Symmetry Methods for Differential Equations and Conservation Laws Peter J. Olver University of Minnesota http://www.math.umn.edu/ ∼ olver Varna, June, 2012
Symmetry Groups of Differential Equations System of differential equations ∆( x, u ( n ) ) = 0 G — Lie group acting on the space of independent and dependent variables: ( � x, � u ) = g · ( x, u ) = (Ξ( x, u ) , Φ( x, u ))
G acts on functions u = f ( x ) by transforming their graphs: g �− → Definition. G is a symmetry group of the system ∆ = 0 if � f = g · f is a solution whenever f is.
Infinitesimal Generators Vector field: v | ( x,u ) = d dε g ε · ( x, u ) | ε =0 In local coordinates: p q ξ i ( x, u ) ∂ ϕ α ( x, u ) ∂ � � v = ∂x i + ∂u α i =1 α =1 generates the one-parameter group dx i du α dε = ξ i ( x, u ) dε = ϕ α ( x, u )
Example. The vector field v = − u ∂ ∂x + x ∂ ∂u generates the rotation group x = x cos ε − u sin ε u = x sin ε + u cos ε � � since d � x d � u dε = − � u dε = � x
Jet Spaces x = ( x 1 , . . . , x p ) — independent variables u = ( u 1 , . . . , u q ) — dependent variables ∂ k u α u α J = — partial derivatives ∂x j 1 . . . ∂x k ( x, u ( n ) ) = ( . . . x i . . . u α . . . u α J . . . ) ∈ J n — jet coordinates � p + n � dim J n = p + q ( n ) = p + q n
Prolongation to Jet Space Since G acts on functions, it acts on their derivatives, leading to the prolonged group action on the jet space: u ( n ) ) = pr ( n ) g · ( x, u ( n ) ) ( � x, � = ⇒ formulas provided by implicit differentiation Prolonged vector field or infinitesimal generator: � J ( x, u ( n ) ) ∂ ϕ α pr v = v + ∂u α J α,J
The coefficients of the prolonged vector field are given by the explicit prolongation formula: p � J = D J Q α + ξ i u α ϕ α J,i i =1 Q = ( Q 1 , . . . , Q q ) — characteristic of v p ξ i ∂u α � Q α ( x, u (1) ) = ϕ α − ∂x i i =1 ⋆ Invariant functions are solutions to Q ( x, u (1) ) = 0 .
Symmetry Criterion Theorem. (Lie) A connected group of transforma- tions G is a symmetry group of a nondegenerate system of differential equations ∆ = 0 if and only if pr v (∆) = 0 ( ∗ ) whenever u is a solution to ∆ = 0 for every infinitesi- mal generator v of G . (*) are the determining equations of the symmetry group to ∆ = 0. For nondegenerate systems, this is equivalent to � pr v (∆) = A · ∆ = A ν ∆ ν ν
Nondegeneracy Conditions Maximal Rank: � � · · · ∂ ∆ ν · · · ∂ ∆ ν rank · · · = max ∂u α ∂x i J Local Solvability: Any each point ( x 0 , u ( n ) 0 ) such that ∆( x 0 , u ( n ) 0 ) = 0 there exists a solution u = f ( x ) with = pr ( n ) f ( x 0 ) u ( n ) 0 Nondegenerate = maximal rank + locally solvable
Normal: There exists at least one non-characteristic di- rection at ( x 0 , u ( n ) 0 ) or, equivalently, there is a change of variable making the system into Kovalevskaya form ∂ n u α ∂t n = Γ α ( x, � u ( n ) ) Theorem. (Finzi) A system of q partial differential equations ∆ = 0 in q unknowns is not normal if and only if there is a nontrivial integrability condition: � D ∆ = D ν ∆ ν = Q order Q < order D + order ∆ ν
Under-determined: The integrability condition follows from lower order derivatives of the equation: � D ∆ ≡ 0 Example: ∆ 1 = u xx + v xy , ∆ 2 = u xy + v yy D x ∆ 2 − D y ∆ 1 ≡ 0 Over-determined: The integrability condition is genuine. Example: ∆ 1 = u xx + v xy − v y , ∆ 2 = u xy + v yy + u y D x ∆ 2 − D y ∆ 1 = u xy + v yy
A Simple O.D.E. u xx = 0 Infinitesimal symmetry generator: v = ξ ( x, u ) ∂ ∂x + ϕ ( x, u ) ∂ ∂u Second prolongation: v (2) = ξ ( x, u ) ∂ ∂x + ϕ ( x, u ) ∂ ∂u + + ϕ 1 ( x, u (1) ) ∂ ∂ + ϕ 2 ( x, u (2) ) , ∂u x ∂u xx
ϕ 1 = ϕ x + ( ϕ u − ξ x ) u x − ξ u u 2 x , ϕ 2 = ϕ xx + (2 ϕ xu − ξ xx ) u x + ( ϕ uu − 2 ξ xu ) u 2 x − − ξ uu u 3 x + ( ϕ u − 2 ξ x ) u xx − 3 ξ u u x u xx . Symmetry criterion: ϕ 2 = 0 whenever u xx = 0 .
Symmetry criterion: ϕ xx + (2 ϕ xu − ξ xx ) u x + ( ϕ uu − 2 ξ xu ) u 2 x − ξ uu u 3 x = 0 . Determining equations: ϕ xx = 0 2 ϕ xu = ξ xx ϕ uu = 2 ξ xu ξ uu = 0 = ⇒ Linear! General solution: ξ ( x, u ) = c 1 x 2 + c 2 xu + c 3 x + c 4 u + c 5 ϕ ( x, u ) = c 1 xu + c 2 u 2 + c 6 x + c 7 u + c 8
Symmetry algebra: v 1 = ∂ x v 5 = u∂ x v 2 = ∂ u v 6 = u∂ u v 7 = x 2 ∂ x + xu∂ u v 3 = x∂ x v 8 = xu∂ x + u 2 ∂ u v 4 = x∂ u Symmetry Group: � ax + bu + c � hx + ju + k, dx + eu + f ( x, u ) �− → hx + ju + k = ⇒ projective group
Prolongation Infinitesimal symmetry v = ξ ( x, t, u ) ∂ ∂x + τ ( x, t, u ) ∂ ∂t + ϕ ( x, t, u ) ∂ ∂u First prolongation pr (1) v = ξ ∂ ∂x + τ ∂ ∂t + ϕ ∂ ∂u + ϕ x ∂ + ϕ t ∂ ∂u x ∂u t Second prolongation ∂ ∂ ∂ pr (2) v = pr (1) v + ϕ xx + ϕ xt + ϕ tt ∂u xx ∂u xt ∂u tt
where ϕ x = D x Q + ξu xx + τu xt ϕ t = D t Q + ξu xt + τu tt ϕ xx = D 2 x Q + ξu xxt + τu xtt Characteristic Q = ϕ − ξu x − τu t
ϕ x = D x Q + ξu xx + τu xt = ϕ x + ( ϕ u − ξ x ) u x − τ x u t − ξ u u 2 x − τ u u x u t ϕ t = D t Q + ξu xt + τu tt = ϕ t − ξ t u x + ( ϕ u − τ t ) u t − ξ u u x u t − τ u u 2 t ϕ xx = D 2 x Q + ξu xxt + τu xtt = ϕ xx + (2 φ xu − ξ xx ) u x − τ xx u t + ( φ uu − 2 ξ xu ) u 2 x − 2 τ xu u x u t − ξ uu u 3 x − − τ uu u 2 x u t + ( ϕ u − 2 ξ x ) u xx − 2 τ x u xt − 3 ξ u u x u xx − τ u u t u xx − 2 τ u u x u xt
Heat Equation u t = u xx Infinitesimal symmetry criterion ϕ t = ϕ xx whenever u t = u xx
Determining equations Coe ffi cient Monomial 0 = − 2 τ u u x u xt 0 = − 2 τ x u xt u 2 0 = − τ uu x u xx − ξ u = − 2 τ xu − 3 ξ u u x u xx ϕ u − τ t = − τ xx + ϕ u − 2 ξ x u xx u 3 0 = − ξ uu x u 2 0 = ϕ uu − 2 ξ xu x − ξ t = 2 ϕ xu − ξ xx u x ϕ t = ϕ xx 1
General solution ξ = c 1 + c 4 x + 2 c 5 t + 4 c 6 xt τ = c 2 + 2 c 4 t + 4 c 6 t 2 ϕ = ( c 3 − c 5 x − 2 c 6 t − c 6 x 2 ) u + α ( x, t ) α t = α xx
Symmetry algebra v 1 = ∂ x space transl. v 2 = ∂ t time transl. v 3 = u∂ u scaling v 4 = x∂ x + 2 t∂ t scaling v 5 = 2 t∂ x − xu∂ u Galilean v 6 = 4 xt∂ x + 4 t 2 ∂ t − ( x 2 + 2 t ) u∂ u inversions v α = α ( x, t ) ∂ u linearity
Potential Burgers’ equation u t = u xx + u 2 x Infinitesimal symmetry criterion ϕ t = ϕ xx + 2 u x ϕ x
Determining equations Coe ffi cient Monomial 0 = − 2 τ u u x u xt 0 = − 2 τ x u xt u 2 − τ u = − τ u xx u 2 − 2 τ u = − τ uu − 3 τ u x u xx − ξ u = − 2 τ xu − 3 ξ u − 2 τ x u x u xx ϕ u − τ t = − τ xx + ϕ u − 2 ξ x u xx u 4 − τ u = − τ uu − 2 τ u x u 3 − ξ u = − 2 τ xu − ξ uu − 2 τ x − 2 ξ u x u 2 ϕ u − τ t = − τ xx + ϕ uu − 2 ξ xu + 2 ϕ u − 2 ξ x x − ξ t = 2 ϕ xu − ξ xx + 2 ϕ x u x ϕ t = ϕ xx 1
General solution ξ = c 1 + c 4 x + 2 c 5 t + 4 c 6 xt τ = c 2 + 2 c 4 t + 4 c 6 t 2 ϕ = c 3 − c 5 x − 2 c 6 t − c 6 x 2 + α ( x, t ) e − u α t = α xx
Symmetry algebra v 1 = ∂ x v 2 = ∂ t v 3 = ∂ u v 4 = x∂ x + 2 t∂ t v 5 = 2 t∂ x − x∂ u v 6 = 4 xt∂ x + 4 t 2 ∂ t − ( x 2 + 2 t ) ∂ u v α = α ( x, t ) e − u ∂ u Hopf-Cole w = e u maps to heat equation.
Symmetry–Based Solution Methods Ordinary Differential Equations • Lie’s method • Solvable groups • Variational and Hamiltonian systems • Potential symmetries • Exponential symmetries • Generalized symmetries
Partial Differential Equations • Group-invariant solutions • Non-classical method • Weak symmetry groups • Clarkson-Kruskal method • Partially invariant solutions • Differential constraints • Nonlocal Symmetries • Separation of Variables
Integration of O.D.E.’s First order ordinary differential equation du dx = F ( x, u ) Symmetry Generator: v = ξ ( x, u ) ∂ ∂x + ϕ ( x, u ) ∂ ∂u Determining equation ϕ x + ( ϕ u − ξ x ) F − ξ u F 2 = ξ ∂F ∂x + ϕ ∂F ∂u ♠ Trivial symmetries ϕ ξ = F
Method 1: Rectify the vector field. v | ( x 0 ,u 0 ) � = 0 Introduce new coordinates y = η ( x, u ) w = ζ ( x, u ) near ( x 0 , u 0 ) so that v = ∂ ∂w These satisfy first order p.d.e.’s ξ η x + ϕ η u = 0 ξ ζ x + ϕ ζ u = 1 Solution by method of characteristics: dx ϕ ( x, u ) = dt du ξ ( x, u ) = 1
The equation in the new coordinates will be invariant if and only if it has the form dw dy = h ( y ) and so can clearly be integrated by quadrature.
Method 2: Integrating Factor If v = ξ ∂ x + ϕ ∂ u is a symmetry for P ( x, u ) dx + Q ( x, u ) du = 0 then 1 R ( x, u ) = ξ P + ϕ Q is an integrating factor. ♠ If ϕ ξ = − P Q then the integratimg factor is trivial. Also, rectification of the vector field is equivalent to solving the original o.d.e.
Higher Order Ordinary Differential Equations ∆( x, u ( n ) ) = 0 If we know a one-parameter symmetry group v = ξ ( x, u ) ∂ ∂x + ϕ ( x, u ) ∂ ∂u then we can reduce the order of the equation by 1.
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