Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations.

Conservation Laws: Systematic Construction, Noether s Theorem, Applications, and Symbolic Computations. Alexey Shevyakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada Physics seminar November 10, 2011 A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 1 / 33

Outline 1 Conservation Laws Definition and Examples Applications 2 Variational Principles 3 Symmetries and Noether s Theorem Symmetries and Variational Symmetries Noether s Theorem, Examples Limitations of Noether s Theorem 4 Direct Construction Method for Conservation Laws The Idea Completeness A Detailed Example 5 Examples of Construction of Conservation Laws Symbolic Software (Maple) KdV Surfactant Dynamics Equations 6 Conclusions A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 2 / 33

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 D x[xg] [xg(u(x, y))] x = x [xg] + g [xg] u [g(u)] u(x, y). x A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 4 / 33

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 D tθ(x, u,...) + D xψ 1 (x, u,...) + D y Ψ 2 (x, u,...) + = 0. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 5 / 33

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 D tθ(x, u,...) + D xψ 1 (x, u,...) + D y Ψ 2 (x, u,...) + = 0. Time-independent: D i Ψ i div x,y,...ψ = 0. Time-dependent: A conserved quantity: D tθ + div x,y,...ψ = 0. D t Θ dv = 0. V A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 5 / 33

ODE Models: Conserved Quantities An ODE: Dependent variable: u = u(t); A conservation law D tf (t, u, u,...) = d dt 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: ẍ(t) + ω 2 x(t) = 0; ω 2 = k/m = const. ( d mẋ 2 (t) Conservation law: + kx ) 2 (t) = 0. dt 2 2 Conserved quantity: energy. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 6 / 33

ODE Models: Conserved Quantities Example: ODE integration An ODE: Three independent conserved quantities: K (x) = 2 (K (x)) 2 K(x) (K (x)) 2 K (x). K(x)K (x) KK (K ) 2 = C1, KK ln K (K ) 2 ln K = C 2, yield complete ODE integration. xkk + KK (K ) 2 x = C 3. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 7 / 33

PDE Models Example 1: small string oscillations, 1D wave equation Independent variables: x, t; dependent: u(x, t). u tt = c 2 u xx, c 2 = T /ρ. Conservation laws: Momentum: D t(ρu t) D x(tu x) = 0; Conserved quantity: total momentum M = ρu tdx = const. ) D x(tu tu x) = 0; ( ρu 2 Energy: D t t 2 + Tu2 x 2 Conserved quantity: total energy E = ( ρut 2 2 + Tu2 x 2 ) dx = const. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 8 / 33

PDE Models Example 2: Adiabatic motion of an ideal gas in 3D Independent variables: t; x = (x 1, x 2, x 3 ) D R 3. Dependent: ρ(x, t), v 1 (x, t), v 2 (x, t), v 3 (x, t), p(x, t). Equations: D tρ + D j (ρv j ) = 0, ρ(d t + v j D j )v i + D i p = 0, i = 1, 2, 3, ρ(d t + v j D j )p + γρpd j v j = 0. Conservation laws: Mass: D tρ + D j (ρv j ) = 0, Momentum: D t(ρv i ) + D j (ρv i v j + pδ ij ) = 0, i = 1, 2, 3, Energy: D t(e) + D j ( v j (E + p) ) = 0, E = 1 2 ρ v 2 + p γ 1. Angular momentum, etc. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 9 / 33

Applications of Conservation Laws ODEs Constants of motion. Integration. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 10 / 33

Applications of Conservation Laws ODEs Constants of motion. Integration. PDEs Direct physical meaning. Constants of motion. Differential constraints (div B = 0, etc.). Analysis: existence, uniqueness, stability. Nonlocally related PDE systems, exact solutions. Potentials, stream functions, etc.: V = (u, v), div V = u x + v y = 0, { u = Φy, v = Φ x. Modern numerical methods often require conserved forms. An infinite number of conservation laws can indicate integrability / linearization. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 10 / 33

Variational Principles Action integral J[U] = L(x, U, U,..., k U)dx. Ω 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,..., k U + ε k v) L(x, U, U,..., k U) = ( ) L[U] ε U v σ + L[U] v σ σ Uj σ j + + L[U] v Uj σ j σ 1 j k + O(ε 2 ) 1 j k = ε(v σ E U σ (L[U]) +...) + O(ε 2 ), where E U σ are the Euler operators. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 12 / 33

Variational Principles Action integral J[U] = L(x, U, U,..., k U)dx. Ω 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: E u σ (L[u]) = L[u] u σ + + ( 1)k D j1 D jk L[u] u σ j 1 j k = 0, σ = 1,..., m. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 12 / 33

Variational Principles Example 1: Harmonic oscillator, x = x(t) L = 1 2 mẋ 2 1 2 kx 2, E xl = m(ẍ + ω 2 x) = 0. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 13 / 33

Variational Principles Example 1: Harmonic oscillator, x = x(t) L = 1 2 mẋ 2 1 2 kx 2, E xl = m(ẍ + ω 2 x) = 0. Example 2: Wave equation for u(x, t) L = 1 2 ρut 2 1 2 T ux 2, E ul = ρ(u tt c 2 u xx) = 0. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 13 / 33

Variational Principles Example 1: Harmonic oscillator, x = x(t) L = 1 2 mẋ 2 1 2 kx 2, E xl = m(ẍ + ω 2 x) = 0. Example 2: Wave equation for u(x, t) L = 1 2 ρut 2 1 2 T ux 2, E ul = ρ(u tt c 2 u xx) = 0. Example 3: Klein-Gordon nonlinear equations for u(x, t) L = 1 2 utux + 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: its linearization must be self-adjoint (symmetric). Relatively few models are! A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 13 / 33

Symmetries of Differential Equations Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N with variables x = (x 1,..., x n ), u = u(x) = (u 1,..., u m ). Definition A transformation x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ). depending on a parameter a is a point symmetry of R σ [u] if the equation is the same in new variables x, u. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 15 / 33

Symmetries of Differential Equations Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N with variables x = (x 1,..., x n ), u = u(x) = (u 1,..., u m ). Definition A transformation x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ). depending on a parameter a is a point symmetry of R σ [u] if the equation is the same in new variables x, u. Example 1: translations The translation leaves KdV invariant: x = x + C, t = t, u = u u t + uu x + u xxx = 0 = u t + u u x + u x x x. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 15 / 33

Symmetries of Differential Equations Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N with variables x = (x 1,..., x n ), u = u(x) = (u 1,..., u m ). Definition A transformation x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ). depending on a parameter a is a point symmetry of R σ [u] if the equation is the same in new variables x, u. Example 2: scaling Same for the scaling: One has x = αx, t = α 3 t, u = αu. u t + uu x + u xxx = 0 = u t + u u x + u x x x. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 15 / 33

Variational Symmetries Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N that follows from a variational principle with J[U] = Ω L(x, U, U,..., k U)dx. Definition A symmetry of R σ [u] given by x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ). is a variational symmetry of R σ [u] if it preserves the action J[U]. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 16 / 33

Variational Symmetries Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N that follows from a variational principle with J[U] = Ω L(x, U, U,..., k U)dx. Definition A symmetry of R σ [u] given by x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ). is a variational symmetry of R σ [u] if it preserves the action J[U]. Example 1: translations for the wave equation u tt = (T /ρ) 2 u xx, L = 1 2 ρut 2 1 2 T ux 2. The translation x = x + C, t = t, u = u is evidently a variational symmetry. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 16 / 33

Variational Symmetries Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N that follows from a variational principle with J[U] = Ω L(x, U, U,..., k U)dx. Definition A symmetry of R σ [u] given by x = f (x, u; a) = x + aξ(x, u) + O(a 2 ), u = g(x, u; a) = u + aη(x, u) + O(a 2 ). is a variational symmetry of R σ [u] if it preserves the action J[U]. Example 2: scaling for the wave equation u tt = (T /ρ) 2 u xx, L = 1 2 ρut 2 1 2 T ux 2. The scaling x = x, t = t, u = u/α is not a variational symmetry: L = α 2 L, J = α 2 J. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 16 / 33

Noether s Theorem (restricted to point symmetries) Theorem Given: a PDE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N, following from a variational principle; a variational symmetry (x i ) = f i (x, u; a) = x i + aξ i (x, u) + O(a 2 ), (u σ ) = g σ (x, u; a) = u σ + aη σ (x, u) + O(a 2 ). Then the system R σ [u] has a conservation law D i Φ i [u] = 0. In particular, D i Φ i [u] Λ σ[u]r σ [u] = 0, where the multipliers are given by Λ σ = η σ (x, u) uσ x i ξ i (x, u). A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 17 / 33

Noether s Theorem: Examples Example: translation symmetry for the harmonic oscillator Equation: ẍ(t) + ω 2 x(t) = 0, ω 2 = k/m. Symmetry: t = t + a, ξ = 1; x = x, η = 0, Multiplier (integrating factor): Λ = η ẋ(t)ξ = ẋ; Conservation law: ΛR = ẋ(ẍ(t) + ω 2 x(t)) = 1 m ( d mẋ 2 (t) + kx ) 2 (t) = 0. dt 2 2 A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 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! Example 1: The use of multipliers. The PDE u tt + H (u x)u xx + H(u x) = 0, as written, does not admit a variational principle. However, the equivalent PDE e x [u tt + H (u x)u xx + H(u x)] = 0, does follow from a variational principle! A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 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 PDE e x u tt e 3x (u + u x) 2 (u + 2u x + u xx) = 0, as written, does not admit a variational principle. But the point transformation x = x, t = t, u (x, t ) = y(x, t) = e x u(x, t), maps it into the self-adjoint PDE y tt (y x) 2 y xx = 0, which is the Euler Lagrange equation for the Lagrangian L[Y ] = y 2 t /2 y 4 x /12. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 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 equation u t + uu x + u xxx = 0, as written, does not admit a variational principle. However, a substitution u = v x yields a variational PDE v xt + v xv xx + v xxxx = 0. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 19 / 33

Limitations of Noether s Theorem (ctd.)... The use of an artificial additional equation. Any PDE system can be made variational, by appending an adjoint equation! Example: The diffusion equation u t u xx = 0 is dissipative, hence not self-adjoint. Its adjoint equation: w t + w xx = 0. But the PDE system u t u xx = 0, is self-adjoint! Lagrangian: ũ t + ũ xx = 0 L = u tw w tu + 2u xw x. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 20 / 33

The Idea of the Direct Construction Method Definition The Euler operator with respect to U j : E U j = D U j i U j i + + ( 1) s D i1... D is U j i 1...i s +, j = 1,..., m. Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, variables x = (x 1,..., x n ), u = u(x) = (u 1,..., u m ). σ = 1,..., N with Theorem The equations E U j F (x, U, U,..., s U) 0, hold for arbitrary U(x) if and only if j = 1,..., m for some functions Ψ i (x, U,...). F (x, U, U,..., s U) D i Ψ i A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 22 / 33

The Idea of the Direct Construction Method Consider a general DE system R σ [u] = R σ (x, u, u,..., k u) = 0, σ = 1,..., N with variables x = (x 1,..., x n ), u = u(x) = (u 1,..., u m ). Direct Construction Method Specify dependence of multipliers: Λ σ = Λ σ(x, U,...), σ = 1,..., N. Solve the set of determining equations E U j (Λ σr σ ) 0, j = 1,..., m, for arbitrary U(x) (off of solution set!) to find all such sets of multipliers. Find the corresponding fluxes Φ i (x, U,...) satisfying the identity Λ σr σ D i Φ i. Each set of fluxes and multipliers yields a local conservation law D i Φ i (x, u,...) = 0, holding for all solutions u(x) of the given PDE system. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 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! Λ σr σ D i Φ i. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 24 / 33

A Detailed Example Consider a nonlinear telegraph system for u 1 = u(x, t), u 2 = v(x, t): R 1 [u, v] = v t (u 2 + 1)u x u = 0, R 2 [u, v] = u t v x = 0. Multiplier ansatz: Λ 1 = ξ(x, t, U, V ), Λ 2 = φ(x, t, U, V ). Determining equations: [ E U ξ(x, t, U, V )(Vt (U 2 + 1)U x U) + φ(x, t, U, V )(U t V ] x) 0, [ E V ξ(x, t, U, V )(Vt (U 2 + 1)U x U) + φ(x, t, U, V )(U t V ] x) 0. Euler operators: E U = U Dx D t, U x U t E V = V Dx D t. V x V t A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 25 / 33

A Detailed Example Consider a nonlinear telegraph system for u 1 = u(x, t), u 2 = v(x, t): R 1 [u, v] = v t (u 2 + 1)u x u = 0, R 2 [u, v] = u t v x = 0. Multiplier ansatz: Λ 1 = ξ(x, t, U, V ), Λ 2 = φ(x, t, U, V ). Determining equations: [ E U ξ(x, t, U, V )(Vt (U 2 + 1)U x U) + φ(x, t, U, V )(U t V ] x) 0, [ E V ξ(x, t, U, V )(Vt (U 2 + 1)U x U) + φ(x, t, U, V )(U t V ] x) 0. Split determining equations: φ V ξ U = 0, φ U (U 2 + 1)ξ V = 0, φ x ξ t Uξ V = 0, (U 2 + 1)ξ x φ t Uξ U ξ = 0. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 25 / 33

A Detailed Example Consider a nonlinear telegraph system for u 1 = u(x, t), u 2 = v(x, t): R 1 [u, v] = v t (u 2 + 1)u x u = 0, R 2 [u, v] = u t v x = 0. Multiplier ansatz: Λ 1 = ξ(x, t, U, V ), Λ 2 = φ(x, t, U, V ). Solution: five sets of multipliers (ξ, φ) = 0 1 t x 1 2 t2 1 t e x+ 1 2 U2 +V e x+ 1 2 U2 V Ue x+ 1 2 U2 +V Ue x+ 1 2 U2 V A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 25 / 33

A Detailed Example Consider a nonlinear telegraph system for u 1 = u(x, t), u 2 = v(x, t): R 1 [u, v] = v t (u 2 + 1)u x u = 0, R 2 [u, v] = u t v x = 0. Multiplier ansatz: Λ 1 = ξ(x, t, U, V ), Λ 2 = φ(x, t, U, V ). Resulting five conservation laws: D tu D xv = 0, D t[(x 1 2 t2 )u + tv] + D x[( 1 2 t2 x)v t( 1 3 u3 + u)] = 0, D t[v tu] + D x[tv ( 1 3 u3 + u)] = 0, D t[e x+ 2 1 u2 +v ] + D x[ ue x+ 1 2 u2 +v ] = 0, D t[e x+ 1 2 u2 v ] + D x[ue x+ 1 2 u2 v ] = 0. To obtain further conservation laws, extend the multiplier ansatz... A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 25 / 33

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); A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 27 / 33

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=*****); A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 28 / 33

Examples of Symbolic Computation of Conservation Laws Example 1 KdV equation: u t + uu x + u xxx = 0. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 29 / 33

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 R 3 : Φ(x) = 0 }. Unit normal: n = Φ. Concentration: c. Φ Surfactant Dynamics Equations: Incompressibility of the flow: Interface transport by the flow: u = 0. Surfactant transport: Φ t + u (Φ) = 0. c t + u i c x i cn i n j u i x j α(δ ij n i n j ) x j ( (δ ik n i n k ) c ) = 0. x k A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 30 / 33

Example 2: Conserved Form of Surfactant Dynamics Equations Result The surfactant dynamics system can be written in a fully conserved form: where u = 0. Φ t + u (Φ) = 0. (c F(Φ) Φ ) + ( ) A i F(Φ) Φ t x i A i = cu i α and F is an arbitrary sufficiently smooth function. = 0, ( (δ ik n i n k ) c ), i = 1, 2, 3, x k A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 31 / 33

Conclusions and Open Problems Conclusions Divergence-type conservation laws are useful in analysis and numerics. Conservation laws can be obtained systematically through the Direct Construction Method, 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 conservation laws. 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 a given DE system? For complicated DEs and multipliers, computation of fluxes/densities can become challenging. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 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 of differential 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. A. Shevyakov (Math&Stat) Conservation Laws 10 Nov. 2011 33 / 33