A Recursion Formula for the Construction of Local Conservation Laws of Differential Equations

A Recursion Formula for the Construction of Local Conservation Laws of Differential Equations Alexei Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada December 4, 2016 A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 1 / 18

Outline 1 Local Conservation Laws 2 The Basic Recursion Formula 3 The General Recursion Formula 4 Examples: PDE, ODE 5 Relationship Between the Recursion Formula and CL-Symmetry Action 6 Discussion A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 2 / 18

Collaborators R. Naz, Lahore School of Economics, Pakistan A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 3 / 18

Notation Variables: Independent: x = (x 1, x 2,..., x n ) or (t, x 1, x 2,...) or (t, x, y,...). Dependent: u = (u 1 (x), u 2 (x),..., u m (x)) or (u(x), v(x),...). A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 5 / 18

Notation Variables: Independent: x = (x 1, x 2,..., x n ) or (t, x 1, x 2,...) or (t, x, y,...). Dependent: u = (u 1 (x), u 2 (x),..., u m (x)) or (u(x), v(x),...). Derivatives: u µ Notation: = u µ = u µ x i x i i. All p th -order partial derivatives: p u. Total derivative operators: D i = x i + u µ i u + µ uµ ii 1 u µ + u µ ii 1 i 2 i 1 u µ +. i 1 i 2 A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 5 / 18

Notation Variables: Independent: x = (x 1, x 2,..., x n ) or (t, x 1, x 2,...) or (t, x, y,...). Dependent: u = (u 1 (x), u 2 (x),..., u m (x)) or (u(x), v(x),...). Derivatives: u µ Notation: = u µ = u µ x i x i i. All p th -order partial derivatives: p u. Total derivative operators: D i = x i + u µ i u + µ uµ ii 1 u µ + u µ ii 1 i 2 i 1 u µ +. i 1 i 2 A DE system R[u] = 0: R σ [u] R σ (x, u, u,..., k u) = 0, σ = 1,..., N. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 5 / 18

Local Conservation Laws Conservation laws A local divergence-type conservation law of R[u] = 0: a divergence expression that vanishes on its solutions, D i Φ i [u] div Φ i [u] = 0. For models involving time: D t Θ[u] + div x Ψ[u] = 0. Θ[u]: conserved density; Ψ[u]: flux vector. Globally conserved quantities The rate of change of total conserved density amount is given by boundary terms: d dt M V = d Θ[u] dv = Ψ[u] ds. dt Vanishing total flux dm V /dt = 0. Other types of CLs exist in 2D, 3D,...: circulation-type, etc. V V A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 6 / 18

Local Conservation Laws Conservation laws A local divergence-type conservation law of R[u] = 0: a divergence expression that vanishes on its solutions, D i Φ i [u] div Φ i [u] = 0. For models involving time: D t Θ[u] + div x Ψ[u] = 0. Θ[u]: conserved density; Ψ[u]: flux vector. Trivial and equivalent CLs Trivial, Type I: density and flux vanish on solutions of the DE system. Trivial, Type II: CL 0 as a differential identity. Equivalent CLs: differ by a trivial one. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 6 / 18

Local Conservation Laws Conservation laws A local divergence-type conservation law of R[u] = 0: a divergence expression that vanishes on its solutions, D i Φ i [u] div Φ i [u] = 0. For models involving time: D t Θ[u] + div x Ψ[u] = 0. Θ[u]: conserved density; Ψ[u]: flux vector. Characteristics, direct CL construction For a locally solvable DE system R[u] = 0, every local CL is equivalent to one in a characteristic form: D i Φ i [u] = Λ σ[u]r σ [u]. Characteristics: {Λ σ[u]}. Usually, trivial CL trivial characteristics. Direct CL construction: Euler operators E u i (Λ σ[u]r σ [u]) 0. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 6 / 18

The Basic Recursion Formula Consider a PDE system with two independent variables x, y and dependent variable(s) u(x, y). Suppose it has a nontrivial local conservation law D xa[u] + D y B[u] = 0. We can write a formal divergence expression D x(ya[u]) + D y (yb[u] ) B[u] dy = 0. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 7 / 18

The Basic Recursion Formula: Example A wave equation u tt u xx = 0, u = u(x, t). Conservation law form: D tu t D xu x = 0. Application of formula with respect to x and t yields two independent local CLs D t(xu t) D x(xu x u) = 0, D t(tu t u) D x(tu x) = 0. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 8 / 18

The Basic Recursion Formula: Example A wave equation u tt u xx = 0, u = u(x, t). Conservation law form: D tu t D xu x = 0. Application of formula with respect to x and t yields two independent local CLs D t(xu t) D x(xu x u) = 0, D t(tu t u) D x(tu x) = 0. With respect to xt: another local CL D t(x(tu t u)) D x(t(xu x u)) = 0. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 8 / 18

The Basic Recursion Formula: Example A wave equation u tt u xx = 0, u = u(x, t). Conservation law form: D tu t D xu x = 0. Application of formula with respect to x and t yields two independent local CLs D t(xu t) D x(xu x u) = 0, D t(tu t u) D x(tu x) = 0. With respect to xt: another local CL D t(x(tu t u)) D x(t(xu x u)) = 0. With respect to x 2 : a nonlocal expression D t(x 2 u t) D x (x(xu x u) ) (xu x u) dx = 0. Spatial flux: Ψ = x(xu x u) w. A nonlocal variable w defined by w x = xu x u. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 8 / 18

The General Recursion Formula Theorem Suppose that a DE system R[u] = 0 admits a nontrivial local CL D i Φ i [u] = 0. Then for an arbitrary differentiable function f = f (x), the formal divergence expression ) D i Ξ i D i (f Φ i f [u] Φ i [u] dx i = 0 ( ) x i on solutions of the given system. Locality conditions The formula ( ) yields a local conservation law if and only if E u j (f (x) D i Φ i [u]) = 0, j = 1,..., m. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 9 / 18

The General Recursion Formula Theorem Suppose that a DE system R[u] = 0 admits a nontrivial local CL D i Φ i [u] = 0. Then for an arbitrary differentiable function f = f (x), the formal divergence expression ) D i Ξ i D i (f Φ i f [u] Φ i [u] dx i = 0 ( ) x i on solutions of the given system. Nontriviality and linear independence For normal systems, when ( ) is a local CL, it is nontrivial, and linearly independent of the given one. Original CL in characteristic form: D i Φ i [u] = Λ σ[u] R σ [u]. New CL in characteristic form: D i Ξ i = f (x)λ σ[u] R σ [u]. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 9 / 18

Example 1: Nonlinear Wave Equation PDE: u tt = (c 2 (u)u x) x, u = u(x, t). A conservation law as it stands. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 10 / 18

Example 1: Nonlinear Wave Equation PDE: u tt = (c 2 (u)u x) x, u = u(x, t). A conservation law as it stands. Can apply the formula to get local CLs for f (x, t) = 1, t, xt, x: D t(u t) D x(c 2 (u)u x) = 0, D t(tu t u) D x(tc 2 (u)u x) = 0, D t(xu t) D x (xc 2 (u)u x ) c 2 (u)du = 0, ) ( D t (x[tu t u] D x t [ xc 2 (u)u x c 2 (u)du ]) = 0. Locality condition: f tt c 2 (u)f xx = 0, which, for an arbitrary u, has four linearly independent solutions f (t, x) = 1, t, x, xt. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 10 / 18

Example 2: Vorticity-Type Equations PDEs: div N = 0, N t + curl M = 0, N = (N 1, N 2, N 3), M = (M 1, M 2, M 3) depending on t, x, y, z. A part of several physical PDE models: Maxwell s equations, N-S and Euler vorticity dynamics, MHD. Cartesian components of the vector PDE: D tn 1 + D y M 3 D zm 2 = 0, D tn 2 + D zm 1 D xm 3 = 0, D tn 3 + D xm 2 D y M 1 = 0. Can apply the formula with f 1, f 2, f 3; local when (f 1, f 2, f 3) = grad F (x, y, z). Potential vorticity CLs: (N grad F ) t + div (M grad F F t N) = 0. Locality holds in a more general context, when F is an arbitrary differential function. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 11 / 18

Example 3: Wave Propagation in a Hyper-Viscoelastic Fiber-Reinforced Material A third-order PDE governing anti-plane shear fiber-aligned displacements G(t, x) of a fiber-reinforced viscoelastic solid [C., Ganghoffer (2015)]: G = G(x, t), G tt = ( Gx 2 + 1 ) G xx [ ] +η G x 2 (4αGx 2 + 3)G xxg tx + (2αGx 2 + 3)G xg txx [ +ζgx 3 12(2αGx 2 + 1)G xxg tx + (4αGx 2 + 3)G xg txx ], α, η, ζ = const. Local CL form: D t ( G t [ η(3 + 2αG 2 x ) + ζ(3 + 4αG 2 D t ( G t A(G x)g xx ) D x (B(G x) ] ) x )Gx 2 G 2 x G xx ) = 0. ( [1 D x + 1 G ] ) 2 3 x Gx A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 12 / 18

Example 3: Wave Propagation in a Hyper-Viscoelastic Fiber-Reinforced Material A third-order PDE governing anti-plane shear fiber-aligned displacements G(t, x) of a fiber-reinforced viscoelastic solid [C., Ganghoffer (2015)]: G = G(x, t), G tt = ( Gx 2 + 1 ) G xx [ ] +η G x 2 (4αGx 2 + 3)G xxg tx + (2αGx 2 + 3)G xg txx [ +ζgx 3 12(2αGx 2 + 1)G xxg tx + (4αGx 2 + 3)G xg txx ], α, η, ζ = const. Local CL form: D t ( G t [ η(3 + 2αG 2 x ) + ζ(3 + 4αG 2 D t ( G t A(G x)g xx ) D x (B(G x) ] ) x )Gx 2 G 2 x G xx ) = 0. ( [1 D x + 1 G ] ) 2 3 x Gx Can apply the formula ( ) with f = t: obtain a 2nd local, linearly independent CL ( D t tg t G t [ ] ) η(3 + 2αGx 2 ) + ζ(3 + 4αGx 2 )Gx 2 G 2 x G xx +D x (η [ 1 + 2 αg ] 2 5 x G 3 x + ζ [ 3 + 4 αg ] 2 5 7 x G 5 x t [ 1 + 1 G ] ) 2 3 x Gx = 0. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 12 / 18

Example 4: A New First Integral for a Nonlinear ODE A third-order nonlinear ODE arising in symmetry classification, K = K(x): K = 2 (K ) 2 K (K ) 2 K KK. Can be shown to admit a first integral ( ) KK D x (K ) 2 = 0. The formula ( ) with f = x yields an independent local first integral: ( ) ( KK KK xkk 0 = D x x (K ) 2 (K ) dx = D 2 x (K ) + K ) 2 K x. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 13 / 18

Example 4: A New First Integral for a Nonlinear ODE A third-order nonlinear ODE arising in symmetry classification, K = K(x): K = 2 (K ) 2 K (K ) 2 K KK. Can be shown to admit a first integral ( ) KK D x (K ) 2 = 0. The formula ( ) with f = x yields an independent local first integral: ( ) ( KK KK xkk 0 = D x x (K ) 2 (K ) dx = D 2 x (K ) + K ) 2 K x. The total of three independent integrals lead to full ODE integration: KK (K ) = C1, xkk 2 (K ) + K 2 K x = C2, KK ln K ln K = C 3. (K ) 2 A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 13 / 18

Symmetry Action on CLs Vast literature on symmetry mappings CL CL [Caviglia & Morro; Khamitova; Vinogradov; Olver; Popovych & Ivanova; Bluman, Anco & Temuerchaolu,...]. Suppose the given DE system has a local CL D i Φ i [u] = 0, and a point symmetry with an infinitesimal operator Then X = ξ i (x, u) x i + η µ (x, u) u µ. Ω i [u] = M i X[Φ] X (r) Φ i + (D j ξ i )Φ j (D j ξ j )Φ i are fluxes of a local conservation law D i Ω i [u] = 0. The new conservation law may be trivial or linearly dependent on the original one... or may be new. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 14 / 18

Relationship of Symmetry Action and Recursion Formula Nonlinear wave equation as a given CL: D tθ + D xψ D t(u t) D x(c 2 (u)u x) = 0. Two CLs obtained by recursion formula with f = t and f = x: D tθ (1) + D xψ (1) = D t(tu t u) D x(tc 2 (u)u x) = 0, D tθ (2) + D xψ (2) = D t(xu t) D x (xc 2 (u)u x ) c 2 (u)du = 0. Time and space translation symmetries: X 1 = / t, X 2 = / x. Then one has M X1 ( Θ(1), Ψ (1) ) = MX2 ( Θ(2), Ψ (2) ) = ( ut, c 2 (u)u x ) = ( Θ, Ψ ). Recursion formula can be thus related to inverse symmetry-cl action. A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 15 / 18

Summary A simple conservation law recursion formula is presented. May be used to quickly derive linearly independent local CLs of a DE system from a given CL, without the need to solve determining equations. Immediately yields basic CLs for many physical systems. Can be handy for complicated models. The formula is related to the action of point symmetries on local CLs. May be generalized to use a differential function f [u] harder to verify locality. Possible meaning of nonlocal expressions? A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 17 / 18

Some references Olver, P.J. (2000) Applications of Lie Groups to Differential Equations. Springer. Anco, S.C. & Bluman, G. (2002) Direct construction method for conservation laws of partial differential equations. Parts I,II. EJAM 13 (5), 545 585. Bluman, G.W., Cheviakov, A.F., and Anco, S.C. (2010) Applications of Symmetry Methods to Partial Differential Equations. Springer. Cheviakov, A.F. & Ganghoffer, J.-F. (2015) One-dimensional nonlinear elastodynamic models and their local conservation laws with applications to biological membranes. JMBBM 58: 105 121. Cheviakov, A.F. & Naz, R. (2016) A recursion formula for the construction of local conservation laws of differential equations. JMAA (in press). + references therein... A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 18 / 18

Some references Olver, P.J. (2000) Applications of Lie Groups to Differential Equations. Springer. Anco, S.C. & Bluman, G. (2002) Direct construction method for conservation laws of partial differential equations. Parts I,II. EJAM 13 (5), 545 585. Bluman, G.W., Cheviakov, A.F., and Anco, S.C. (2010) Applications of Symmetry Methods to Partial Differential Equations. Springer. Cheviakov, A.F. & Ganghoffer, J.-F. (2015) One-dimensional nonlinear elastodynamic models and their local conservation laws with applications to biological membranes. JMBBM 58: 105 121. Cheviakov, A.F. & Naz, R. (2016) A recursion formula for the construction of local conservation laws of differential equations. JMAA (in press). + references therein... Thank you for your attention! A. Cheviakov (U. Saskatchewan) A Conservation Law Recursion Formula CMS Meeting, December 4, 2016 18 / 18