Systematic Construction of Conservation Laws of Nonlinear ...
Transcript of Systematic Construction of Conservation Laws of Nonlinear ...
Systematic Construction of Conservation Lawsof Nonlinear Partial Differential equations
Alexei Cheviakov
Department of Mathematics and Statistics,University of Saskatchewan, Saskatoon, Canada
Physics seminar11 Haziran 2012
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 1 / 33
Outline
1 Conservation LawsDefinition and ExamplesApplications
2 Variational Principles
3 Symmetries and Noether’s TheoremSymmetries and Variational SymmetriesNoether’s Theorem, ExamplesLimitations of Noether’s Theorem
4 Direct Construction Method for Conservation LawsThe IdeaCompletenessA Detailed Example
5 Examples of Construction of Conservation LawsSymbolic Software (Maple)KdVSurfactant Dynamics Equations
6 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 2 / 33
Outline
1 Conservation LawsDefinition and ExamplesApplications
2 Variational Principles
3 Symmetries and Noether’s TheoremSymmetries and Variational SymmetriesNoether’s Theorem, ExamplesLimitations of Noether’s Theorem
4 Direct Construction Method for Conservation LawsThe IdeaCompletenessA Detailed Example
5 Examples of Construction of Conservation LawsSymbolic Software (Maple)KdVSurfactant Dynamics Equations
6 Conclusions
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Conservation Laws: Some Definitions
Notation:
The total derivative by x :
The partial derivative by x assuming all chain rules carried out as needed.
Dependent variables should be taken care of.
Example: u = u(x , y), g = g(u), then
Dx [xg ] ≡ ∂
∂x[xg(u(x , y))]
=∂
∂x[xg ] +
∂
∂g[xg ]
∂
∂u[g(u)]
∂
∂xu(x , y).
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Conservation Laws: Some Definitions
Conservation law:
Given: some model.Independent variables: x = (t, x , y , ...); dependent variables: u = (u, v , ...).
A conservation law: a divergence expression equal to zero
DtΘ(x,u, ...) + DxΨ1(x,u, ...) + DyΨ2(x,u, ...) + · · · = 0.
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Conservation Laws: Some Definitions
Conservation law:
Given: some model.Independent variables: x = (t, x , y , ...); dependent variables: u = (u, v , ...).
A conservation law: a divergence expression equal to zero
DtΘ(x,u, ...) + DxΨ1(x,u, ...) + DyΨ2(x,u, ...) + · · · = 0.
Time-independent:
DiΨi ≡ divx,y,...Ψ = 0.
Time-dependent:
DtΘ + divx,y,...Ψ = 0.
A conserved quantity:
Dt
∫V
Θ dV = 0.
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ODE Models: Conserved Quantities
An ODE:
Dependent variable: u = u(t);A conservation law
DtF (t, u, u′, ...) =d
dtF (t, u, u′, ...) = 0.
yields a conserved quantity (constant of motion):
F (t, u, u′, ...) = C = const.
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ODE Models: Conserved Quantities
An ODE:
Dependent variable: u = u(t);A conservation law
DtF (t, u, u′, ...) =d
dtF (t, u, u′, ...) = 0.
yields a conserved quantity (constant of motion):
F (t, u, u′, ...) = C = const.
Example: Harmonic oscillator, spring-mass system
Independent variable: t, dependent: x(t).
ODE: x(t) + ω2x(t) = 0; ω2 = k/m = const.
Conservation law:d
dt
(mx2(t)
2+
kx2(t)
2
)= 0.
Conserved quantity: energy.
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ODE Models: Conserved Quantities
Example: ODE integration
An ODE:
K ′′′(x) =−2K(x) (K ′′(x))
2 − (K ′(x))2
K ′′(x)
K(x)K ′(x).
Three independent conserved quantities:
KK ′′
(K ′)2= C1,
KK ′′ ln K
(K ′)2− ln K ′ = C2,
xKK ′′ + KK ′
(K ′)2− x = C3.
yield complete ODE integration.
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PDE Models
Example 1: small string oscillations, 1D wave equation
Independent variables: x , t; dependent: u(x , t).
utt = c2uxx , c2 = T/ρ.
Conservation laws:
Momentum: Dt(ρut)−Dx(Tux) = 0;
Conserved quantity: total momentum M =∫ρutdx = const.
Energy: Dt
(ρu2
t
2+
Tu2x
2
)−Dx(Tutux) = 0;
Conserved quantity: total energy E =∫ (ρu2
t
2+
Tu2x
2
)dx = const.
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PDE Models
Example 2: Adiabatic motion of an ideal gas in 3D
Independent variables: t; x = (x1, x2, x3) ∈ D ⊂ R3.
Dependent: ρ(x , t), v 1(x , t), v 2(x , t), v 3(x , t), p(x , t).
Equations:Dtρ+ Dj(ρv j) = 0,
ρ(Dt + v jDj)v i + Dip = 0, i = 1, 2, 3,
ρ(Dt + v jDj)p + γρpDjvj = 0.
Conservation laws:
Mass: Dtρ+ Dj(ρv j) = 0,
Momentum: Dt(ρv i ) + Dj(ρv iv j + pδij) = 0, i = 1, 2, 3,
Energy: Dt(E) + Dj
(v j(E + p)
)= 0, E = 1
2ρ|v |2 +
p
γ − 1.
Angular momentum, etc.
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Applications of Conservation Laws
ODEs
Constants of motion.
Integration.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 10 / 33
Applications of Conservation Laws
ODEs
Constants of motion.
Integration.
PDEs
Direct physical meaning.
Constants of motion.
Differential constraints (divB = 0, etc.).
Analysis: existence, uniqueness, stability.
Nonlocally related PDE systems, exact solutions. Potentials, stream functions, etc.:
V = (u, v), divV = ux + vy = 0,
u = Φy ,v = −Φx .
Modern numerical methods often require conserved forms.
An infinite number of conservation laws can indicate integrability / linearization.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 10 / 33
Outline
1 Conservation LawsDefinition and ExamplesApplications
2 Variational Principles
3 Symmetries and Noether’s TheoremSymmetries and Variational SymmetriesNoether’s Theorem, ExamplesLimitations of Noether’s Theorem
4 Direct Construction Method for Conservation LawsThe IdeaCompletenessA Detailed Example
5 Examples of Construction of Conservation LawsSymbolic Software (Maple)KdVSurfactant Dynamics Equations
6 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 11 / 33
Variational Principles
Action integral
J[U] =
∫Ω
L(x ,U, ∂U, . . . , ∂kU)dx .
Principle of extremal action
Variation of U: U(x)→ U(x) + εv(x).
Require: variation of action δJ ≡ J[U + εv ]− J[U] =∫
Ω(δL) dx = O(ε2).
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Variational Principles
Action integral
J[U] =
∫Ω
L(x ,U, ∂U, . . . , ∂kU)dx .
Principle of extremal action
Variation of U: U(x)→ U(x) + εv(x).
Require: variation of action δJ ≡ J[U + εv ]− J[U] =∫
Ω(δL) dx = O(ε2).
Variation of Lagrangian
δL = L(x ,U + εv , ∂U + ε∂v , . . . , ∂kU + ε∂kv)− L(x ,U, ∂U, . . . , ∂kU)
= ε
(∂L[U]
∂Uσvσ +
∂L[U]
∂Uσj
vσj + · · ·+ ∂L[U]
∂Uσj1···jk
vσj1···jk
)+ O(ε2)
= ε(vσEUσ (L[U]) + ...) + O(ε2),
where EUσ are the Euler operators.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 12 / 33
Variational Principles
Action integral
J[U] =
∫Ω
L(x ,U, ∂U, . . . , ∂kU)dx .
Principle of extremal action
Variation of U: U(x)→ U(x) + εv(x).
Require: variation of action δJ ≡ J[U + εv ]− J[U] =∫
Ω(δL) dx = O(ε2).
Euler-Lagrange equations:
Euσ (L[u]) =∂L[u]
∂uσ+ · · ·+ (−1)kDj1 · · ·Djk
∂L[u]
∂uσj1···jk= 0,
σ = 1, . . . ,m.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 12 / 33
Variational Principles
Example 1: Harmonic oscillator, x = x(t)
L =1
2mx2 − 1
2kx2, ExL = −m(x + ω2x) = 0.
Example 2: Wave equation for u(x , t)
L =1
2ρut
2 − 1
2T ux
2, EuL = −ρ(utt − c2uxx) = 0.
Example 3: Klein-Gordon nonlinear equations for u(x , t)
L = −1
2utux + h(u), EuL = utx + h′(u) = 0.
Many other non-dissipative systems have variational formulations.
A practical way to tell whether a DE system has a variational formulation: itslinearization must be self-adjoint (symmetric).
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 13 / 33
Variational Principles
Example 1: Harmonic oscillator, x = x(t)
L =1
2mx2 − 1
2kx2, ExL = −m(x + ω2x) = 0.
Example 2: Wave equation for u(x , t)
L =1
2ρut
2 − 1
2T ux
2, EuL = −ρ(utt − c2uxx) = 0.
Example 3: Klein-Gordon nonlinear equations for u(x , t)
L = −1
2utux + h(u), EuL = utx + h′(u) = 0.
Many other non-dissipative systems have variational formulations.
A practical way to tell whether a DE system has a variational formulation: itslinearization must be self-adjoint (symmetric).
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 13 / 33
Variational Principles
Example 1: Harmonic oscillator, x = x(t)
L =1
2mx2 − 1
2kx2, ExL = −m(x + ω2x) = 0.
Example 2: Wave equation for u(x , t)
L =1
2ρut
2 − 1
2T ux
2, EuL = −ρ(utt − c2uxx) = 0.
Example 3: Klein-Gordon nonlinear equations for u(x , t)
L = −1
2utux + h(u), EuL = utx + h′(u) = 0.
Many other non-dissipative systems have variational formulations.
A practical way to tell whether a DE system has a variational formulation: itslinearization must be self-adjoint (symmetric).
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 13 / 33
Outline
1 Conservation LawsDefinition and ExamplesApplications
2 Variational Principles
3 Symmetries and Noether’s TheoremSymmetries and Variational SymmetriesNoether’s Theorem, ExamplesLimitations of Noether’s Theorem
4 Direct Construction Method for Conservation LawsThe IdeaCompletenessA Detailed Example
5 Examples of Construction of Conservation LawsSymbolic Software (Maple)KdVSurfactant Dynamics Equations
6 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 14 / 33
Symmetries of Differential Equations
Consider a general DE system
Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
with variables x = (x1, ..., xn), u = u(x) = (u1, ..., um).
Definition
A transformationx∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2),
depending on a parameter a is a point symmetry of Rσ[u] if the equation is the samein new variables x∗, u∗.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 15 / 33
Symmetries of Differential Equations
Consider a general DE system
Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
with variables x = (x1, ..., xn), u = u(x) = (u1, ..., um).
Definition
A transformationx∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2),
depending on a parameter a is a point symmetry of Rσ[u] if the equation is the samein new variables x∗, u∗.
Example 1: translations
The translationx∗ = x + C , t∗ = t, u∗ = u
leaves KdV invariant:
ut + uux + uxxx = 0 = u∗t∗ + u∗u∗x∗ + u∗x∗x∗x∗ .
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 15 / 33
Symmetries of Differential Equations
Consider a general DE system
Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
with variables x = (x1, ..., xn), u = u(x) = (u1, ..., um).
Definition
A transformationx∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2),
depending on a parameter a is a point symmetry of Rσ[u] if the equation is the samein new variables x∗, u∗.
Example 2: scaling
Same for the scaling:x∗ = αx , t∗ = α3t, u∗ = α−2u.
One hasut + uux + uxxx = 0 = u∗t∗ + u∗u∗x∗ + u∗x∗x∗x∗ .
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 15 / 33
Variational Symmetries
Consider a general DE system
Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
that follows from a variational principle with J[U] =∫
ΩL(x ,U, ∂U, . . . , ∂kU)dx .
Definition
A symmetry of Rσ[u] given by
x∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2).
is a variational symmetry of Rσ[u] if it preserves the action J[U].
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 16 / 33
Variational Symmetries
Consider a general DE system
Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
that follows from a variational principle with J[U] =∫
ΩL(x ,U, ∂U, . . . , ∂kU)dx .
Definition
A symmetry of Rσ[u] given by
x∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2).
is a variational symmetry of Rσ[u] if it preserves the action J[U].
Example 1: translations for the wave equation
utt = (T/ρ)2uxx , L =1
2ρut
2 − 1
2T ux
2.
The translation x∗ = x + C , t∗ = t, u∗ = u is evidently a variational symmetry.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 16 / 33
Variational Symmetries
Consider a general DE system
Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N
that follows from a variational principle with J[U] =∫
ΩL(x ,U, ∂U, . . . , ∂kU)dx .
Definition
A symmetry of Rσ[u] given by
x∗ = f (x , u; a) = x + aξ(x , u) + O(a2),u∗ = g(x , u; a) = u + aη(x , u) + O(a2).
is a variational symmetry of Rσ[u] if it preserves the action J[U].
Example 2: scaling for the wave equation
utt = (T/ρ)2uxx , L =1
2ρut
2 − 1
2T ux
2.
The scaling x∗ = x , t∗ = t, u∗ = u/α is not a variational symmetry: L∗ = α2L,J∗ = α2J.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 16 / 33
Noether’s Theorem (restricted to point symmetries)
Theorem
Given:
a PDE system
Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N,
following from a variational principle;
a variational symmetry
(x i )∗ = f i (x , u; a) = x i + aξi (x , u) + O(a2),(uσ)∗ = gσ(x , u; a) = uσ + aησ(x , u) + O(a2).
Then the system Rσ[u] has a conservation law DiΦi [u] = 0.
In particular,DiΦ
i [u] ≡ Λσ[u]Rσ[u] = 0,
where the multipliers are given by
Λσ = ησ(x , u)− ∂uσ
∂xiξi (x , u).
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 17 / 33
Noether’s Theorem: Examples
Example: translation symmetry for the harmonic oscillator
Equation: x(t) + ω2x(t) = 0, ω2 = k/m.
Symmetry:t∗ = t + a, ξ = 1;x∗ = x , η = 0,
Multiplier (integrating factor): Λ = η − x(t)ξ = −x ;
Conservation law:
ΛR = −x(x(t) + ω2x(t)) = − 1
m
d
dt
(mx2(t)
2+
kx2(t)
2
)= 0.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 18 / 33
Limitations of Noether’s Theorem
The given DE system, as written, must be variational.
Numbers of PDEs and dependent variables must coincide.
Dissipative systems are not variational.
If single PDE, must be of even order.
If a PDE system is not variational, artifices sometimes can make it variational!
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 19 / 33
Limitations of Noether’s Theorem
The given DE system, as written, must be variational.
Numbers of PDEs and dependent variables must coincide.
Dissipative systems are not variational.
If single PDE, must be of even order.
If a PDE system is not variational, artifices sometimes can make it variational!
Example 1: The use of multipliers.
The PDEutt + H ′(ux)uxx + H(ux) = 0,
as written, does not admit a variational principle. However, the equivalent PDE
ex [utt + H ′(ux)uxx + H(ux)] = 0,
does follow from a variational principle!
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 19 / 33
Limitations of Noether’s Theorem
The given DE system, as written, must be variational.
Numbers of PDEs and dependent variables must coincide.
Dissipative systems are not variational.
If single PDE, must be of even order.
If a PDE system is not variational, artifices sometimes can make it variational!
Example 2: The use of a transformation.
The PDEexutt − e3x(u + ux)2(u + 2ux + uxx) = 0,
as written, does not admit a variational principle. But the point transformation x∗ = x ,t∗ = t, u∗(x∗, t∗) = y(x , t) = exu(x , t), maps it into the self-adjoint PDE
ytt − (yx)2yxx = 0,
which is the Euler–Lagrange equation for the Lagrangian L[Y ] = y 2t /2− y 4
x /12.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 19 / 33
Limitations of Noether’s Theorem
The given DE system, as written, must be variational.
Numbers of PDEs and dependent variables must coincide.
Dissipative systems are not variational.
If single PDE, must be of even order.
If a PDE system is not variational, artifices sometimes can make it variational!
Example 3: The use of a differential substitution.
The KdV equationut + uux + uxxx = 0,
as written, does not admit a variational principle. However, a substitution u = vx yields avariational PDE
vxt + vxvxx + vxxxx = 0.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 19 / 33
Limitations of Noether’s Theorem (ctd.)
. . .
The use of an artificial additional equation. Any PDE system can be madevariational, by appending an adjoint equation!
Example:
The diffusion equation ut − uxx = 0 is dissipative, hence not self-adjoint.
Its adjoint equation: wt + wxx = 0.
But the PDE systemut − uxx = 0,
ut + uxx = 0
is self-adjoint!
Lagrangian:L = utw − wtu + 2uxwx .
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 20 / 33
Outline
1 Conservation LawsDefinition and ExamplesApplications
2 Variational Principles
3 Symmetries and Noether’s TheoremSymmetries and Variational SymmetriesNoether’s Theorem, ExamplesLimitations of Noether’s Theorem
4 Direct Construction Method for Conservation LawsThe IdeaCompletenessA Detailed Example
5 Examples of Construction of Conservation LawsSymbolic Software (Maple)KdVSurfactant Dynamics Equations
6 Conclusions
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 21 / 33
The Idea of the Direct Construction Method
Definition
The Euler operator with respect to U j :
EU j =∂
∂U j−Di
∂
∂U ji
+ · · ·+ (−1)sDi1 . . .Dis
∂
∂U ji1...is
+ · · · , j = 1, . . . ,m.
Consider a general DE system Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N withvariables x = (x1, ..., xn), u = u(x) = (u1, ..., um).
Theorem
The equationsEU j F (x ,U, ∂U, . . . , ∂sU) ≡ 0, j = 1, . . . ,m
hold for arbitrary U(x) if and only if
F (x ,U, ∂U, . . . , ∂sU) ≡ DiΨi
for some functions Ψi (x ,U, ...).
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 22 / 33
The Idea of the Direct Construction Method
Consider a general DE system Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N withvariables x = (x1, ..., xn), u = u(x) = (u1, ..., um).
Direct Construction Method
Specify dependence of multipliers: Λσ = Λσ(x ,U, ...), σ = 1, ...,N.
Solve the set of determining equations
EU j (ΛσRσ) ≡ 0, j = 1, . . . ,m,
for arbitrary U(x) (off of solution set!) to find all such sets of multipliers.
Find the corresponding fluxes Φi (x ,U, ...) satisfying the identity
ΛσRσ ≡ DiΦi .
Each set of fluxes and multipliers yields a local conservation law
DiΦi (x , u, ...) = 0,
holding for all solutions u(x) of the given PDE system.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 23 / 33
Completeness
Completeness of the Direct Construction Method
For the majority of physical DE systems (in particular, all systems in solved form),all conservation laws follow from linear combinations of equations!
ΛσRσ ≡ DiΦi .
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 24 / 33
A Detailed Example
Consider a nonlinear telegraph system for u1 = u(x , t), u2 = v(x , t):
R1[u, v ] = vt − (u2 + 1)ux − u = 0,R2[u, v ] = ut − vx = 0.
Multiplier ansatz: Λ1 = ξ(x , t,U,V ), Λ2 = φ(x , t,U,V ).
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 25 / 33
A Detailed Example
Consider a nonlinear telegraph system for u1 = u(x , t), u2 = v(x , t):
R1[u, v ] = vt − (u2 + 1)ux − u = 0,R2[u, v ] = ut − vx = 0.
Multiplier ansatz: Λ1 = ξ(x , t,U,V ), Λ2 = φ(x , t,U,V ).
Determining equations:
EU
[ξ(x , t,U,V )(Vt − (U2 + 1)Ux − U) + φ(x , t,U,V )(Ut − Vx)
]≡ 0,
EV
[ξ(x , t,U,V )(Vt − (U2 + 1)Ux − U) + φ(x , t,U,V )(Ut − Vx)
]≡ 0.
Euler operators:
EU =∂
∂U−Dx
∂
∂Ux−Dt
∂
∂Ut,
EV =∂
∂V−Dx
∂
∂Vx−Dt
∂
∂Vt.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 25 / 33
A Detailed Example
Consider a nonlinear telegraph system for u1 = u(x , t), u2 = v(x , t):
R1[u, v ] = vt − (u2 + 1)ux − u = 0,R2[u, v ] = ut − vx = 0.
Multiplier ansatz: Λ1 = ξ(x , t,U,V ), Λ2 = φ(x , t,U,V ).
Determining equations:
EU
[ξ(x , t,U,V )(Vt − (U2 + 1)Ux − U) + φ(x , t,U,V )(Ut − Vx)
]≡ 0,
EV
[ξ(x , t,U,V )(Vt − (U2 + 1)Ux − U) + φ(x , t,U,V )(Ut − Vx)
]≡ 0.
Split determining equations:
φV − ξU = 0, φU − (U2 + 1)ξV = 0,
φx − ξt − UξV = 0, (U2 + 1)ξx − φt − UξU − ξ = 0.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 25 / 33
A Detailed Example
Consider a nonlinear telegraph system for u1 = u(x , t), u2 = v(x , t):
R1[u, v ] = vt − (u2 + 1)ux − u = 0,R2[u, v ] = ut − vx = 0.
Multiplier ansatz: Λ1 = ξ(x , t,U,V ), Λ2 = φ(x , t,U,V ).
Solution: five sets of multipliers (ξ, φ) =
0 1
t x − 12t2
1 −t
ex+ 12U2+V Uex+ 1
2U2+V
ex+ 12U2−V −Uex+ 1
2U2−V
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 25 / 33
A Detailed Example
Consider a nonlinear telegraph system for u1 = u(x , t), u2 = v(x , t):
R1[u, v ] = vt − (u2 + 1)ux − u = 0,R2[u, v ] = ut − vx = 0.
Multiplier ansatz: Λ1 = ξ(x , t,U,V ), Λ2 = φ(x , t,U,V ).
Resulting five conservation laws:
Dtu −Dxv = 0,
Dt [(x − 12t2)u + tv ] + Dx [( 1
2t2 − x)v − t( 1
3u3 + u)] = 0,
Dt [v − tu] + Dx [tv − ( 13u3 + u)] = 0,
Dt [ex+ 1
2u2+v ] + Dx [−uex+ 1
2u2+v ] = 0,
Dt [ex+ 1
2u2−v ] + Dx [uex+ 1
2u2−v ] = 0.
To obtain further conservation laws, extend the multiplier ansatz...
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Outline
1 Conservation LawsDefinition and ExamplesApplications
2 Variational Principles
3 Symmetries and Noether’s TheoremSymmetries and Variational SymmetriesNoether’s Theorem, ExamplesLimitations of Noether’s Theorem
4 Direct Construction Method for Conservation LawsThe IdeaCompletenessA Detailed Example
5 Examples of Construction of Conservation LawsSymbolic Software (Maple)KdVSurfactant Dynamics Equations
6 Conclusions
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Symbolic Software for Computation of Conservation Laws
Example of use of the GeM package for Maple for the KdV.
Use the module: with(GeM):
Declare variables: gem_decl_vars(indeps=[x,t], deps=[U(x,t)]);
Declare the equation:
gem_decl_eqs([diff(U(x,t),t)=U(x,t)*diff(U(x,t),x)
+diff(U(x,t),x,x,x)],
solve_for=[diff(U(x,t),t)]);
Generate determining equations:
det_eqs:=gem_conslaw_det_eqs([x,t, U(x,t), diff(U(x,t),x),
diff(U(x,t),x,x), diff(U(x,t),x,x,x)]):
Reduce the overdetermined system:
CL_multipliers:=gem_conslaw_multipliers();
simplified_eqs:=DEtools[rifsimp](det_eqs, CL_multipliers, mindim=1);
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Symbolic Software for Computation of Conservation Laws
Example of use of the GeM package for Maple for the KdV.
Solve determining equations:
multipliers_sol:=pdsolve(simplified_eqs[Solved]);
Obtain corresponding conservation law fluxes/densities:
gem_get_CL_fluxes(multipliers_sol, method=*****);
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Examples of Symbolic Computation of Conservation Laws
Example 1
KdV equation:ut + uux + uxxx = 0.
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Example 2: Conserved Form of Surfactant Dynamics Equations
Ref.: C. Kallendorf, A.S., M. Oberlack, Y.Wang, 2011
Surfactants = surface active agents.
Two-phase interface described by a level set function Φ; S =
x ∈ R3 : Φ(x) = 0
.
Unit normal: n = − ∇Φ
|∇Φ| . Concentration: c.
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Example 2: Conserved Form of Surfactant Dynamics Equations
Ref.: C. Kallendorf, A.S., M. Oberlack, Y.Wang, 2011
Surfactants = surface active agents.
Two-phase interface described by a level set function Φ; S =
x ∈ R3 : Φ(x) = 0
.
Unit normal: n = − ∇Φ
|∇Φ| . Concentration: c.
Surfactant Dynamics Equations:
Incompressibility of the flow:∇ · u = 0.
Interface transport by the flow:
Φt + u · ∇ (Φ) = 0.
Surfactant transport:
ct + ui ∂c
∂x i− cninj
∂ui
∂x j− α(δij − ninj)
∂
∂x j
((δik − nink)
∂c
∂xk
)= 0.
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Example 2: Conserved Form of Surfactant Dynamics Equations
Result
The surfactant dynamics system can be written in a fully conserved form:
∇ · u = 0.
Φt + u · ∇ (Φ) = 0.
∂
∂t(c F(Φ) |∇Φ|) +
∂
∂x i
(Ai F(Φ) |∇Φ|
)= 0,
where
Ai = cui − α(
(δik − nink)∂c
∂xk
), i = 1, 2, 3,
and F is an arbitrary sufficiently smooth function.
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Conclusions and Open Problems
Conclusions
Divergence-type conservation laws are useful in analysis and numerics.
Conservation laws can be obtained systematically through the Direct ConstructionMethod, which employs multipliers and Euler operators.
The method is implemented in a symbolic package GeM for Maple.
For variational DE systems, conservation laws correspond to variational symmetries.
Noether’s theorem is usually not a preferred way to derive unknown conservationlaws.
Open problems
Computations become hard for conservation laws of high orders.Can we tell anything in advance about highest order of the conservation law for agiven DE system?
For complicated DEs and multipliers, computation of fluxes/densities can becomechallenging.
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Conclusions and Open Problems
Conclusions
Divergence-type conservation laws are useful in analysis and numerics.
Conservation laws can be obtained systematically through the Direct ConstructionMethod, which employs multipliers and Euler operators.
The method is implemented in a symbolic package GeM for Maple.
For variational DE systems, conservation laws correspond to variational symmetries.
Noether’s theorem is usually not a preferred way to derive unknown conservationlaws.
Open problems
Computations become hard for conservation laws of high orders.Can we tell anything in advance about highest order of the conservation law for agiven DE system?
For complicated DEs and multipliers, computation of fluxes/densities can becomechallenging.
A. Cheviakov (Math&Stat) Conservation Laws 11 June 2012 32 / 33
Some references
Bluman, G.W., Cheviakov, A.F., and Anco, S.C. (2010).Applications of Symmetry Methods to Partial Differential Equations.Springer: Applied Mathematical Sciences, Vol. 168.
Anco, S.C. and Bluman, G.W. (1997).Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78,2869–2873.
Anco, S.C. and Bluman, G.W. (2002).Direct construction method for conservation laws of partial differential equations.Part I: Examples of conservation law classifications. Eur. J. Appl. Math. 13,545–566.
Cheviakov, A.F. (2007).GeM software package for computation of symmetries and conservation laws ofdifferential equations. Comput. Phys. Commun. 176, 48–61.
Kallendorf, C., Cheviakov, A.F., Oberlack, M., and Wang, Y.Conservation Laws of Surfactant Transport Equations. Submitted, 2011.
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