Symmetries and integrability conditions for difference equations - PowerPoint PPT Presentation

Symmetries and integrability conditions for difference equations A.V. Mikhailov University of Leeds, UK Joint work with Jing Ping Wang and P. Xenitidis DART IV, Beijing, 2010 1

• Difference equation Q = 0 and Dynamical variables. • Difference fields F Q , F 0 , F s , F t , the elimination map. • Symmetries and conservation laws. • Recursion operators for difference equations. • Formal difference series, the difference Adler Theorem. • Canonical conservation laws: integrability conditions. • Recursion and co-recursion operator for the Viallet equation. 2

Difference equations on Z 2 can be seen as a discrete analogue of partial differential equations with two inde- pendent variables. Let us denote by u = u ( n, m ) a complex-valued function u : Z 2 �→ C where n and m are “independent variables” and u will play the rˆ ole of a “dependent” variable in a difference equation. Instead of partial derivatives we have two commuting shift maps S and T defined as S : u �→ u 1 , 0 = u ( n +1 , m ) , T : u �→ u 0 , 1 = u ( n, m +1) For uniformity of notations, we denote u as u 0 , 0 . 3

In the theory of difference equations we shall treat symbols u p,q as commuting variables . We denote U = { u p,q | ( p, q ) ∈ Z 2 } . For a function f = f ( u p 1 ,q 1 , . . . , u p k ,q k ) S i T j ( f ) = f i,j = f ( u p 1 + i,q 1 + j , . . . , u p k + i,q k + j ) . A quadrilateral difference equation can be defined as Q ( u 0 , 0 , u 1 , 0 , u 0 , 1 , u 1 , 1 ) = 0 , where Q ( u 0 , 0 , u 1 , 0 , u 0 , 1 , u 1 , 1 ) is an irreducible polyno- mial of the “dependent variable” u = u 0 , 0 and its shifts. ( p, q ) ∈ Z 2 . Q p,q = Q ( u p,q , u p +1 ,q , u p,q +1 , u p +1 ,q +1 ) = 0 , 4

We shall assume that Q is an irreducible affine-linear polynomial which depends non-trivially on all variables: ∂ 2 ∂ u i,j Q � = 0 , u i,j Q = 0 , i, j ∈ { 0 , 1 } . Example: The Viallet equation Q := a 1 u 0 , 0 u 1 , 0 u 0 , 1 u 1 , 1 + a 2 ( u 0 , 0 u 1 , 0 u 0 , 1 + u 1 , 0 u 0 , 1 u 1 , 1 + u 0 , 1 u 1 , 1 u 0 , 0 + u 1 , 1 u 0 , 0 u 1 , 0 ) + a 3 ( u 0 , 0 u 1 , 0 + u 0 , 1 u 1 , 1 ) + a 4 ( u 1 , 0 u 0 , 1 + u 0 , 0 u 1 , 1 ) + a 5 ( u 0 , 0 u 0 , 1 + u 1 , 0 u 1 , 1 ) + a 6 ( u 0 , 0 + u 1 , 0 + u 0 , 1 + u 1 , 1 ) + a 7 = 0 , where a i are free complex parameters, such that Q is irreducible. 5

Let C [ U ], U = { u p,q | ( p, q ) ∈ Z 2 } be the ring of polyno- mials. S , T ∈ Aut C [ U ] and thus C [ U ] is a difference ring . The difference ideal J Q = �{ Q p,q | ( p, q ) ∈ Z 2 }� is prime and thus the quotient ring C [ U ] /J Q is an inte- gral domain. Solutions of the difference equation are points of the affine variety V ( J Q ). 6

Rational functions of variables u p,q on V ( J Q ) form a field F Q = { [ a ] / [ b ] | a, b ∈ C [ U ] , b �∈ J Q } , where [ a ] denotes the class of equivalent polynomials (two polynomials f, g ∈ C [ U ] are equivalent, denoted by f ≡ g , if f − g ∈ J Q ). For a, b, c, d ∈ C [ U ] , b, d �∈ J Q , rational functions a/b and c/d represent the same element of F Q if ad − bc ∈ J Q . The fields of rational functions of variables U s = { u n, 0 | n ∈ Z } , U t = { u 0 ,n | n ∈ Z } , U 0 = U s ∪ U t . are denoted respectively as F s = C ( U s ) , F t = C ( U t ) , F 0 = C ( U 0 ) . 7

In the affine-linear case we can uniquely resolve equa- tion Q = 0 with respect to each variable u 0 , 0 = F ( u 1 , 0 , u 0 , 1 , u 1 , 1 ) , u 1 , 0 = G ( u 0 , 0 , u 0 , 1 , u 1 , 1 ) , u 0 , 1 = H ( u 0 , 0 , u 1 , 0 , u 1 , 1 ) , u 1 , 1 = M ( u 0 , 0 , u 1 , 0 , u 0 , 1 ) . We can recursively and uniquely express any variable u p,q in terms of the variables U 0 = U s ∪ U t . For example (H1 or potential KdV equation): Q = ( u 0 , 0 − u 1 , 1 )( u 1 , 0 − u 0 , 1 ) − α, α � = 0 , α ∈ C , α u 0 , 0 = u 1 , 1 + = F ( u 1 , 0 , u 0 , 1 , u 1 , 1 ) , u 1 , 0 − u 0 , 1 α u 1 , 0 = u 0 , 1 + = G ( u 1 , 0 , u 0 , 1 , u 1 , 1 ) . u 0 , 0 − u 1 , 1 8

Definition 1. For elements of U the elimination map E : U �→ C ( U 0 ) is defined recursively: E ( u 0 ,p ) = u 0 ,p , E ( u p, 0 ) = u p, 0 , ∀ p ∈ Z , if p > 0 , q > 0 , E ( u p,q ) = M ( E ( u p − 1 ,q − 1 ) , E ( u p,q − 1 ) , E ( u p − 1 ,q )) , if p < 0 , q > 0 , E ( u p,q ) = H ( E ( u p,q − 1 ) , E ( u p +1 ,q − 1 ) , E ( u p +1 ,q )) , if p > 0 , q < 0 , E ( u p,q ) = G ( E ( u p − 1 ,q ) , E ( u p − 1 ,q +1 ) , E ( u p,q +1 )) , if p < 0 , q < 0 , E ( u p,q ) = F ( E ( u p +1 ,q ) , E ( u p,q +1 ) , E ( u p +1 ,q +1 )) . For polynomials f ( u p 1 ,q 1 , . . . , u p k ,q k ) ∈ C [ U ] the elimina- tion map E : C [ U ] �→ C ( U 0 ) is defined as E : f ( u p 1 ,q 1 , . . . , u p k ,q k ) �→ f ( E ( u p 1 ,q 1 ) , . . . , E ( u p k ,q k )) ∈ C ( U 0 ) . For rational functions a/b, a, b ∈ C [ U ] , b �∈ J Q the elim- ination map E is defined as E : a/b �→ E ( a ) / E ( b ) . 9

Variables U 0 we shall call the dynamical variables. E : C [ U ] �→ C ( U 0 ) is a difference ring homomorphism Ker E = J Q , Im E ∼ C [ U ] /J Q . The field C ( U 0 ) is a difference field with automor- phisms E ◦ S and E ◦ T . The map E : F Q �→ C ( U 0 ) is a difference field isomor- phism . Two rational functions f, g of variables U are equivalent (i.e. represent the same element of F Q ): f ≡ g ⇔ E ( f ) = E ( g ) 10

Symmetries and conservation laws of difference equations Definition 2. Let Q = 0 be a difference equation. Then K ∈ F Q is called a symmetry (a generator of an in- finitesimal symmetry) of the difference equation if D Q ( K ) ≡ 0 . Here D Q is the Fr´ echet derivative of Q defined as Q u i,j S i T j , ∑ D Q = Q u i,j = ∂ u i,j Q. i,j ∈ Z One has to check is that E ( D Q ( K )) = 0. 11

If K is a symmetry and u = u ( n, m ) is a solution of a difference equation Q = 0, then the infinitesimal trans- formation of solution u : ˆ u = u + ǫK satisfies equation u 1 , 1 ) ≡ O ( ǫ 2 ) . Q (ˆ u 0 , 0 , ˆ u 1 , 0 , ˆ u 0 , 1 , ˆ If the difference equation Q = 0 admits symmetries, then they form a Lie algebra. With a symmetry K ∈ F Q we associate an evolution- ary derivation ( S X K = X K S , T X K = X K T ) of F Q (or a vector field on F Q ): ∂ K p,q = S p T q ( K ) ∑ X K = K p,q , ∂u p,q ( p,q ) ∈ Z 2 12

For any a ∈ J Q we have E ( X K ( a )) = 0 and thus the evolutionary derivation X K is defined correctly on F Q . X F X G − X G X F = X H , where H = [ F, G ] is also a symmetry, with [ F, G ] denot- ing the Lie bracket [ F, G ] = X F ( G ) − X G ( F ) = D G ( F ) − D F ( G ) ∈ F Q . The Lie algebra of symmetries of the difference equa- tion Q = 0 will be denoted as A Q . Existence of an infinite dimensional Lie algebra A Q for the field F Q is a characteristic property of integrable equations and can be taken as a definition of integra- bility . 13

• With a difference equation Q = 0 we associate the ideal J Q = �S n T m Q | ( n, m ) ∈ Z 2 � of C [ U ] generated by the polynomial Q and all its shifts. It is a differ- ence ideal . • Solution of a difference equation is a point on the difference affine variety V Q = V ( J Q ). • A good difference equation , i.e. an equation with the property of uniqueness of its solutions, gives rise to a prime difference ideal J Q . Therefore C [ U ] /J Q is integral domain and we define the corresponding difference field of fractions F Q . 14

• Continuous (or Lie–B¨ acklund) symmetries of a good difference equation Q = 0 are elements of the Lie algebra A of derivations S p T q ( K ) ∂ u pq , ∑ ∂ K = K ∈ F Q (1) ( p,q ) ∈ Z 2 of the difference field F Q . • A good difference equation Q = 0 we call inte- grable if the Lie algebra A has an infinite dimen- sional Abelian subalgebra.

Definition 3. (1) A pair ( ρ, σ ) ∈ ˆ F Q is called a conser- vation law for the difference equation Q = 0 , if ( T − 1 )( ρ ) ≡ ( S − 1 )( σ ) . Functions ρ and σ will be referred to as the density and the flux of the conservation law and 1 denotes the identity map. (2) A conservation law is called trivial , if functions ρ and σ are components of a (difference) gradient of some element H ∈ F Q , i.e. ρ = ( S − 1 )( H ) , σ = ( T − 1 )( H ) . (3) If ρ 1 − ρ 2 ∈ Im( S − 1 ) , then ρ 1 ∼ = ρ 2 . 15

Typically conserved densities belong to F s or F t . Euler’s operator gives a criteria to determine whether two elements of F s are equivalent or not. Definition 4. Let f ∈ F s has order ( N 1 , N 2 ) , then the variational derivative δ s of f is defined as N 2 ( ) ∂f S − k ∑ δ s ( f ) = . ∂u k, 0 k = N 1 For ρ, ̺ ∈ F s , ρ ∼ = ̺ ⇔ δ s ( ρ ) = δ s ( ̺ ) . If ρ is trivial then δ s ( ρ ) = 0. The order of a density ρ ∈ F s is defined as ord δ s ( ρ ) = N 2 − N 1 , where ( N 1 , N 2 ) = ord( δ s ( ρ )). 16

Recursion operators for difference equations Definition 5. (1) Elements of F Q [ S ] are called s -difference operators. (2) Elements of F Q ( S ) are called s -pseudo-difference operators. Similarly one can define t -difference and t -pseudo-difference operators. The action of a difference operator A ∈ F Q [ S ] on ele- ments of F Q is naturally defined and Dom( A ) = F Q . The domain of a pseudo-difference operator B ∈ F Q ( S ) is defined as Dom( B ) = { a ∈ F Q | B ( a ) ∈ F Q } . For instance, if B = FG − 1 where F, G ∈ F Q [ S ], then Dom( B ) = Im G . 17