1

Conservation Laws for Fifth – order Nonlinear

Evolution Equations

By

Tha'er Jamil Hammad Abu Khammash

Supervisor

Dr. Gharib Mousa

This Thesis was Submitted in Partial Fulfillment of the Requirements

for the Master’s Degree of Science in Mathematics

Faculty of Graduate Studies

Zarqa University

Zarqa – Jordan

2-11-2015

2

COMMITTEE DECISION

This Thesis / Dissertation (Conservation Laws for Fifth – order

Nonlinear Evolution Equations) was successfully defended and approved

on ________________

Signature Examination Committee

______________) Supervisor(Gharib Mousa Dr.

Assoc. Prof. of Mathematics

(Member) ______________A'babneh Osama Dr.

Assoc. Prof. of Mathematics

(Member) ______________Khaled Jaber Dr.

Assoc. Prof. of Mathematics

(Member) ______________ Mousa AbushaarDr.

Assoc. Prof. of Mathematics

ACKNOWLEDGEMENT

3

I am truly thankful to Allah for His ultimate blessings and guidance at each step of this

work. Peace and prayer for the bearer of the torch of wisdom Muhammad (peace be

upon him) (P B U H).

I offer my sincere thanks to my supervisor Dr. Gharib Mousa for his motivating

guidance, beneficial suggestions and support during my work.

I express my thanks to my father, who remains desirous for my success; who always

prays to see me flourishing and progressing. I am also thankful to my sisters and

brothers for their prayer and true wishes.

My most sincere thanks are due to Dr.Osama A'babneh and Dr .Khaled Jaber and

Dr.Mousa Abushaar for reviewing this work and for their interesting remarks,

comments and suggestions.

4

CONTENTS

COMMITTEE DECISION ii

ACKNOWLEDGEMENT iii

CONTENTS iv

v ABSTRACT

NTRODUCTION 1

Chapter 1 : PRELIMINARIES

1.1 Pseudo-spherical Surfaces 3

1.2 The AKNS system and the backland transformations 6

1.3: Conservation law 10

Chapter 2: Conservation Laws for some Nonlinear Evolution Equations which

Describe Pseudo-Spherical Surfaces

2.1. Introduction 12

2.2: Inverse scattering problem and DEs which describe pss 18

2.3. Infinite number of conservation Laws for some NLEEs 24

2.4. Conservation laws for some NLEEs which describe pss 35

2.5. Conclusion 45

Chapter 3: Conservation Laws for Fifth – order Nonlinear Evolution Equations

3.1. Introduction 46

3.2: conservation Laws by using Riccati equations 51

3.3: Conclusions 57

3.4 References: 59

5

ABSTRACT

In this thesis we implemented the inverse scattering problem (ISP)

and conservation laws for some nonlinear evolution equations

(NLEEs) which describe pseudo spherical surfaces (pss). The thesis

consists of an introduction and there chapters, together with Arabic

and English summaries and is organized as follows:

Introduction: The introduction includes a short historical discussion of the early

geometers' ideas on "integrable geometric constructions" followed by a

quick hint for the importance of conservation laws

Chapter 1: This chapter explains the basis of pss and conservation laws for some

NLEEs, the general basic consideration for the models considered in this

thesis together with the necessary preliminaries. In this chapter, we

obtained all local conservation laws for some NLPDEs. The conservation

Laws does not depend on the system having a Lagrangian formulation, in

contrast to Noerther's theorem, which requires a Lagrangian.

Chapter 2:

In this chapter, Ageneralized inverse scattering method (ISM) and the

fundamental equations of pss are given by extending the results of Konno,

Wadati (1975) and Sasaki (1979) respectively. An infinite number of

conserved quantities are also obtained by solving a set of coupled Riccati

equations. We obtained the inverse scattering method.

Chapter 3: In this chapter, we find the conservation laws for Fifth – Order Nonlinear Evolution Equations that describe pss, based on a geometrical property of

these surfaces.

6

الخامسةقوانين البقاء لممعادالت المتطورة الغير خطية من الدرجة Conservation Laws for Fifth – order Nonlinear Evolution

Equations

إعداد )ثائر جميل حماد ابوخماش(

(70093002)الجامعي الرقم المشرف الرئيس

)د.غريب موسى غريب (

مشروع خطة رسالة ماجستير قدمت استكماال لمتطمبات الحصول عمى درجة الماجستير في )الرياضيات(

مية الدراسات العمياك جامعة الزرقاء

األردن -الزرقاء

7-00-7002الفصل الدراسي األول /

7

INTRODUCTION

This introduction, gives a brief survey of how mathematicians and

physicists noticed and began to work on certain problems of mutual

interest. The theory of integrable systems has been an active area of

mathematics for the past thirty years. Different aspects of the subject have

fundamental relations with mechanics and dynamics, applied mathematics,

algebraic structures, theoretical physics, analysis including spectral theory

and geometry. Most differential geometers have some information and

experience with finite dimensional integrable systems as they appear in

sympectic geometry (mechanics) or Ordinary Differential Equations

(ODEs), although the reformulation of part of this theory as algebraic

geometry is not commonly known [1-7]. There are two quite separate

methods of extension of these ideas to Partial Differential Equations

(PDEs); one based on algebraic constructions and one based on spectral

theory and analysis. These are less familiar still to geometers.

This introduction contains a short historical discussion of the early

geometers' ideas on "integrable geometric constructions" followed by a

short hint for the importance of the conservation laws and a brief survey of

pss. Conservation laws are central to the analysis of physical field

equations by providing conserved quantities, such as energy, momentum,

and angular momentum [8-11]. For a given field equation, local

conservation laws are well-known to arise through multipliers, analogous

to integrating factors of ODEs, with the product of the multiplier and the

field equation being a total divergence expression. Such divergences

correspond to a conserved current vector for solutions of the field equation

whenever the multiplier is non-singular. If a field equation possesses a

Lagrangian, Noether's theorem shows that the multipliers for local

conservation laws consist of symmetries of the field equation such as the

8

action principle which is invariant (within a boundary term). Moreover,

the variational relation between the Lagrangian and the field equation

yields an explicit formula for the resulting conserved current vector. This

characterization of multipliers for a Lagrangian field equation has a

generalization to any field equation by means of adjoint-symmetries,

whether or not a Lagrangian formulation exists [3-9].

9

Chapter 1

PRELIMINARIES

In this chapter, the basic definitions and background material, are

necessary to deal with in the following chapters

1.1 Pseudo-Spherical Surfaces:

The explicit study of surfaces of constant negative total curvature goes

back to the work of minding [1-3] in 1838. Thus in that year, Minding's

Theorem established the important result that these surfaces are isometric,

that is, points on two such surfaces can be placed in one-to-one

correspondence in a way that, the metric is preserved. Beltrami [4-7]

subsequently gave the term Pseudo Spherical to these surfaces and made

important connections with Lobachevski's Non- Euclidean geometry. It

was Bour [8], in 1862, who seems to have first set down what is now

termed the sine-Gordon equation arising out of the compatibility

conditions for the Gauss equations for Pseudo-Spherical Surfaces (pss)

expressed in asymptotic coordinates. In 1879, Bianchi in his habilitations

thesis presented in mathematical terms, a geometric construction for pss.

Thus the result was extended by backend in 1883 to incorporate a key

parameter that allows the iterative construction of such pss. The BT was

subsequently shown By Darboux (1882), in 1885, to be associated with an

elegant invariance of the Sine- Gordon equations.

11

This invariance has become known as the BT for the Sine-Gordan

equations. It includes an earlier parameter-independent result of Darboux

(1889). The BT has important applications in soliton theory, indeed. It

appears that the property of invariance under backlund and associated

Darboux transformations as originated in (1897) are enjoyed by all soliton

equations. The contribution of Bianchi and Darboux to the geometry of

surfaces and, in particular, the role of BTs preserving certain geometric

properties have been discussed by Sym et al. (1984), Chern (1985) and

Sym in [9-11].

We consider the following definition.

Let denote the positon vector of a generic point P on a

surface M2 in R

3. Then, the vectors and are tangential to M

2 at P and,

at such points at which they are linearly independent,

(1.1)

| |

Determines the unit normal to M2 .The 1

st and 2

nd fundamental forms of

M2 are defined by

Where

11

The Gaussian (total) curvature K defended by

(1.4)

If the Gaussian curvature of M2is negative, that is, if M

2 is hyperbolic

surface, then the asymptotic lines on M2

may be taken as parametric

curves. Then and the Gaussian curvature (1.4) reduces to,

(1.5)

In the particular case when =

<0 is constant, M

2 is termed a

pseudo- spherical surface.

Definition (1.1)

An evolution equation is a PDE for an unknown function of

the form

(1.6)

Where is an expression involving only u and its derivatives with

respect to , if this expression is nonlinear, equation (1.6) is called a

Nonlinear Evolution Equation (NLEE) [12-15].

12

Definition (1.2)

A scalar differential equation

in two independent variables is of pseudo-spherical type (or, it is said

to describe pss) if there exists one forms ≠0

= ( ) ( ) (1.7)

whose coefficients are differentiable functions, such that the one-forms

satisfy the structure equations

(1.8)

Whenever is a solution of (1.6).

1.2 The Abowitz-Kaup-Newell-Segur (AKNS) system and the

backland transformations (BTs):

The Backland transformation (BT) technique is one of the direct

techniques for generating a new solution of a NLEE from (a known

solution of that equation) see, for example, [13-17]. Konno and Wadati

(1975), for example, had derived some BTs for NLEEs of the Abowitz-

Kaup-Newell-Segur (AKNS) class [18]. These BTs explicitly express the

new solutions in terms of the known solutions of the associated AKNS

system [19]. The AKNS system is a liner eigenvalue problem in the form

of a system of first-order PDEs. Therefore, the problem of obtaining new

solutions by BT is equivalent to obtaining the wave function [11-13]. It is

known that many NLEEs can be derived from the AKNS system [7-9].

13

(

)

Where denotes exterior differentiation, is a column vector and

the 2 × 2 matrix is traceless

(

)

Then

(

)

from equations (1.9) and (1.11), we obtain

where and are two 2 × 2 null-trace matrices

( )

(

)

Here is a parameter, independent of and , while and are functions

of and .

Now

14

which requires the vanishing of the two forms

Θ

or in acomponent form

or

where

15

By suitably choosing ), we will obtain

various NLEEs which q must satisfy. Kannon and Wadati (1975)

introduced the following:

Γ

This function first appeared used and explained in the geometric context of

pseudo spherical equations in [11, 13], and see also the classical papers by

Sasaki[63], and Chern–Tenenblat[19]. Then Eq. (1.12) is reduced to the

Riccati equations:

Soliton solutions to equations like the Korteweg-de Vries (KdV) equations

[14]. Using a BT for a NLPDE, one obtains a new solution to the equation

from a known one. For example, the system:

,

2sin2)(

uuuu x

(1.22)

(1.23) ,2

sin2

)(uu

tuu

where ≠0 is an arbitrary constant, defines a BT uu of the Sine-

Gordon equation:

16

In fact, if u is a solution to the Sine-Gordon equation, then the system

above is integrable and yields a new solution u to the Sine-Gordon

equation. In particular, from the trivial solution we obtain the so-

called one soliton solution [48]

(1.25) ).

1exp(arctan4),( txtxu

Where is also an arbitrary constant and repeated application of this

procedure yield the so-called multiple soliton solutions.

1.3 Conservation law

Definition (1.3)

A conservation law associated to differential equation in two dimensions

is an expression of the form:

,0)(

u

xt

(1.26)

which is satisfied for all solutions of (1.6), where the conserved

density, and )( u the associated flux, in general are functions of ,

and the partial derivatives of (with respect to ). denotes the total

derivatives with respect to the total derivative with respect to .

If is a polynomial in derivatives is exclusively, then is

17

called a local polynomial conserved density. If )( u is also such a

polynomial, then (1.26) is called a local polynomial conservation law.

Example 1: The most famous scalar evolution equation from soliton

theory, the Korteweg-de Vries (KdV) equation,

(1.27) , 06 xxxxt

uuuuuF

is known to have infinitely many polynomial conservation laws. The first

three polynomial conservation laws are given by:

0)3()( 2 xxxt

uuu ,

)28.1(,0)2

2

1()

2

1( 322 xxxxt uuuuu

.0)6

12

2

3

3

1()

6

1

3(

224223

xxxxxxxxxxtx uuuuuuuuuu

Example 2: the Sine – Gordon equation

)29.1( ,0sin uuuF xt

is known to have infinitely many polynomial conservation laws. The first

three polynomial conservation laws are given by:

.0)cos()4

1(

)30.1(,0)1(cos)2

1(

,0)2

1()cos1(

224

2

2

xxtxxx

xtx

xtt

uuuu

uu

uu

The first two express conservation of momentum and energy,

respectively, and are relatively easy to compute by hand. The third one,

which is less obvious, requires more work.

18

Chapter 2

Conservation Laws for some Nonlinear Evolution Equations

which Describe Pseudo-Spherical Surfaces.

2.1. Introduction

In 1979 Sasaki observed that a class of NLPDEs, such as KdV, mKdV and

Sine- Gordon (SG) equations which can be solved by the AKNS 2 × 2

inverse scattering method (ISM) [14-18], was related to pss. The

geometric notion of a differential equation. For a real function, which

describes a pss was actually introduced in the literature by Chern and

Tenenblat in (1986), where equations of type

were studied systematically. Later, in [19], this concept was applied to

other types of DEs. A generic solution of such an equation provides a

metric defined on an open subset in R2, for which the Gaussian curvature

is -1. Such a DE is characterized as being the integrability condition of a

linear problem of the form:

(

(

⁄

⁄

)

)

19

where is aparameter, is a 2 × 2 traceless matrix and A is a 2 × 2 of

diagonal matrix depending on , u and its derivatives. Examine if this

class of equations are (real) equations of the AKNS type. Other examples,

which are not AKNS, can be found in [20]. Geometric interpretation of

special properties (such as infinite number of conservation laws and BTs)

for solutions DEs which describes pss have been systematically exploited

in [21]. In 1995, Kamran and Tenenbalt extending the results of Chern and

Tenenbalt (1986), gave a complete classification of the evolution

equations (EEs) of type which describe pss by

considering equations which are the integrability condition of a linear

problem of the form given above. Moreover, they proved that there exists,

under a technical assumption, a smooth mapping transforming any generic

solution of such equations into a solution of the other. This geometric

notion of scalar DEs was also generalized to DEs of the type

by Reyes recently in [22].

Let g be a Riemannian metric on M2, the corresponding Levi-Civita

connection on the tangent bundle T M2, e1,e2 be a moving orthonormal

frame on some open domain U ⊂ M2 and a corresponding

moving coframe. The relations, je define the connection

one-form matrix with respect to the frame e1, e2.

21

The orthogonality of this frame implies that ).(,0 3

1

2

2

1

2

2

1

1

Hence the Levi-Civita connection one-form on the tangent bundle T M2

with respect to the moving frame e1,e2 is :

(

)

It yields the following structural equations:

(2.1) ,312 d 231 d

The Gaussian curvature K of the space M2 is defined by the Gauss

equations:

(2.2) .123 kd

Sasaki (1979) gave a formula some local connection on a two-dimensional

real Riemannian manifold M2[Riemannian manifold: is a real smooth manifold M

equipped with an inner product g on the tangent plane T.],which is quite relevant

in the theory of nonlinear integral partial differential equations:

(2.3) =

,

231

312

as a new connection for some (non-specified) bundle over M2. The key

property of the matrix one-form Ω is that it satisfies the curvature

condition ,0 d

21

If on U. In some older (for example [23]) and many subsequent

papers (some of the most recent are [24] different matrix one-forms were

discussed, depending on function (or some functions) of some

independent variables, such that the curvature condition =0 for this form

is equivalent to one of well-known NLPDEs having an infinite number of

conservation laws and symmetry groups. The generalizations to higher

dimensions are given in [25].

The condition =0 depends only on relation (2.1) and the commulative

relations in the algebra SL (2, R).

The one-form Ω may be written as [26]:

,

0

0

0

21

23

13

which contains the Levi-Civita connection form

,

03

3

o

as a direct summand and avoids the surprising factor 1/2 in (2.3).

As a consequence, each solution of the DE provides a metric on M2,

whose Gaussain curvature is constant, equal to . Moreover, the above

22

definition is equivalent to saying that the DE for u is the integrability

condition for the problem:

)4.2(,,

3

2

1

v

v

v

vvdv

where denotes exterior differentiation, v a vector and the 3×3 matrix Ω

) is traceless

(2.5)

and consists of a one-paramter ( the eigenvalue) family of one-forms in

the independent variables the dependent variable and its

derivatives. Integrability of equations (2.4) requires:

,)(0 2 vddvvdvd

or the vanishing of the two-form

(2.6)

which corresponds, by construction, to the original NLEE to be solved.

Equations (2.4) correspond to 3 equations and only selected solutions are

possible, i.e. those satisfying (2.6). This was of course, equally true in that

Sasaki formulation.

,0 d

23

The main aim of this chapter is to extend somewhat the inverse scattering

problem in the reference [18] by considering v as a three component

vector and Ω as a traceless 3×3 matrix one-form. An application of the

original Chern and Tenenblat construction of conservation laws is given

for the nonlinear Schrödinger equation (NLSE) [7-11], Liouville's

equation [12] a mKdV equation, the(-)[Beals, Rabelo, Tenenblat](BRT)

equations[45], the Ibragimov-Shabat (IS) equation, the Camassa-Holm

(CH) equation and Hunter-Sexton (HS) equation, which are very useful in

several areas of physics as may be seen from the numerous references.

The chapter is organized as follows. In Section 2.2, we introduce the

inverse scattering problem and apply the geometrical method to several

PDEs which describe pss. In Section 2.3, we obtain an infinite number of

conserved densities for some NLEEs which describe pss using a theorem

of Cavalcante and Tenenblat (1988). In Section 2.4, we obtain

conservation laws by extending the classical discussion of Wadati,

Sanuki and Konno (1975). Section 2.5 contains the conclusion.

24

2.2 Inverse scattering problem and DEs which describe pss

Some NLPDEs invariant to translations in , can be solved exactly

by an ISM with the set of linear equations [19- 26]

)7.2(,,, 21113311323112131 vvfvvvfvvfvfv xxx

)8.2(.,, 222121332212323212231 vfvfvvfvfvvfvfvttt

The functions , depend on and its

derivatives. They can also be functions of the parameters . We have

restricted ourselves to the case where is a parameters. The

compatibility conditions for equations (2.7) and (2.8), obtained by cross

differentiation, are:

(2.10)

(2.11)

In order, to solve (2.9)-(2.11) for in general, one finds that another

condition still has to be satisfied. This latter condition is the Evolution

Equation (EE). In terms of exterior differential forms, the inverse

scattering problem can be formulated as follows: equations (2.7) and (2.8)

can be rewritten in matrix form by using equations (2.4), and Ω is a

traceless 3×3 matrix of one-forms given by

,31123211,22 fffff x

.122211,31,32 fffff tx

)12.2(

0

0

0

222111

223231

21113231

dtfdxdtfdxf

dtfdxdtfdxf

dtfdxfdtfdxf

25

In this scheme (2.6), yielding the EE, must be satisfied for the existence of

a solution . Whenever the functions are

real, Sasaki (1979) gave a geometrical interpretation for the problem.

Consider the one-forms defined by

.,, 323132221212111 dtfdxfdtfdxfdtfdxf

We say that a DE for describes a pss if it is a necessary and

sufficient condition for the existence of functions

, depending on and its derivatives, ,such that the

one-forms in equations (2.2), satisfy the structure equations (2.3) of a pss.

It follows from this definition that for each nontrivial solution u of the DE,

one gets a metric defined on M2, whose Gaussian curvature is .

It has been known, for a long time, that the SG equation describes a pss.

More recently other equations, such as KdV and MKdV equations, were

also shown to describe such surfaces [26-29]. Here we show that other

equations such as the NLSE, Liouvilles equations ,the mKdV equation, the

(-) and (+) BRT equations proved to be of great important in many

physical applications [30]. The procedure is clarified in the following

examples:

Let M2 be a differentiable surface, parametrized by coordinates

(a) NLSE

Consider

,]()([)(2

1 ***

1 dtuuiuuidxuu xx

)13.2(,]2[ *2

2 dtiuuidx

.](*)([*)(2

1 *

3 dtuuiuuidxuu xx

26

M2 is a pss iff satisfies the NLSE

(b) Liouville's equation

Consider

(2.15)

Then M2 is a pss iff satisfies Liouville's equations

(c) The (-) BRT equations

Consider

,1 dtg

(2.17) ,)(2

2 dtg

dx

.)(3 dtugdxu xx

.3

,2

,1

dt

ue

dt

ue

dx

dxx

u

27

Then M2 is a pss iff satisfies the (-) BRT equation

xxt uugu ]))(([ ),(ug

,,where g(u) is a differentiable function of u which satisfies

, gg

and are real constants, such that .22

d) The (+) BRT equation )

Consider

dtugdu xxx )(1

,)(2

2 dtg

dx

.3 dtg

Then M2 is a pss iff satisfies the (+) BRT equation

xxt uugu ]))(([ )(ug

where g(u) is a differentiable function of u which satisfies

andwheregg ,,,

are real constants, such that

.22

(e)The mKdV equation

Consider

,3

21 dtux

)21.2(,)

3(

23

2 dtau

dx

28

.)3

(3

2

3

2 32

3 dtauuu

uudx xx

Then M2 is a pss iff satisfies the mKdV equation

)22.2(,)( 2

xxxxt uuauu

where is a constant.

(f) The IS equation

Consider

(

) (

)

(2.23)

.)4( 4

3 dtuuuu

udx x

xx

Then M2 is a pss iff satisfies IS equation

(2.24).393 422

xxxxxxxt uuuuuuuu

g) A CH equation)

Consider

,)1()( 22

2

2

21 dtuuu

uuu

dxuu xx

xx

xx

)( 1

2 xuudx

.)()1( 22

2223 dtuu

uuuuu

dxuu xx

xx

xx

Then M2 is a pss iff satisfies the CH equation

,)4( 4

2 dtuuuu

udx x

xx

29

in which the parameters and are constrained by the relation

(h) A HS equation

Consider

,)11

()(21 dtuuu

uudxu xx

xx

xx

(

)

.)1

()1(23 dtuuu

uudxu xx

xx

xx

Then M2 is a pss iff satisfies the HS equation

in which the parameters and are constrained by the relation

122

31

2.3. Infinite number of conservation laws for some NLEEs

In this section we apply the Cavalcante and Tenenblat method to obtain an

infinite number of conserved densities for some NLEEs. To each equation

we associate functions satisfying the following theorem [31-35]:

Theorem (2.1) Let , be differentiable

functions of such that

then the following statements are valid:

i) The following system is completely integrable for

,cossin 211131 fffx (2.32)

.cossin 221232 ffft

(ii) For solution of (2.32)

( ) ( )

is a closed one-form.

(iii) If are analytic functions of a parameter at the origin then the

solutions ),,( tx of (2.32) and the one-form ω are also analytic in at the

origin.

Proof: i) follows from Frobenius theorem. In fact, a straightforward

computation shows that (2.31) implies that xttx ·

31

ii) is proved by showing that (2.31) and (2.32) imply that the

differential of is zero.

iii) Suppose are analytic functions of the parameter . Each equation

of the system (2.32) can be considered as an ordinary differential equation.

It follows from

a theorem for ODEs ([43], p.36) on the dependence of solutions on

parameters, that ),,( tx is an analytic function of , for in an

appropriate neighborhood of zero.

Cavalcante and Tenenblat supposed [44],

.),(),,(0

k

k

k

ijij txftxf

(2.34)

Then the solution of (2.32) is of the form

.),(0

j

j

j tx

(2.35)

They consider the following functions of , for fixed :

(2.36)

).),(sinsin)(0

j

j

j tiB

It follows form (2.36) that

A(0)=cos 0, B(0)= sin 0,

,)0()!1()0(1

0

jk

k

ii

i

k

k

d

Bidkk

d

Ad

(2.37)

).),(coscos)(0

j

j

j tiA

32

,)0()!1()0(1

0

jk

k

ii

i

k

k

d

Aidkk

d

Bd

For k ≥ 1. Finally, they define the functions of :

),0()0( 2 ij

iji

kij

i

ik

ji

kd

Bdf

d

iAdjfH

(2.38)

),0()0( 21 ij

i

kij

iji

k

ij

kd

iAdjf

d

BdfL

,!

1

! 1

01

1

3

rn

k

n

r

rn

r

k

n

r

n

knk Lrn

Hr

rnfF

where are non-negative integers such that

The functions

and

defined above depend on ;,,, 110 j

the functions F1k and Fnk depend on 0 and 110 ,,, n , respectively.

Corollary (2.2): Let t, ), be

differentiable functions of , analytic at 0 , that satisfy (2.31). Then,

with the above notation, the following statements hold.

(a) The solution of (2.32) is analytic at 0 , then 0 is determined by

)39.2(,, 00

2

0

32,0

00

1

0

31,0 LfLf tx

and, for j are recursively determined by the system

., 2

00

2,1

00

1, Jjtjjjxj FHFH (2.40)

33

(b) For any such solution and integer j ≥ 1,

),()!(

121

0

dtHdxHij

ijijj

i

j

(2.41)

is a closed one-form.

Now, we consider NLPDEs for which describes a pss. There exist

functions , which depend on and

its derivatives such that, for any solution of the EE, fij satisfy (2.31).

Then it follows form theorem (2.1) that (2.32) is completely integrable for

. Suppose to be analytic functions of a parameter , then the

solutions of (2.32) and the one form ω, given by (2.33), are analytic

in .Their coefficients j and j , as functions if ,are determined by

(2.39) - (2.41). Therefore, the closed one-forms j provide a sequence of

conservation laws for the PDE, with conserved density and flux given

respectively by:

,)!(

11

0

ijj

i

j Hij

D

(2.42)

0j,)!(

12

0

ijj

i

j Hij

F .

We consider the following examples:

34

(a) NLSE

For equation (2.14) we consider the following functions of defined

by

],()([),(2

1 **

12

*

11 xx uuiuuifuuf

],2[, *2

2221 iuuiff

)].()([),(2

1 **

32

*

31 xx uuiuuinfuuf

As corresponding to equation (2.13). For any solution u of equation (2.14),

the above functions satisfy (2.31). Applying the corollary, we have a

sequence of functions j determined by (2.36) and (2.37). It follows from

(2.43) that (2.39) reduces to:

,sin)(2

1)(

2

10

**

,0 uuuux

,cos2sin)()( 0

*

0

**

,0 iuuuuiuui xxxxt (2.44)

and (2.37) we obtain recursively

1),1( 1 jdxeFe s

j

s

j (2.45)

where

S= dxuu ocos)(2

1 *

and

.)0(2

)(

!

1)0(

)!1(

1 *1

11

1

1 iji

ij

ij

j

jd

Aduu

i

ij

jd

Ad

jF

35

The sequence of conserved densities for NLSE is given by:

,cos)(2

1 *

ouu

1),0()!1(

1)0(

2

)(

!

11

1*

jd

Bd

jd

Aduu

j j

j

j

j

(2.46)

solving the integrable system of equation (2.44), then from 0 we obtain

j , 1j defined by (2.45).

(b) Liouvilles equation

For equation (2.16) we consider the functions defined by

,0, 1211 fufix

,, 2221

ue

ff

.0 3231

ueff

For any solution of equation (2.16), the above functions satisfy (2.31).

As in the preceding example, we obtain a sequence of conserved densities

for (2.16) by using theorem (2.1). Substituting (2.47) into (2.32), we

obtain the system of equations

.cossin xx u

).cos1(

u

t

e

This is completely integrable whenever is a solution of equation (2.16).

From the first equation of (2.48) one concludes that is analytic with

respect to . Therefore consider:

36

,),(0

j

j

j tx

(2.49)

Equation (2.48) reduces to:

,....,5,3,0,sin, nnoouxxo

)50.2(,1),1( 1

jdxeFe h

j

h

j

where

,1),( 11 Ftgudxuh xx

and

.2,)0(!

1)0(

)!1(

1 1

11

1

1

jd

Adu

i

ij

jd

Ad

jF iji

i

x

j

ij

j

j

Using (2.37) in the above expression, we obtain j in terms of . we

display only the first terms of the series:

,....,5,3,0,0 nn

)51.2(),1(1 dxee hh

,2

he

etc

37

The sequence of conserved densities for Liouvilles equations is given by

0cosxu

)52.2(.1),0()!1(

1)0(

! 1

1

jd

Bd

jdj

Aduj

j

j

j

x

Using the expressions in equation (2.37) and the functions j given by

(2.50), we obtain the first conserved densities

,xu

.

)53.2(),22(22

etc

uteuueu u

ttt

u

x

(c) The (-) BRT equation

For any solution of the (-) BRT equation (2.18), the functions:

,,0 1211 gff

)54.2(,,2

2221

gff

,)(, 3231 xx ugfuf

satisfy (2.31), by applying the corollary (2.2),we obtain j

defended by

),(0 thudxux

)55.2(.1,)0()!1(

11

1

jdxd

Ad

j j

j

j

Using (2.37) in the above expressions we obtain j . The first terms are:

38

),(thuo

)56.2(,cos1 dxo

,sin12 dxo

etc.

The conserved densities are given by:

)57.2(.1),0(1

1

jd

Bdj

j

Using (2.37), we obtains the first terms

,sin 0

)58.2(,cos 01

,sincos2 0

2

102

,etc

where the j are given by (2.55).

39

(d) The (+) BRT equation

For equations (2.20) we consider the functions of defined by:

,)(12,11 xx ugfuf

)59.2(,,2

2221

gff

.,0 3231 gff

For any solutions of equations (2.20), the above functions fij satisfy

(2.31). Applying the corollary (2.2), we have a sequence of functions j

determined by (2.36) and (2.38). It's following from (2.59) that reduces to

,sin 0,0 xx u

)60.2(,0cos)( 0

2 gg

and from (2.37) we obtain reclusively

)61.2(,1),1( 1

jdxeFe h

j

h

j

where

dxuh x 0cos

and

.)0(!

1)0(

)!1(

1 1

11

1

1 iji

ij

i

xj

j

jd

Adu

i

ij

jd

Ad

jF

The sequence of conserved densities for the (+) BRT equation is given by

41

,cos 0 xu

)62.2(.1),0()!1(

1)0(

! 1

1

jd

Bd

jd

Ad

j

uj

j

j

j

x

Solving the integrable system of equation (2.60), from 0 we obtain j ,

defended by (2.61).

(e) The mKdV equation

For any solution of mKdV equation (2.22), the functions

,3

2,0 1211 xuff

)63.2(),3

(,2

3

2221

au

ff

),3

(3

2,

3

2 32

3231 auuu

ufuf xx

satisfy (2.31). Applying the corollary (2.2), we obtain j , 0j defended

by

,3

20 udx

)64.2(.1,)0()!1(

11

1

jdxd

Ad

j j

j

j

Using (2.37) in the above expressions we obtain j. The first terms are

,3

20 udx

41

,cos 01 dx

,sin 02 dx

The conserved densities are given by

.1),0(1

1

jd

Bdj

j

Using (2.37), we obtain the first ones

,

,sincos2

)65.2(,cos

,sin

0

2

102

01

0

etc

where the j are given by (2.64).

42

2.4. Conservation laws for some NLEEs which describe pss

One of the most widely accepted definitions of integrability of PDEs

requires the existence of soltion solutions i.e., of a special kind of

traveling wave solutions that interact elastically, without changing their

shapes. The analytic construction of soliton solutions is based on the

general ISM. In the formulation of Zakharov and Shabat (1974), all known

integrable system supporting solutions can be realized as the integrablility

condition of a linear problem of the form [30-40]

)66.2(,, QvvPvv tx

where the matrices P and Q are two 2 × 2 null-trace matrices

)67.2(.,

2

2

AC

BA

Q

r

qP

Here is a parameter, independent of and . Thus an equation (2.66) is

kinematically integrable if it is equivalent to the curvature condition

)68.2(.0],[ QPQP tx

Konno and Wadati (1975) introduced the functions

Γ

and for each of the NLEE, derived a BT with the following from:

(Γ )

where is a new solution of the corresponding NLEE. As mentioned in

the previous sections, Sasaki (1979), Chern and Tenenblat (1986), and

Cavalcante and Tenenblat (1988) have given a geometrical method for

constructing conservation laws of equations describing pss. The formal

content of this method is contained in the following theorem, which may

43

be seen as generalizing the classical discussion on conservation laws

appearing in Wadati, Sanuki and Konno (1975).

Theorem (2.3):

Suppose that ( ) or more generally

is an EE describing pss. The systems:

)71.2(,)( 2

111 q

qDqrD x

x

)72.2(),()2

( 11

q

BADD xt

and

)73.2(,)( 2

222 r

rDqeD x

x

)74.2(),()2

( 22

r

CADD xt

in which xD and tD are the total derivative operator defined by :

,0

1

K k

kxu

ux

D

,)(,0 kK

K

xtu

fDt

D

are integrable on solutions of equation or

generally

Proof: The equation or more

generally , is the necessary and sufficient

44

condition for the integrability of the linear problem (2.66). Equivalently,

by (2.68), the functions satisfy the equations

Set ,2

1

v

v and define 1=

Γ , 2= Γ. Straightforward computations

using equation (2.75)-(2.77) allow one check that if

2

1

v

vv is a non-trivial

solution of the linear system 1, vdv is a solution of the system (2.71),

(2.72) and 2 is s solution of the system of equations (2.73) and (2.74).

This theorem (2.3) provides one with at least one -dependent

conservation law of the EE or

, to wit, equations (2.71), (2.72) or ((2.73),

(2.74)). One obtains a sequence of -independent conservations laws by

expanding 1 or 2 in inverse powers of [35-40].

)78.2(,1

)(

22

n

n

n

Consideration of equation (2.73) yields the recursion relations:

)79.2(,)1(

2 qr

)80.2(,1,)(

2

1

1

2

)(

2

)(

2

)1(

2

nDr

rD inn

i

in

x

nxn

which in turn, by replacing into (2.74), yields the sequence of

conservations laws of equations integrable by AKNS inverse scattering

45

found by Wadati, Sanuki and Konno (1975). This section ends with the

examples:

(a) NLSE

For equation (2.14) one considers the functions of defined by

).()(

)81.2(],[,2

1

,2

1,

2

1

**

*2

*

x

x

uiuiC

uiBiuuiA

uqur

Equation (2.73) becomes:

)82.2(.)(4

1 2

22*

**

2 u

uDuuD x

x

Assume that 2 can be expanded in a series of the from (2.78). Equation

(2.81) implies that 2 is determined by the recursion relation:

)84.2(,1,

)83.2(,4

1

)(

2

1

1

)(

2

)(

2

)(

2*

*

)1(

2

*)1(

2

nDu

uD

uu

inn

i

in

x

nxn

whenever is a solution of the NLSE. This recursion relation yields a

sequence of conserved densities given by the coefficients of the series in

)85.2(,2 1

)(

2

n

n

n

which one obtains from equation (2.74).

46

(b) The Liouvilles equation

For (2.16) one considers the functions defended by

)86.2(,2

.2

,2

,2

1,

2

1

u

u

u

xx

eB

eC

eA

uqur

Equation (2.73) becomes

)87.2(.)(4

1 2

22

2

2

x

xx

xxu

uDuD

Assume that 2 can be expanded in series of the form (2.78). Equation

(2.87) implies that 2 is determined by the recursion relation

)89.2(,1,

)88.2(,4

1

)(

2

1

1

)(

2

)(

2

)2(

2

)1(

2

2)1(

2

nDu

uD

u

inn

i

in

x

x

xxn

x

whenever is a solution of Liouville's equation, and it follows from

equation (2.74) that the coefficients of the series in , given by (2.85), are

sequence of conserved densities for Liouville's equation.

(c) The (-) BRT equation

For any solution u of the (-) BRT equation (2.18), we consider the

functions

47

).)((2

1

)90.2(),)((2

1),(

2

1

,2

,2

2

gugC

uggBg

A

uq

ur

x

x

xx

Equation (2.73) becomes

)91.2(.)(4

1 2

22

22

2 x

xx

xxu

uDuD

Assume that 2 can be expanded in a series of the form (2.78). Equation

implies that 2 is determined by the recursion relation

)93.2(.1,

)92.2(,4

1

)(

2

1

1

)(

2

)(

2

)(

2

)1(

2

22)1(

2

nDu

uD

u

inn

i

in

x

n

x

xxn

x

Whenever u(x, t) is a solution of the (-) BRT equation, and it follows from

equation (2.74) that the coefficients of the series in , given by (2.85), are

sequence of conserved densities for the (-) BRT equation.

(d) The (+) BRE equation

For the equations (2.20) we consider the functions of u(x, t) defined by

).)((2

1

)94,2(),)((2

1),(

2

1

2,

22

gugC

uggBg

A

uq

ur

x

x

xx

Equation (2.73) becomes

)95.2(.)(4

1 2

22

22

2

x

xx

xxu

uDuD

48

Assume that 2 can be expanded in a series of the form (2.78). Equation

(2.95) implies that 2 is determined by the recursion relations

)96.2(.1,

,4

1

)(

2

1

0

)(

2

)(

2

)(

2

)1(

2

22)1(

2

nDu

uD

u

inn

i

in

x

n

x

xxn

x

Whenever is a solution of the (+) BRT equation, and it follows

from equation (2.74) that the coefficients of the series in , given by

(2.85), are a sequence of conserved densities for the (+) BRT equation.

(e) A mKdV equation

For any solution u of the mKdV equation (2.22), the functions:

,6

,6

uq

ur

)97.2(),3

(6

1),

3(

2

1 32

23 auu

uuuBa

uA xxx

).3

(6

1 32 auu

uuuC xxx

Equation (2.73) becomes:

)98.2(.)(6

1 2

22

2

2 u

uDuD x

x

Assume that 2 can be expanded in a series of the form (2.78).

Equation (2.98) implies that 2 is determined by the recursion relation

49

)100.2(.1,

)99.2(,6

1

)(

2

1

1

)(

2

)(

2

)(

2

)1(

2

2)1(

2

nDu

uD

u

inn

i

in

x

nxn

Whenever is a solution of the mKdV equation, and it follows from

equation (2.74) that the coefficients of series in , given in (2.85), are

sequence of conserved densities for the mKdV equation.

(f) A IS equation

For any solution u of the IS equation (2.24), one considers the functions

),4(2

),(2

1),(

2

1

4

22

xxx

xx

uuuu

uA

uu

uqu

u

ur

(2.101)

)],4()958[(

2

1 4326

x

xx

xxxx

xxx uuuu

uuuuuuu

u

uB

Equation (2.73) becomes:

)102.2(.)

)(

)(

()_)((4

1 2

222

2

222

2

uu

u

uu

uD

uu

uD

x

xx

xx

Assume that 2 can be expanded in a series of the form (2.78). Equation

(2.102) implies that 2 is determined by the recursion relation:

)103.2(),)((4

1 222)1(

2

uu

ux

)].4()958[(2

1 4326

x

xx

xxxx

xxx uuuu

uuuuuuu

u

uC

51

)104.2(.1,

)(

)()(

2

1

1

)(

2

/)(

2

)(

22

2

)1(

2

nD

uu

u

uu

uD

inn

i

in

x

n

x

xx

n

Whenever u(x, t) is a solution of the IS equation, and it follows from

equation (2.74) that the coefficients of the series in , given (2.85), are a

sequence of conserved densities for the IS equation.

(g) A CH equation

For any solution u of the CH equation (2.26), we consider the functions:

),1(2

1),

2

1

222( 2

2

2

2

quur xx

)105.2(),22

1(),(

2

12

1

uuuuBuuA x

x

),2

1)

11)(1(

2(

22

22

uuuuuC x

xx

in which the parameters and are constrained by the relation (2.27).

Equation (2.73) becomes:

)106.2(.)

)2

1

222(

)((

)1)(2

1

222(

2

1

2

222

2

2

2

2

22

uu

uuD

uuD

xx

xxx

xxx

Assume that 2 can be expanded in a series of the form (2.78). Equation

(2.106) implies that 2 is determined by the recursion relation:

)107.2(),1)(2

1

222(

2

1 2

2

2

2

)1(

2

uuxx

51

)108.2(.1,

)2

1

222(

)((

1

1

)(

2

)(

2

)(

2

)(

22

2

)1(

2

nD

uu

uuD n

i

inin

x

n

xx

xxxn

Whenever is solution of the CH equation, and it follows from

equation (2.74) that the coefficients of the series in , given in (2.85), are

a sequence of conserved densities of the CH equation.

(h) A HS equation

For any solution u of the HS equation (2.29), one considers the functions:

),2

11(

)109.2(),1(2

1),

1(

2

1

),1(2

1),

2

1

2(

2

uuuu

uuC

uuBuuA

qur

xx

xx

x

xx

in which the parameters and are constrained by the relation (2.30).

Equation (2.73) becomes:

)110.2(.)

)2

1

2(

()1)(2

1

2(

2

1 2

222

xx

xxxxxx

u

uDuD

Assume that 2 can be expanded in a power series of the form (2.78).

Equation (2.110) implies that 2 is determined by the recursion relations:

)111.2(),1)(2

1

2(

2

1)1(

2

xxu

)112.2(.1,

)2

1

2(

)(

2

1

1

)(

2

)(

2

)(

2

)1(

2

nD

u

uD inn

i

in

x

n

xx

xxxn

52

Whenever is solution of the HS equation, and follows from

equation (2.74) that the coefficients of the series on , given in (2.85), are

a sequence of conserved densities for the HS equation.

2.5. Conclusion

The inverse scattering method [41-47] may be rewritten by considering

as a three component Vector and Ω as traceless 3x3 matrix one-form [48].

The latter yields directly the curvature condition (Gaussian curvature equal

to -1, corresponding to pseudo-spherical surfaces). Thus geometrical

method is considered for several NLPDEs which describe pss: NLSE,

Liouville's equation, a mKdV equation, the (-) and (+) BRT equation, a IS,

a CH and a HS equation. Next an infinite number of conservation laws is

derived for the first five of the NLPDEs just mentioned using a theorem

by Cavalcante and Tenenblat (1988). This geometrical method allows

some further generalization of the work on conservation laws given by

Wadati, Sanuki and Konno (1975). An infinite number of conservation

laws for all eight NLPDEs mentioned above are derived in this way.

53

Chapter 3

Conservation laws for Fifth – order

Nonlinear Evolution Equations

3.1. Introduction

A differential equation for a real-valued function describes pseudo

spherical surfaces (pss) if there exists one-forms

where

are functions of and a finite number of its derivatives, such that the

structure equations of a surface of constant curvature , namely,

(3.2)

The notion of a differential equation which describes pseudo spherical

surfaces was introduced by Chern and Tenenblat[19], who also performed

a complete classification of the evolution equations

which describe pseudo spherical surfaces,

under the assumption that This classification included a

constructive procedure to obtain the linear problem associated with each

of these differential equations. A similar study for different classes of

differential equations was carried out by Jorge and Tenenblat[61, 19].

Examples of the classes mentioned so far are given by the Korteweg–de

Vries, modified Korteweg–de Vries, Sine-Gordon, Sinh- Gordon, Burgers

and Liouville equations [49-55].

An important problem is to determine all evolution equations with

which are independent of . In the case of this problem was solved by Rabelo and Tenenblat[60]. We

address a problem along these lines, working with fifth-order evolution

equations

Equations of the form which describe

pseudo spherical surfaces include [56-59]

54

The Kaup–Kupershmidt equation

(3.3)

and the new nonlinear evolution equation

(a) The Kaup–Kupershmidt equation

The fact that equation is of

considerably high order which makes its study in full generality very

difficult, demanding that include some further assumptions on the

associated linear problem. In attempting to determine which such an

assumption should be, we were essentially motivated by the fact that the

fifth-order The Kaup–Kupershmidt equation

describes pseudo spherical surfaces, with G independent of , and

associated one-forms , satisfying

(3.2), if and only if,

(

)

(

)

(

)

(

)

(

)

( )

(3.5)

where are differentiable functions

with . The functions are given by

55

(

)

(

)

(

)

( )

.

Putting

in the class of equations described in equations (3.5) and (3.6) ,we obtain

that the Kaup–Kupershmidt equation,

describes pseudo spherical equations with associated one-forms

given by

56

(

)

(

)

.

(b) The new nonlinear evolution equation

An evolution equation describes

pseudospherical surfaces, with G independent of , and associated one-

forms , satisfying (3.2), if and

only if,

Where is a differentiable functions with . The

functions are given by

(

)

57

By setting in the class of equations

described in equations (3.8) and (3.9), we obtain that the new nonlinear

evolution equation

describes pseudo spherical equations with associated 1-forms

given by

(

)

(

)

(

)

58

3.2: conservation laws by using Riccati equations

The inverse scattering transform method allows one to linearize a large

class of nonlinear evolution equations and can be considered as a

nonlinear version of the Fourier transform [12-15]. An essential

prerequisite of inverse scattering transform method is the association of

the nonlinear evolution equation with a pair of linear problems (Lax pair ),

a linear eigenvalue problem , and a second associated linear problem, such

that the given equation results as a compatibility condition between them

[16-22]. Consider the following Abowitz-Kaup-Newell-Segur (AKNS)

eigenvalues problem

(

) (3.11)

where denotes exterior differentiation, is a column vector and the 2 ×

2 matrix is traceless

(

)

Take

(

) (3.12)

from Eqs. (3.11) and (3.12), we obtain

(3.13)

where and are two 2 × 2 null-trace matrices

( ) (3.14)

(

) (3.15)

59

Here is a parameter, independent of and , while and are functions

of and . Now

which requires the vanishing of the two forms

Θ (3.16)

or in component form

(3.17)

or

(3.18)

where

(3.19)

Chern and Tenenblat [19], obtained Eq. (3.17) directly from the structure

equations (3.2). By suitably choosing and in (3.17), one shall

obtain various fifth order nonlinear evolution equation which must

satisfied. Konno and Wadati introduced the function [62],

Γ

(3.20)

61

This function first is used and explained in the geometric context of

pseudo spherical equations in [11, 13], and see also the classical papers by

Sasaki [63], and Chern–Tenenblat [19]. Then Eq. (3.13) is reduced to the

Riccati equations [60]:

(3.21)

(3.22)

Equations (3.21) and (3.22) imply that

(3.23)

to both sides and using the expression Adding

from (3.17), equation (3.23) takes the form

(3.24)

let us show how an infinite number of conservation laws come out from

these results. The Riccati equation (3.21) can be rearranged to take the

form

. (3.25)

A similar pair of equations can be obtained for the derivatives. Expand

into a power series in the inverse of so that

∑ , (3.26)

the are unknown at this point, however a recursion

relation can be obtained for the by using (3.25), substituting (3.26)

into the equation in (3.25), we find that:

∑

∑

*∑

+

61

Appling the Cauchy product formula

∑

∑ ∑

in (3.27), then we

obtain

∑ ∑ (∑ )

∑

Now equate powers of on both sides of this expression to produce the

set of recursions,

(3.29)

∑

Substituting (3.26) into (3.24), the following system of conservation laws

appears:

∑

* ∑

+ (3.30)

This procedure generates an infinite number of conservation laws for any

nonlinear evolution equations and conservation laws for the equation

under examination. To obtain conservation laws, using (3.30) in a

particular examples using this procedure, let us consider the following two

examples:

(a) The Kaup–Kupershmidt equation

From the equations (3.19) and (3.7), one obtains the functions

for the Kaup–Kupershmidt equation as follows:

62

(

)

(3.31)

(

)

Substituting (3.31) into (3.17), one obtains the Kaup–Kupershmidt

equation (3.3). Putting (3.31) into (3.30), it is found that:

∑

[ ∑

]

Such as

(b) The new nonlinear evolution equation

From the equations (3.19) and (3.10), one obtains the functions

for the new nonlinear evolution equation as follows:

63

(

)

* (

)

+

(

) (3.32)

[ (

)

]

(

)

Substituting (3.32) into (3.17), one obtains the new nonlinear evolution

equation (3.4). Putting (3.32) into (3.30), it is found that

∑

[ ∑

]

such as :

64

(

)

3.3: Conclusions

The nonlinear evolution equations play a central role in the field of

integrable systems and also play a fundamental role in several other areas

of mathematics and physics. In this paper, we show how the geometrical

properties of a pss may be applied to obtain analytical results for the

Kaup–Kupershmidt equation and the new nonlinear evolution equation

which describe pss. An infinite number of conservation laws are derived

for the Kaup–Kupershmidt equation and the new nonlinear evolution

equation using by the Riccati equations. Conservation law plays a vital

role in the study of nonlinear evolution equations, particularly with regard

to integrability, linearization and constants of motion. Conserved

quantities are not unique, since one can always add a constant to a

conserved quantity. Since most laws of physics express some kind of

conservation, conserved quantities commonly exist in mathematical

models of real systems. For example, any classical mechanics model will

have energy as a conserved quantity so long as the forces involved are

conservative. In physics, a conservation law states that a particular

measurable property of an isolated physical system does not change as the

system evolves.( In dynamical systems with intrinsic chaos, with many

degrees of freedom, and many conserved quantities, a fundamental issue is

the statistical relevance of suitable subsets of these conserved quantities in

appropriate regimes). The Galerkin truncation of the Burgers-Hopf

equation has been introduced recently as a prototype model with solutions

exhibiting intrinsic stochasticity and a wide range of correlation scaling

65

behavior that can be predicted successfully by simple scaling arguments.

Here it is established that the truncated Burgers-Hopf model is a

Hamiltonian system with Hamiltonian given by the integration of the third

power [61].

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72

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