Alexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, PDF Free Download

Direct Method of Construction of Conservation Laws for Nonlinear Differential Equations, its Relation with Noether s Theorem, Applications, and Symbolic Software Alexei F. Cheviakov University of Saskatchewan, Saskatoon, Canada INPL seminar June 09, 2011 A. Cheviakov (UofS) Conservation Laws Nancy 2011 1 / 26

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 Symbolic Computation of Conservation Laws Symbolic Sequence Some Examples 6 Conclusions A. Cheviakov (UofS) Conservation Laws Nancy 2011 2 / 26

Conservation Laws: Definition and Examples Conservation law: A divergence expression equal to zero: d dt Ψ + d dx Φ1 + d dy Φ2 + = 0. Example 1: Harmonic oscillator Independent variable: t, dependent: x(t). ODE: ẍ(t) + ω 2 x(t) = 0; Conservation law: ω 2 = k/m = const. d ( ) mẋ 2 (t) + kx2 (t) = 0 2 2 dt A. Cheviakov (UofS) Conservation Laws Nancy 2011 4 / 26

Conservation Laws: Definition and Examples 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. Cheviakov (UofS) Conservation Laws Nancy 2011 4 / 26

Conservation Laws: Definition and Examples Example 3: Korteweg-de Vries equation (KdV) u = u(x, t), u t + uu x + u xxx = 0. Conservation laws: D t(u) + D x ( 1 2 u2 + u xx ) = 0, D t ( 1 2 u2) + D x ( 1 3 u3 + uu xx 1 2 u2 x ) = 0, D t ( 1 6 u3 1 2 u2 x ) + Dx ( 1 8 u4 uu 2 x + 1 2 (u2 u xx + u 2 xx) u xu xxx ) = 0,..., infinite countable set of conservation laws. A. Cheviakov (UofS) Conservation Laws Nancy 2011 4 / 26

Applications of Conservation Laws Direct physical meaning. Analysis: existence, uniqueness, stability. Modern finite element and finite volume numerical methods often require conserved forms. Nonlocally related PDE systems, exact solutions. An infinite number of conservation laws can be an indication of integrability and/or linearization. A. Cheviakov (UofS) Conservation Laws Nancy 2011 5 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 7 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 7 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 8 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 8 / 26

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 wave 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 linearized version must be self-adjoint. A. Cheviakov (UofS) Conservation Laws Nancy 2011 8 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 10 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 10 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 10 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 11 / 26

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 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. Cheviakov (UofS) Conservation Laws Nancy 2011 11 / 26

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 wave equation u tt = (T /ρ) 2 u xx, L = 1 2 ρut 2 1 2 T ux 2. The scaling x = αx, t = αt, u = u is not a variational symmetry. A. Cheviakov (UofS) Conservation Laws Nancy 2011 11 / 26

Noether s Theorem 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. Cheviakov (UofS) Conservation Laws Nancy 2011 12 / 26

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: 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. Cheviakov (UofS) Conservation Laws Nancy 2011 13 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 14 / 26

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 an extremum Y = y of the action integral with Lagrangian L[Y ] = Yt 2 /2 Yx 4 /12. A. Cheviakov (UofS) Conservation Laws Nancy 2011 14 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 14 / 26

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 evidently self-adjoint! ũ t + ũ xx = 0 A. Cheviakov (UofS) Conservation Laws Nancy 2011 15 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 17 / 26

The Idea of the Direct Construction Method 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 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) 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. Cheviakov (UofS) Conservation Laws Nancy 2011 18 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 19 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 20 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 20 / 26

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 1, φ 1) = (0, 1), (ξ 2, φ 2) = (t, x 1 2 t2 ), (ξ 3, φ 3) = (1, t), (ξ 4, φ 4) = (e x+ 2 1 U2 +V, Ue x+ 2 1 U2 +V ), (ξ 5, φ 5) = (e x+ 2 1 U2 V, Ue x+ 2 1 U2 V ). Resulting five conservation laws: D tu + D x[ v] = 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+ 1 2 u2 +v ] + D x[ ue x+ 1 2 u2 +v ] = 0, D t[e x+ 2 1 u2 v ] + D x[ue x+ 1 2 u2 v ] = 0. A. Cheviakov (UofS) Conservation Laws Nancy 2011 20 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 22 / 26

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. Cheviakov (UofS) Conservation Laws Nancy 2011 23 / 26

Examples of Symbolic Computation of Conservation Laws Example 1 KdV equation: u t + uu x + u xxx = 0. A. Cheviakov (UofS) Conservation Laws Nancy 2011 24 / 26

Examples of Symbolic Computation of Conservation Laws Example 2 Euler equations of 2D incompressible fluid dynamics u = u(t, x, y), v = v(t, x, y), P = P(t, x, y); div v u x + v y = 0, u t + uu x + vu y + P x = 0, v t + uv x + vv y + P y = 0. A. Cheviakov (UofS) Conservation Laws Nancy 2011 24 / 26

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 not a simplest 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. Cheviakov (UofS) Conservation Laws Nancy 2011 25 / 26

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. Bluman, G.W., Cheviakov, A.F., and Anco, S.C. (2009). Construction of Conservation Laws: How the Direct Method Generalizes Noether s Theorem. Proceedings of 4th Workshop Group Analysis of Differential Equations & Integrability 1-23. A. Cheviakov (UofS) Conservation Laws Nancy 2011 26 / 26

Alexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011