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

2. 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
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3. ( I ) Conservation laws and nonlocally related PDE systems
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4. 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 [ u ] + x 2 : n = .
For any physical PDE system (in the solved form), look for multipliers that yield conservation laws: [ u ] R D i = :
Symmetry-2009

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

7. 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.
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8. ( II ) Nonlocally related PDE systems of Planar Gas Dynamics
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9. 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: = 1 q
Local conservation laws: assume i = ( y ; s V P Q )
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10. Conservation laws and potential systemsf y ; s v p q g : 8
< = + B ( )
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11. Conservation laws and potential systems1 f y ; s v p q w g : 8
>
< = + B ( )
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12. 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 = q x = w 1 ; t s , E A 1 f x ; t v p g : 8
>
< = + ( ) B
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13. Euler system ® E A f x ; t v p ½ ® g : 8 > < ¡ = + ( ) B E f x ;1 f x ; t v p g : 8
>
< = + ( ) B
Exclude => nonlocally related subsystem (Euler system) 1 E f x ; t v p g : 8
< + ( ) = B 1
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14. Conservation laws and potential systems2 f y ; s v p q w g : 8
>
< = + B ( )
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15. Conservation laws and potential systems3 f y ; s v p q w g : 8
>
< = + B ( )
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16. Conservation laws and potential systems4 f y ; s v p q w g 8
>
< : = S ( ) + B
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17. Conservation laws and potential systems5 f y ; s v p q w g 8
>
< : = 2 + K ( ) B
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18. Other nonlocally related subsystemsf y ; s v p q g : 8
< = + B ( )
Exclude v L f y ; s p q g : + = B ( )
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19. Other nonlocally related subsystemsW 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
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20. Tree for the Lagrange PGD system
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21. ( III ) Nonlocal symmetries for Planar Gas Dynamics
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22. 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 ) @ +
is a nonlocal symmetry of
if one or more of
depend on nonlocal variables. R f x ; t u g , ( x ; t u v )
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 .
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23. Tree for the Lagrange PGD system
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24. Nonlocal symmetries of the Lagrange PGD system
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25. Nonlocal symmetries of the Lagrange PGD system
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26. Nonlocal symmetries of the Lagrange PGD system
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27. ( IV ) Exact solutions arising from nonlocal symmetries
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28. 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 ) @ + :
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29. The standard algorithm for invariant solutionsx = t u v
(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

30. 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

31. 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.
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32. 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).
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33. 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 )
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34. 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 :
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35. 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 :
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36. Exact solutions: Extended (combined) approachF 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

37. 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
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38. 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.
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39. 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),
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),
A. Cheviakov, An extended procedure for finding exact solutions of partial differential equations arising from potential symmetries, J. Math. Phys. 49 (2008),
Thank you for your attention!
Symmetry-2009