University of Saskatchewan An extended procedure for finding exact solutions of PDEs arising from potential symmetries Alexei F. Cheviakov University of Saskatchewan Symmetry-2009

Talk plan Local conservation laws of PDE systems Nonlocally related PDE systems Potential systems, Subsystems Trees of nonlocally related systems Example: Planar Gas Dynamics (PGD) equations Conservation laws; Tree of nonlocally related systems Nonlocal (potential) symmetries Construction of exact solutions from potential symmetries Standard algorithm Three refinements New exact solutions for PGD equations Symmetry-2009

( I ) Conservation laws and nonlocally related PDE systems Symmetry-2009

Local conservation laws A PDE system: R i [ u ] ´ ( x ; @ : k ) = 1 N x = ( 1 ; : n ) u m . A conservation law: D i © [ u ] ´ x 1 + : n = . Time-dependent systems: D t ª [ u ] + x 2 © : n = . For any physical PDE system (in the solved form), look for multipliers that yield conservation laws: ¤ ¾ [ u ] R ´ D i © = : Symmetry-2009

Conservation laws and potential equations Example: wave equation U f x ; t u g : = c 2 ( ) Conservation law: @ t ( c ¡ 2 x ) u = ½ v x = c ¡ 2 ( ) u t ; : Potential equations: Potential system: potential equations plus remaining equations U V f x ; t u v g : ½ = c ¡ 2 ( ) V is nonlocal variable Solution set: equivalent to that of the given system. Symmetry-2009

Subsystems U f x ; t u g : = c ( ) U V f x ; t u v g : ½ = c ( ) V f x Nonlocally related subsystems: exclude dependent variables using differential relations. Given: U V f x ; t u v g : ½ = c ¡ 2 ( ) Nonlocally related subsystems: U f x ; t u g : = c 2 ( ) V f x ; t v g : = ( c 2 ) Symmetry-2009

Tree of nonlocally related systems Construction of the tree of nonlocally related systems: [Bluman & Cheviakov, JMP 46 (2005); Bluman, Cheviakov & Ivanova, JMP 47 (2006)] For a given PDE system, construct local conservation laws. Construct potential systems. (include ones with pairs, triplets, quadruplets of potentials,…) Construct nonlocally related subsystems. Find further conservation laws. Continue. Symmetry-2009

( II ) Nonlocally related PDE systems of Planar Gas Dynamics Symmetry-2009

Planar Gas Dynamics equations Lagrange PDE system of planar gas dynamics: L f y ; s v p q g : 8 < ¡ = + B ( ) Lagrangian coordinates (initial positions) of fluid particles: y Time: s Velocity: v Density: ½ = 1 q Local conservation laws: assume ¤ i = ( y ; s V P Q ) Symmetry-2009

Conservation laws and potential systems f y ; s v p q g : 8 < ¡ = + B ( ) Symmetry-2009

Conservation laws and potential systems 1 f y ; s v p q w g : 8 > < = + B ( ) Symmetry-2009

Euler system ® ; v p ½ = q x = w ; t s L W f y ; s v p q w g : 8 > 1 f y ; s v p q w g : 8 > < = + B ( ) Change of variables. Dependent: Independent: ® 1 ; v p ½ = q x = w 1 ; t s , E A 1 f x ; t v p ½ ® g : 8 > < ¡ = + ( ) B Symmetry-2009

Euler system ® E A f x ; t v p ½ ® g : 8 > < ¡ = + ( ) B E f x ; 1 f x ; t v p ½ ® g : 8 > < ¡ = + ( ) B Exclude => nonlocally related subsystem (Euler system) ® 1 E f x ; t v p ½ g : 8 < + ( ) = B 1 Symmetry-2009

Conservation laws and potential systems 2 f y ; s v p q w g : 8 > < ¡ = + B ( ) Symmetry-2009

Conservation laws and potential systems 3 f y ; s v p q w g : 8 > < = + ¡ B ( ) Symmetry-2009

Conservation laws and potential systems 4 f y ; s v p q w g 8 > < : = S ( ) + B Symmetry-2009

Conservation laws and potential systems 5 f y ; s v p q w g 8 > < : = 2 + K ( ) ¡ B Symmetry-2009

Other nonlocally related subsystems f y ; s v p q g : 8 < ¡ = + B ( ) Exclude v L f y ; s p q g : ½ + = B ( ) Symmetry-2009

Other nonlocally related subsystems W 4 f y ; s v p q w g 8 > < : = S ( ) + B Exclude v L W 4 f y ; s p q w g : 8 > < + = S ( ) B Symmetry-2009

Tree for the Lagrange PGD system Symmetry-2009

( III ) Nonlocal symmetries for Planar Gas Dynamics Symmetry-2009

Nonlocal symmetries L f y ; s v p q g B ( p ; q ) = ° c o n s t . X = Given system: R f x ; t u g Potential system: R V f x ; t u v g A symmetry of R V f x ; t u v g X = » ( x ; t u v ) @ + ¿ ´ ¾ ³ ¹ is a nonlocal symmetry of if one or more of depend on nonlocal variables. R f x ; t u g , » ( x ; t u v ) ¿ ´ ¾ Seek nonlocal symmetries of the Lagrange system in the polytropic case L f y ; s v p q g B ( p ; q ) = ° c o n s t . Symmetry-2009

Tree for the Lagrange PGD system Symmetry-2009

Nonlocal symmetries of the Lagrange PGD system Symmetry-2009

Nonlocal symmetries of the Lagrange PGD system Symmetry-2009

Nonlocal symmetries of the Lagrange PGD system Symmetry-2009

( IV ) Exact solutions arising from nonlocal symmetries Symmetry-2009

The standard algorithm for invariant solutions Given system: Potential system: R f x ; t u g , R V f x ; t u v g (For simplicity: consider scalar ) u ; v : Potential symmetry of R f x ; t u g : X = » ( x ; t u v ) @ + ¿ ´ ³ : Symmetry-2009

The standard algorithm for invariant solutions x » = t ¿ u ´ v ³ (1) Characteristic equations: (2) Solutions (invariants): z = Z ( x ; t u v ) h 1 H 2 (3) Translation coordinate: ^ z = Z ( x ; t u v ) : X ^ Z ( x ; t u v ) = 1 (4) Change variables in the potential system R V f x ; t u v g : ( x ; t u v ) ! z ^ h 1 2 (5) Drop dependence on ^ z : h 1 = ( ) ; 2 (6) Solve ODEs to get h 1 = ( z ) ; 2 : (7) Express u ; v : Symmetry-2009

The first refinement [Pucci & Saccomandi (1993)] ... (4) Change variables in the given system R f x ; t u g , ( x ; t u v ) ! z ^ h 1 2 (5) Drop dependence on ^ z : h 1 = ( ) ; 2 (6) Solve ODEs to get h 1 = ( z ) ; 2 : (7) Express u ; v : The potential variable is sought in the invariant form, but is not a solution of the potential equations. Symmetry-2009

The second refinement [Sjoberg & Mahomed (2004)] ... (4) Change variables in the potential system R V f x ; t u v g : ( x ; t u v ) ! z ^ h 1 2 (5) In the expression for , drop dependence on u ^ z : h 1 = ( ) ; 2 (6) Solve ODEs to get h 1 = ( z ) ; 2 : (7) Express u ; v : The potential variable is not sought in the invariant form. Symmetry-2009

The combined approach Do both: The potential variable is not sought in the invariant form, and is not a solution of the potential equations (i.e., ansatz is substituted into the given system). Symmetry-2009

Example: Exact solutions of the Lagrange system Lagrange polytropic system: L f y ; s v p q g : 8 > < ¡ = + ° Potential system: L W 2 f y ; s v p q w g : 8 > < ¡ = + ° Nonlocal symmetry: J 8 = y 2 @ + p ¡ 3 q ( w v ) Symmetry-2009

Exact solutions: Standard algorithm Nonlocal symmetry: J 8 = y 2 @ + p ¡ 3 q ( w v ) ^ z Invariants: z = s ; h 1 p y 2 3 q w 4 v ¡ : Translation coordinate: ^ z = 1 y : p ( y ; s ) = h 1 q 2 3 v 4 + w : Invariant form: Standard invariant solution: v ( y ; s ) = ¡ C 1 + 3 p q 2 : Symmetry-2009

Exact solutions: Extended (combined) approach Translation coordinate: ^ z = 1 y : p ( y ; s ) = h 1 q 2 3 v 4 + w : Invariant form: Substitute into not L f y ; s v p q g L W 2 f y ; s v p q w g : Symmetry-2009

Exact solutions: Extended (combined) approach F 1 : v ( y ; s ) = ¡ a + 3 p q 2 F 2 : v ( y ; s ) = b 1 + p q ¡ 3 F 3 : 8 > < v ( y ; s ) = c 1 n ¡ + 2 4 p q ( I n t e g r ° = 6 1 . ) Standard invariant solution: v ( y ; s ) = ¡ C 1 + 3 p q 2 : Symmetry-2009

Exact solutions: Extended (combined) approach Theorem: Families and do not arise as invariant solutions of the Lagrange or potential system with respect to any of their point symmetries. Families and only arise from the extended (combined) algorithm and not from first or second refinement. F 2 F 3 F 2 F 3 Symmetry-2009

Conclusions One can systematically seek nonlocal symmetries of PDE systems; If a nonlocal (potential) symmetry of a PDE system is found, an extended procedure (presented in this talk) can yield additional solutions compared to the classical method. Symmetry-2009

Thank you for your attention! Some references G. Bluman, A. Cheviakov, N. Ivanova, Framework for nonlocally related PDE systems and nonlocal symmetries: extension, simplification, and examples, J. Math. Phys. 47 (2006), 113505. A. Sjoberg and F.M. Mahomed, Non-local symmetries and conservation laws for one-dimensional gas dynamics equations. App. Math. Comp. 150 (2004), 379–397 E. Pucci and G. Saccomandi, Potential symmetries and solutions by reduction of partial differential equations. J. Phys. A: Math. Gen. 26 (1993), 681-690. A. Cheviakov, An extended procedure for finding exact solutions of partial differential equations arising from potential symmetries, J. Math. Phys. 49 (2008), 083502. Thank you for your attention! Symmetry-2009