�1utz þ lnu: ð60ÞThus, for m = 1, n = 1, (59) and (60) define a nontrivial conserved vector.

Case 2b Eq. (5) is quasi-self-adjoint, n +m = 1 and v = lnu.For the second case of quasi-self-adjointness of Eq. (5) we have the following C1 from (35):

C1 ¼ u�1uz½ð1� 2mÞzðuþmu�2mu2z � u1�2muzzÞ þ u1�2muz � tu1�m� þ ð3� 2mÞu lnu

or

C1 ¼ ð3� 2mÞu lnuþ Dz½ð1� 2mÞzu� � ð1� 2mÞu� Dz1� 2m

2zu�2mu2

z

� �þ 1� 2m

2u�2mu2

z þ u�2mu2z � tu�muz

where

�tu�muz ¼ �Dz1

1�m tu1�m �

if m–1�Dzðt lnuÞ if m ¼ 1; n ¼ 0:

(

Thus, for n +m = 1 and m– 1, the component C1 has the form:

C1 ¼ 12ð3� 2mÞu�2mu2

z þ ð3� 2mÞu lnu� ð1� 2mÞu: ð61Þ

Eq. (5) is itself a conservation law, which has the density equal toeC1 ¼ u and eC2 ¼ mun�m�1utuz � un�mutz þ un:

For n +m = 1 and m– 1 the component eC2 of the conserved vector (4) has the form:

eC2 ¼ mu�2mutuz � u1�2mutz þ u1�m: ð62Þ

Hence, (61) is the density of a conservation law, which is the linear combination of two conservation laws. We exclude theterm ð1� 2mÞeC1 ¼ ð1� 2mÞu from C1 in (61):


ARTICLE IN PRESS


C1 ¼ ð3� 2mÞ 12u�2mu2

z þ u lnu� �

ð63Þ

and add the term ð1� 2mÞeC2 to the respective C2. Using (37) and also adding the term corresponding for m– 1 to

Dz½ð1� 2mÞzu� � Dz1� 2m

2zu�2mu2

z

� �� Dz

11�m

tu1�m

� �;

we finally obtain:

C2 ¼ ð3� 2mÞ lnu½u1�m � u1�2mutz þmu�2mutuz� þ u�1ut ½tðu1�m þmu�2mutuz � u1�2mutzÞ � ð1� 2mÞzu�þ ð1�mÞu�1uz½tu�2mu2

t � 2u1�2mut þ ð1� 2mÞzu�2mutuz� þ ð�u�2utuz þ u�1utzÞ½tu1�2mut � 2u2�2m

þ ð1� 2mÞzu1�2muz� þ Dt½ð1� 2mÞzu� � Dt1� 2m

2zu�2mu2

z

� �� Dt

11�m

tu1�m

� �þ ð1� 2mÞ½mu�2mutuz � u1�2mutz þ u1�m� ð64Þ

whence

C2 ¼ ð3� 2mÞ lnu½mu�2mutuz � u1�2mutz þ u1�m� þmu�2mutuz � u1�2mutz � m1�m

u1�mn o

:

Thus, it follows from (63) and the latter equation, that if n +m = 1, m– 1, the conserved vector has the followingcomponents:

C1 ¼ 12u�2mu2

z þ u lnu;

C2 ¼ lnu½mu�2mutuz � u1�2mutz þ u1�m� þmu�2mutuz � u1�2mutz � m1�m

u1�m: ð65Þ

Since we assumed here that n– 0, we investigate case m = 1, n = 0 later on.

Remark. The conserved vector in [11], Case (B.5) is a linear combination of two conserved vectors. It should have thecomponents (65).

7. Computation of conservation laws

n = 0, m– 0 and m– 43

In this case Eq. (5) admits a group of transformations with the following operators [11]:

Xa ¼ aðtÞ oot

; X2 ¼ ooz

; eX3 ¼ �mzooz

þ 2uoou

; ð66Þ

where a(t) is an arbitrary function.We have investigated the operator X2 in Section 6.2. Let us have a look at two other operators.

7.1. Operator Xa ¼ aðtÞ oot

For this operator n1 = a(t), n2 = g = 0, therefore W = �aut. The component (21) has the form:

C1 ¼ vf�aut½1þmu�m�1uzz �mðmþ 1Þu�m�2u2z � þ 2mu�m�1uzDzðautÞg � DzðautÞDzðvu�mÞ

or

C1 ¼ �av½ut þmu�m�1utuzz �mðmþ 1Þu�m�2utu2z � 2mu�m�1uzutz� � autzDzðvu�mÞ:

Using (28) when n = 0 we obtain

C1 ¼ �avu�mutzz � autzDzðvun�mÞ ¼ Dzð�avu�mutzÞ:

Thus the operator Xa provides only trivial conservation laws.

7.2. Operator eX3 ¼ �mz ooz þ 2u o

ou

Nonlocal conservation lawThe operator eX3 ¼ X3 � Xa when n = 0 and a(t) = (2 �m)t. It means that we can find the components of conserved vectors

using the formulae (34) and (36) for C12, C

22. They do not include the terms depending on t. Setting n = 0 in (34) and (36) we

obtain the components C1 and C2:


ARTICLE IN PRESS


C1 ¼ vz½�mzðuþmu�m�1u2z � u�muzzÞ þ ð2�mÞu�muz� þ ð2�mÞvu;

C2 ¼ v½�ð2�mÞu�mutz þmð2�mÞu�m�1utuz� þmzuvt � ½2u1�m þmzu�muz�vtz:ð67Þ

Thus, for n = 0, m– 0 and m– 43, a nonlocal conserved vector is defined by (67).

Local conservation lawsSince n = 0 but m– 0, we can take into consideration only the quasi-self-adjointness of Eq. (5). We have the following

possibilities:

ðiÞ m–2; m–1; v ¼ u1�m; ðiiÞ m ¼ 2; v ¼ u�1; ðiiiÞ m ¼ 1; v ¼ lnu:

Case (i) follows from (52) and (53):

C1 ¼ 12ð1�mÞu�2mu2

z þ1

2�mu2�m;

C2 ¼ mu�2mutuz � u1�2mutz;

ð68Þ

Case (ii) from (54) and (55):

C1 ¼ �12u�4u2

z þ lnu;

C2 ¼ 2u�4utuz � u�3utz:

ð69Þ

Case (iii). When m = 1 and n = 0 we obtain from (63) the following C1:

C1 ¼ 12u�2u2

z þ u lnu ð70Þ

Excluding terms with t in (64) and setting m = 1 we have

C2 ¼ lnu½1� u�1utz þ u�2utuz� þ zut þ ð�u�2utuz þ u�1utzÞ½�2� zu�1uz� �Dt ½zu� þDt12zu�2u2

z

� �� ½u�2utuz � u�1utz þ1�

whence

C2 ¼ lnu½�u�1utz þ u�2utuz� þ u�2utuz � u�1utz: ð71Þ

Remark. The conserved vector in [11], Case (B.6) in Table 2, is a linear combination of two conserved vectors. It should havethe components (70) and (71).

8. Computation of conservation laws, n = 0 and m ¼ 43

For these values of m and n Eq. (5) admits a group of transformations with the operators (66) and the additional operator[11] is

X4 ¼ �z2ooz

þ 3zuoou

: ð72Þ

Hence

W ¼ 3zuþ z2uz; DzðWÞ ¼ 3uþ 5zuz þ z2uzz; DtDzðWÞ ¼ 3ut þ 5zutz þ z2utzz:

8.1. Nonlocal conservation law

We obtain C1 for a nonlocal conservation law from (21):

C1 ¼ v ð3zuþ z2uzÞ 1þ 43u�7

3uzz � 289

u�103 u2

z

� �þ 83u�7

3uzð3uþ 5zuz þ z2uzzÞ� �

þ ð3uþ 5zuz þ z2uzzÞDzðvu�43Þ

or

C1 ¼ v 8u�43uz þ z 3uþ 4u�4

3uzz þ 4u�73u2

z

h iþ z2 uz þ 4u�7

3uzuzz � 289

u�103 u3

z

� �� þ ð3uþ 5zuz þ z2uzzÞDzðvu�4

3Þ ð73Þ


ARTICLE IN PRESS


and C2 from (22):

C2 ¼ ð3zuþ z2uzÞ v83u�7

3utz � 569

u�103 utuz

� �� 83Dtðvu�7

3uzÞ � 43Dzðvu�7

3utÞ � DtDz vu�43

� � �þ 43vu�7

3utð3uþ 5zuz þ z2uzzÞ � vu�43ð3ut þ 5zutz þ z2utzzÞ:

Setting in Eq. (5) m ¼ 43 ; n ¼ 0 we obtain:

�u�43utzz þ 8

3u�7

3uzutz þ 43u�7

3utuzz � 289

u�103 utu2

z ¼ �ut:

Therefore, for the following terms with z2v, we have

�u�43utzz þ 8

3u�7

3uzutz þ 43u�7

3utuzz � 569

u�103 utu2

z ¼ �ut � 289

u�103 utu2

z :

Hence

C2 ¼ v u�43ut þ z 3u�4

3utz � 12u�73utuz

� � z2 ut þ 28

9u�10

3 utu2z

� �� 83Dtðvu�7

3 uzÞ � 43Dzðvu�7

3utÞ � DtDz vu�43

� � �: ð74Þ

Thus, for m ¼ 43 ; n ¼ 0, the nonlocal conserved vector has the components (73) and (74).

8.2. Local conservation law

Since n– �m and n +m– 1, we have only one possibility: we can use the quasi-self-adjointness of Eq. (5) when v ¼ u�13:

Invoking (73) we get the following C1:

C1 ¼ 8u�53uz þ z 3u

23 þ 4u�5

3uzz þ 4u�83u2

z

h iþ z2 u�1

3uz þ 4u�83 uzuzz � 28

9u�11

3 u3z

� �� 53u�8

3uzð3uþ 5zuz þ z2uzzÞ

or

C1 ¼ Dz 4zu�53uz � 3

2u�2

3 þ 32z2u

23 þ 7

6z2u�8

3u2z

� �:

Thus the local conservation law is trivial.

8.3. Remark

In [11] the conserved vector with the components

C1 ¼ zeC1 C2 ¼ zeC2 þ u�43ut

is obtained from the conserved vector with the components eC1, eC2, defined by (4), under the action of the operator X4.The operator is admitted by Eq. (5) when n = 0 and m ¼ 4

3. However, for n = 0, the equation and conservation law have theform

Dt u� D2z

u1�m

1�m

� �� ¼ 0; if m–1;

Dt½u� D2z ðlnuÞ� ¼ 0; if m ¼ 1:

Therefore it is evident that for n = 0 it is possible to multiply the conserved vector by any function f(z). The new conservedvector will have the following components:

C1 ¼ f ðzÞu� f 00ðzÞ u1�m

1�m;

C2 ¼ f ðzÞðmu�m�1utuz � u�mutzÞ þ f 0ðzÞu�mut

and

C1 ¼ f ðzÞu� f 0ðzÞ lnu;

C2 ¼ f ðzÞðu�2utuz � u�1utzÞ þ f 0ðzÞu�1ut

for m– 1 and m = 1, respectively.I do not investigate the case n = 0, m = 0 because Eq. (5) becomes linear.


ARTICLE IN PRESS


9. Conclusion

The direct method for calculating conserved vectors of the general magma equation is complicated. On the other hand,the use of the theorem on nonlocal conservation laws [1] and self-evident symmetries of the general magma equation (5)gives a simple regular algorithm for computing conservation laws.

References

[1] Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333(1):311–28.[2] Ibragimov NH. Quasi-self-adjoint differential equations. Preprint in Archives of ALGA, vol. 4. BTH, Karlskrona, Sweden: Alga Publications; 2007. p. 55–

60, ISSN: 1652-4934.[3] Takahashi D, Sachs JR, Satsuma J. Properties of the magma and modified magma equations. J Phys Soc Jpn 1990;59(6):1941–53.[4] Scott DR, Stevenson DJ. Magma solutions. Geophys Res Lett 1984;11:1161–4.[5] Dullien FAL. Porous media fluid transport and pure structure. New York: Academic Press; 1979.[6] Atherton RW, Homsy GM. On the existence and formulation of variational principles for nonlinear differential equations. Studies Appl Math

1975;LIV(1):31.[7] Khamitova R. Self-adjointness and quasi-self-adjointness of an equation modelling melt migration through the Earth’s mantle. Nonlocal conservation

laws. Preprint in Archives of ALGA, vol. 5. BTH, Karlskrona, Sweden: Alga Publications; 2008. p. 143–58, ISSN: 1652-4934.[8] Khamitova R. In: Tenreiro Machado JA, Silva MF, Barbosa RS, Figueiredo LB, editors. Proceedings of the 2nd conference on nonlinear science and

complexity, July 28–31, 2008, Porto, Portugal, ISBN: 978-972-8688-56-1.[9] Barcilon V, Ritcher FM. Nonlinear waves in compacting media. J Fluid Mech 1986;164:429–48.[10] Harris SE. Conservation laws for a nonlinear wave equation. Nonlinearity 1996;9:187–208.[11] Maluleke GH, Mason DP. Derivation of conservation laws for a nonlinear wave equation modelling melt migration using Lie point symmetry

generators. Commun Nonlinear Sci Numer Simulat 2007;12(4):423–33.


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126. Andreas Jansson, 2007. Collective Action Among Shareholder Activists (doktorsav-handling). ISBN: 978-91-7636-573-1.

127. Karl-Olof Lindahl, 2007. On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point (doktorsavhandling) ISBN : 978-91-7636-574-8.

128. Annette Årheim, 2007. När realismen blir orealistisk. Litteraturens ”sanna historier” och unga läsares tolkningsstrategier (doktorsavhandling). ISBN: 978-91-7636-571-7.

129. Marcela Ramírez-Pasillas, 2007. Global spaces for local entrepreneurship: Stret-ching clusters through networks and international trade fairs (doktorsavhandling). ISBN: 978-91-7636-577-9.

130. Daniel Ericsson, Pernilla Nilsson, Marja Soila-Wadman (red.), 2007. Tankelyft och bärkraft: Strategisk utveckling inom Polisen. ISBN: 978-91-7636-580-9.

131. Jan Ekberg (red.), Sveriges mottagning av flyktingar – några exempel. Årsbok 2007 från forskningsprofilen Arbetsmarknad, Migration och Etniska relationer (AMER) vid Växjö universitet. ISBN: 978-91-7636-581-6.

132. Birgitta E. Gustafsson, 2008. Att sätta sig själv på spel. Om språk och motspråk i pe-dagogisk praktik (doktorsavhandling). ISBN: 978-91-7636-589-2.

133. Ulrica Hörberg, 2008. Att vårdas eller fostras. Det rättspsykiatriska vårdandet och traditionens grepp (doktorsavhandling). ISBN: 978-91-7636-590-8.

134. Mats Johansson, 2008. Klassformering och klasskonflikt i Södra och Norra Möre hä-rader 1929 – 1931 (licentiatavhandling). ISBN: 978-91-7636-591-5.

135. Djoko Setijono, 2008. The Development of Quality Management toward Customer Value Creation (doktorsavhandling). ISBN : 978-91-7636-592-2.

136. Elisabeth Björk Brämberg, 2008. Att vara invandrare och patient i Sverige. Ett indi-vidorienterat perspektiv (doktorsavhandling). ISBN: 978-91-7636-594-6.

137. Anne Harju, 2008. Barns vardag med knapp ekonomi. En studie om barns erfarenhe-ter och strategier (doktorsavhandling). ISBN: 978-91-7636-595-3.

138. Johan Sjödin, 2008. Strength and Moisture Aspects of Steel-Timber Dowel Joints in Glulam Structures. An Experimental and Numerical Study (doktorsavhandling). ISBN: 978-91-7636-596-0.

139. Inger von Schantz Lundgren, 2008. Det är enklare i teorin… Om skolutveckling i praktiken. En fallstudie av ett skolutvecklingsprojekt i en gymnasieskola (doktorsav-handling). ISBN: 978-91-7636-600-4.

140. Lena Nordgren, 2008. När kroppen sätter gränser – en studie om att leva med hjärt-svikt i medelåldern (doktorsavhandling). ISBN: 978-91-7636-593-9.

141. Mirka Kans, 2008. On the utilisation of information technology for the management of profitable maintenance (doktorsavhandling). ISBN : 978-91-7636-601-1.

143. Christer Fritzell (red.), 2008. Att tolka pedagogikens språk – perspektiv och diskur-ser. ISBN: 978-91-7636-603-5.

144. Ernesto Abalo, Martin Danielsson, 2008. Digitalisering och social exklusion. Om medborgares användning av och attityder till Arbetsförmedlingens digitala tjänster. ISBN: 978-91-7636-608-0.

145. Patrik Wahlberg, 2008. On time-frequency analysis and pseudo-differential opera-tors for vector-valued functions (doktorsavhandling). ISBN: 978-91-7636-612-7.

146. Morgan Ericsson, 2008. Composition and Optimization (doktorsavhandling). ISBN: 978-91-7636-613-4.

147. Jesper Johansson, 2008. ”Så gör vi inte här i Sverige. Vi brukar göra så här.” Retorik och praktik i LO:s invandrarpolitik 1945-1981 (doktorsavhandling). ISBN: 978-91-7636-614-1.

148. Monika Hjeds Löfmark, 2008. Essays on transition (doktorsavhandling). ISBN: 978-91-7636-617-2.

149. Bengt Johannisson, Ewa Gunnarsson, Torbjörn Stjernberg (red.), 2008. Gemensamt kunskapande – den interaktiva forskningens praktik. ISBN: 978-91-7636-621-9.

150. Sara Hultqvist, 2008. Om brukardelaktighet i välfärdssystemen – en kunskapsöver-sikt. ISBN: 978-91-7636-623-3.

151. Jaime Campos Jeria, ICT tools for e-maintenance (doktorsavhandling). ISBN: 978-91-7636-624-0.

152. Johan Hall, Transition-Based Natural Language Parsing with Dependency and Con-stituency Representations (doktorsavhandling). ISBN: 978-91-7636-625-7.

153. Maria Fohlin, L’adverbe dérivé modifieur de l’adjectif. Étude comparée du français et du suédois (doktorsavhandling). ISBN: 978-91-7636-626-4.

154. Tapio Salonen, Ernesto Abalo, Martin Danielsson, 2008. Myndighet frågar medbor-gare. Brukarundersökningar I offentlig verksamhet. ISBN: 978-91-7636-628-8.

155. Ann-Christin Torpsten, 2008. Erbjudet och upplevt lärande i mötet med svenska som andraspråk och svensk skola (doktorsavhandling). ISBN: 978-91-7636-629-5.

156. Guillaume Adenier, 2008. Local Realist Approach and Numerical simulations of Nonclassical Experiments in Quantum Mechanics (doktorsavhandling). ISBN: 978-91-7636-630-8.

157. Jimmy Johansson, 2008. Mechanical processing for improved products made from Swedish hardwood (doktorsavhandling). ISBN: 978-91-7636-631-8.

158. Annelie Johansson Sundler, 2008. Mitt hjärta, mitt liv: Kvinnors osäkra resa mot häl-sa efter en hjärtinfarkt (doktorsavhandling). ISBN: 978-91-7636-633-2.

159. Attila Lajos, 2008. På rätt sida om järnridån? Ungerska lantarbetare i Sverige 1947-1949. ISBN: 978-91-7636-634-9.

160. Mikael Ohlson, 2008. Essays on Immigrants and Institutional Change in Sweden (doktorsavhandling). ISBN: 978-91-7636-635-6

161. Karin Jonnergård, Elin K. Funck, Maria Wolmesjö (red.), 2008. När den professio-nella autonomin blir ett problem. ISBN: 978-91-7636-636-3

162. Christine Tidåsen, 2008. Att ta över pappas bolag. En studie av affärsförbindelser som triadtransformationer under generationsskiften i familjeföretag (doktorsavhand-ling). ISBN: 878-91-7636-637-0

163. Jonas Söderberg, 2009. Essays on the Scandinavian Stock Market (doktorsavhand-ling). ISBN: 978-91-7636-638-7

164. Svante Lundberg, Ellinor Platzer (red.), 2008. Efterfrågad arbetskraft? Årsbok 2007 från forskningsprofilen Arbetsmarknad, Migration och Etniska relationer (AMER) vid Växjö universitet. ISBN: 978-91-7636-639-4

165. Katarina H. Thorén, 2008 “Activation Policy in Action”: A Street-Level Study of Social Assistance in the Swedish Welfare State. ISBN: 978-91-7636-641-7

166. Lennart Karlsson, 2009. Arbetarrörelsen, Folkets Hus och offentligheten i Bromölla 1905-1960 (doktorsavhandling). ISBN: 978-91-7636-645-5.

167. Anders Ingwald, 2009. Technologies for better utilisation of production process re-sources (doktorsavhandling) ISBN: 978-91-7636-646-2.

168. Martin Estvall, 2009. Sjöfart på stormigt hav – Sjömannen och Svensk Sjöfarts Tid-ning inför den nazistiska utmaningen 1932-1945 (doktorsavhandling). ISBN: 978-91-7636-647-9.

169. Cecilia Axelsson, 2009. En Meningsfull Historia? Didaktiska perspektiv på historie-förmedlande museiutställningar om migration och kulturmöten (doktorsavhandling). ISBN: 978-91-7636-648-6.

170. Raisa Khamitova, 2009. Symmetries and conservation laws (doktorsavhandling). ISBN: 978-91-7636-650-9.

Växjö University Press S-351 95 Växjö SWEDEN www.vxu.se [email protected]

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