Nonlocally Related PDE Systems Alexei F. Cheviakov A PDE system: R f x ; u g : i = 1 N Variables: x = ( 1 ; : n ) u m . PDE system Q f z ; w g : j = ( ) is locally related to R f x ; u g Analysis: linearization, finite-time blowup QM symms? Physics is drawing conclusions from symms in nature. Wigner: classification of particles following from symmetries Symm breaking => approx symms _x=d/dx! if its variables are expressed in terms of local quantities in (x,u)-space: z = ( x ; u @ : ) w Otherwise, the systems are nonlocally related.

Types of nonlocally related systems Nonlocally related PDE systems: Potential systems (augmented) Subsystems (reduced) 1) Potential systems: use a conservation law to introduce a nonlocal variable. E.g.1: nonlinear wave equation R f x ; t u g : = ( c ) Potential system: S f x ; t u v g : ½ = c ( )

Types of nonlocally related systems Nonlocally related PDE systems: Potential systems (augmented) Subsystems (reduced) 1) Potential systems: use a conservation law to introduce a nonlocal variable. E.g.2: Equations of ideal Plasma Equilibrium: R f x ; y z V B P g : ( . c u r l £ ) = Potential system: S f x ; y z V B P ª g : ( . £ = r a d

Types of nonlocally related systems Nonlocally related PDE systems: Potential systems (augmented) Subsystems (reduced) For potential systems: Nonlocal relation with original system (usually); Solution sets are always equivalent (nothing is lost); Solution sets are not isomorphic (one-to-many relation)

Types of nonlocally related systems 2) Subsystems: Exclude dependent variables. R f x ; t u v w g : 8 < ¡ = + 1 2 ¢ E.g. Exclude u: R 1 f x ; t v w g : ( ¡ = + ³ 2 ´ locally related. Exclude v: also locally related.

Types of nonlocally related systems 2) Subsystems: Exclude dependent variables. R f x ; t u v w g : 8 < ¡ = + 1 2 ¢ E.g. Exclude u: R 1 f x ; t v w g : ( ¡ = + ³ 2 ´ locally related. Exclude v: also locally related. Exclude w: by cross-differentiation, hence nonlocally related: R 2 f x ; t u v g : ½ ¡ = + 1 ¢

Types of nonlocally related systems (Basically, a subsystem R is nonlocally related to system R if R is a potential system for R.) Why are we interested in nonlocally related systems? Nonlocally related systems have equivalent solution sets, therefore any general method of analysis (qualitative, perturbation, numerical, etc.) that fails to work for a given PDE system, could be successful when applied to a nonlocally related PDE system. Our examples of interest: Nonlocal symmetries; Nonlocal conservation laws.

R f x ; t u g S f x ; t u v g X = » ( x ; t u @ : ) + ¿ ´ Nonlocal Symmetries The idea: Suppose we have a PDE system and its potential system R f x ; t u g S f x ; t u v g Local symmetries of R: X = » ( x ; t u @ : ) + ¿ ´ Local symmetries of S: X ¤ = » ( x ; t u v : ) @ + ¿ ´ ³ Such symmetry is nonlocal for R if one or more of components essentially depend on v. » ; ¿ ´

Nonlocal Symmetries U f x ; t u g : ¡ ( L ) = U V f x ; t u v g : ½ = Example: nonlinear diffusion equation. Symmetry classification. U f x ; t u g : ¡ ( L ) = admits only 2 conservation laws: D t ( u ) ¡ x ³ L ´ = ; : Potential systems: U V f x ; t u v g : ½ = ( L ) U A f x ; t u a g : ½ = ( L ) ¡ U V A f x ; t u v a g

Nonlocal Symmetries U f x ; t u g : ¡ ( L ) = Symmetry classification for U f x ; t u g : ¡ ( L ) = (Here K=L’ )

Nonlocal Symmetries UV, UA

Nonlocal Symmetries UVA

Further extensions and applications Nonlocal theory can be used in any number of dimensions, for linear and nonlinear equations. Using nonlocally related systems, one can: Construct nonlocal (and new local) conservation laws -> further extend the tree; Find new exact solutions -> invariant solutions w.r.t. nonlocal symmetries; -> solutions by mappings Find non-invertible linearizing mappings.

Conclusions Theory of Nonlocally Related PDE systems extends the set of algorithmically calculable symmetries and conservation laws; New applications are to be considered (asymptotic, numerical methods, etc.) Examples have been worked out for many PDE systems: Nonlinear Diffusion, Wave, Telegraph equations; 1+1-dim. nonlinear elasticity; 1+1-dim. gas dynamics (Euler); For references and papers, see http://www.math.ubc.ca/~alexch/ http://www.math.ubc.ca/~bluman/

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