Recursion operator for the Narita-Itoh-Bogoyavlensky lattice Jing - PowerPoint PPT Presentation

Recursion operator for the Narita-Itoh-Bogoyavlensky lattice Jing Ping Wang School of Mathematics and Statistics University of Kent “Solitons, Collapses and Turbulence: Achievements, Developments and perspectives“ Novosibirsk, June 8, 2012 1

Evolutionary differential-difference equations u t = K ( u q , u q +1 , · · · u p ) , q, p ∈ Z , q ≤ j ≤ p u j = S j u ( n, t ) = u ( n + j, t ) u t = ∂ t u, The order of K is ( q, p ) is ∂ u q K∂ u p K � = 0 and its total order p − q . The Volterra Chain u t = u ( u 1 − u − 1 ) is of order ( − 1 , 1) with total order 2. 2

Motivations • Integrable discretisation of integrable systems Example. The equation u t = u 2 ( u 2 u 1 − u − 1 u − 2 ) − u ( u 1 − u − 1 ) is of oder (-2,2) and it can be interpreted as the Sawada- Kotera equation U τ = U xxxxx + 5 UU xxx + 5 U x U xx + 5 U 2 U x under the following continuous limit at ǫ → 0: 3 + ǫ 2 9 ǫt, τ + 2 ǫ 5 u ( n, t ) = 1 9 U ( x − 4 135 t ) , x = ǫn. ( Alder: arXiv:11035139) 3

• Generalised symmetry of discrete equations Example. The discrete Korteweg-de Vries equation ( u 1 , 1 − u 0 , 0 )( u 1 , 0 − u 0 , 1 ) = α − β possesses a generalised symmetry of order ( − 1 , 1) : 1 u τ = . u 1 , 0 − u − 1 , 0 This can be transformed into the modified Volterra chain v τ = v 2 ( v 1 − v − 1 ) , 1 where v = u 1 , 0 − u − 1 , 0 . 4

• Classification problems are still open The following types have been classified: 1. Volterra type: u t = f ( u − 1 , u, u 1 ); 2. Toda type: u tt = f ( u t , u − 1 , u, u 1 ); 3. Relativistic Toda-Type: u t = f ( u 1 , u, v ) , v t = g ( v − 1 , v, u ) and u tt = f ( u 1 , u, u 1 ,t , u t ) − g ( u, u − 1 , u t , u − 1 ,t ) 5

Complex of variational calculus U s = { u n | n ∈ Z } F s = { smooth functions of variables U s } [ g ] an equivalent class: g ≡ h ⇔ g − h ∈ Im ∆, ∆ = S− 1; F ′ s : the space of equivalent classes Lie algebra h : the space of evolutionary vector fields. [ ∂, S ]=0 ∂ ∂ k ∈ Z S k P · ∂ = � k ∈ Z h k · − → ∂ P = � ∂u k = ⇒ h ∂u k F ′ s is a h -module with a representation as follows: k ∈ Z ( S k P ) ∂g ∂u k ] , P ∈ h , g ∈ F ′ P ◦ g = [ ∂ P ( g )] = [ � s 6

What is the space Ω n ? Ω 0 = F ′ s A natural non-degenerate pairing between ∂ P and k h k · d u k : a vertical 1-form ω = � h ( n ) S n P ] = < S − n h ( n ) , P > . � � < ω, P > = [ n ∈ Z n ∈ Z n S − n h ( n ) d u 0 = ⇒ Ω 1 ω → ξ · d u, ξ = � S − k ∂g d : Ω 0 → Ω 1 = � ⇒ δ ( g ) = ∂u k k 7

Fr´ echet derivatives and Lie derivatives Def . For any objects in the complex O , its Fr´ echet derivative along a vector field P ∈ h is defined as D O [ P ] = d � � ǫ =0 O [ u + ǫP ] . � d ǫ Eg . For H = u ( S − S − 1 ) u , D H [ P ] = P ( S − S − 1 ) u + u ( S − S − 1 ) P . Thm . Let L K denote Lie derivative along K ∈ h . Then L K g = [ D g [ K ]] ∈ F ′ g ∈ F ′ for s ; → conserved density s L K h = [ K, h ] for h ∈ h ; → symmetry L K ξ = D ξ [ K ] + D ⋆ K ( ξ ) for ξ ∈ Ω 1 ; → cosymmetry L K R = D R [ K ] − D K R + R D K for R : h → h ; → recursion Op. K for H : Ω 1 → h ; → Hamiltonian L K H = D H [ K ] − D K H−H D ⋆ L K I = D I [ K ] + D ⋆ K I + I D K for I : h → Ω 1 . → symplectic 8

All results related about concepts for evolutionary partial differential equations are valid for evolutionary differential-difference equations. A recursion operator of Volterra chain ℜ = u S + u + u 1 + u S − 1 + u t ( S − 1) − 1 1 u generating local symmetries of order ( − n, n ) , e.g. u t 1 = u ( u 1 − u − 1 ) u t 2 = uu 1 ( u + u 1 + u 2 ) − u − 1 u ( u − 2 + u − 1 + u ) · · · · · · 9

Conservation laws A pair of functions ( ρ, σ ) is called a conservation law of an equation u t = K if � D t ρ = ( S − 1) σ � u t = K . � The functions ρ and σ are called the density and flux of the conservation law respectively. The Volterra chain u t = ( S − 1) � uu − 1 � ∂ t ln u = u t � u + u − 1 � u = u 1 − u − 1 = ( S − 1) · · · · · · 10

Residues and Adler’s Theorem Consider Laurent formal difference series of order N A = a N S N + a N − 1 S N − 1 · · · The residue res( A ) and the logarithmic residue res ln( A ) are defined as res( A ) = a 0 , res ln( A ) = ln( a N ) . Adler’s Theorem Let A and B be two Laurent formal difference series of order N and M respectively. Then res[ A, B ] = ( S − 1 )( σ ( A, B )) , where N i M i S − k ( a − i ) S i − k ( b i ) − S − k ( b − i ) S i − k ( a i ) . � � � � σ ( A, B ) = i =1 i =1 k =1 k =1 11

Infinitely many conserved densities Thm. Consider an equation u t = K . If there exists a series ℜ L such that D ℜ L [ K ] = [ D K , ℜ L ] , res( ℜ i L ) and res ln( ℜ L ) are its conserved densities. The Volterra chain u t ℜ L = u S + u + u 1 + u S − 1 + S − i � u − i i =1 ρ 0 = res ln( ℜ L ) = ln u ρ 1 res( ℜ L ) = u + u 1 ≡ 2 u L ) = 3 uu 1 + u 1 u 2 + u 2 + u 2 ρ 2 = res( ℜ 2 1 ≡ 4 uu 1 + 2 u 2 � D t ρ 2 = 4( S − 1)( u 2 u − 1 + u − 1 uu 1 ) � · · · · · · 12

Bi-Hamiltonian structures u t = H 1 δ u f = H 2 δ u g, where H 1 , H 2 are Hamiltonian operators and δ u is the variational derivative. The Volterra chain ln u u t = H 1 δ u u = H 2 δ u 2 , H 1 = u ( S − S − 1 ) u, H 2 = ℜH 1 = u (1 + S − 1 )( S u − u S − 1 )(1 + S ) u . 13

Narita-Itoh-Bogoyavlensky lattices (1980’s): p ∈ N p p � � u t = u ( u k − u − k ); k =1 k =1 p p � � v t = v ( v − k ); v k − k =1 k =1 p p w t = w 2 ( � � w k − w − k ) . k =1 k =1 p − 1 p � � u = and u = v k w k . k =0 k =0 For finite lattices, work has been done on Hamiltonian structures, associations with classical Lie algebras and the r -matrix structure etc (Suris, Nijhoff, Papageor- giou...). 14

Discrete Sawada-Kotera equation (dSK) ( Alder: arXiv:11035139): u t = u 2 ( u 2 u 1 − u − 1 u − 2 ) − u ( u 1 − u − 1 ) • Tsujimoto and Hirota (1996): continuous limit of the reduced discrete BKP hierarchy. • Both u t ′ = u ( u 1 − u − 1 ) and u t ′′ = u 2 ( u 2 u 1 − u − 1 u − 2 ) are integrable, but do not commute. • Lax representation: L = ( S + u ) − 1 ( u S + 1) S 2 A = ( u − 1 S + 1 − u − 1 u − 2 + u − 2 S − 1 )( S − S − 1 ). 15

Symmetries of dSK : u t := P 4 + P 2 u 2 ( u 1 u 2 2 u 3 u 4 + u 2 1 u 2 2 u 3 + uu 2 1 u 2 2 + u − 1 uu 2 1 u 2 − u − 2 u 2 − 1 uu 1 − u 2 − 2 u 2 − 1 u − u − 3 u 2 − 2 u 2 − 1 − u − 4 u − 3 u 2 − 2 u − 1 ) + · · · + u ( u 1 u 2 + u 2 1 + u 1 u − uu − 1 − u 2 − 1 − u − 1 u − 2 ) =: Q 7 + Q 5 + Q 3 ⇒ [ P 4 , Q 7 ] = 0; [ P 2 , Q 3 ] = 0 . = Cosymmetries: G 1 = 1 u , G 2 = u 1 u 2 + u 1 u − 1 + u − 1 u − 2 − 1 Questions : Hamiltonian strictures? Recursion opera- tors? The hierarchy dSK (Alder & Postnikov: arXiv:1107.2305) p − 1 p − 1 p p u t = u 2 ( � � � � u i − u − i ) − u ( u i − u − i ) i =1 i =1 i =1 i =1 16

What was known? • p = 1: The Volterra chain • p = 2: Zhang, Tu, Oevel & Fuchssteiner (1991) u t = u ( u 2 + u 1 − u − 1 − u − 2 ) = u ( S 2 + S − S − 1 − S − 2 ) uδ u u has a recursion operator ℜ = u (1 + S − 1 + S − 2 )( S 2 u − u S − 1 )( u S − 1 − S u ) − 1 ( u S − 2 − S u )(1 − S − 2 ) − 1 u − 1 • For arbitrary p , the equation is Hamiltonian: p p S k − S − k ) uδ u u. � � u t = u ( k =1 k =1 17

Main Results Thm. For any p ∈ N , a recursion operator of the Narita- Itoh-Bogoyavlensky lattice is p → p ( S p +1 − i u − u S − i )( S p − i u − u S − i ) − 1 . S − i ) � � ℜ = u ( i =0 i =1 It is a Hamiltonian equation with respect to → ( p − 1) p   ( S p +1 − i u − u S − i )( S p − i u − u S − i ) − 1 S − i ) � � ℜH = u (   i =0 i =1 p ( S u − u S − p )( S i ) u , � i =0 where H = u ( � p k =1 S k − � p k =1 S − k ) u . Indeed, 1 u t = p + 1 H δ u ln u . 18

Example . When p = 2, the equation is bi-Hamiltonian. u t = u ( u 2 + u 1 − u − 1 − u − 2 ) = u ( S 2 + S − S − 1 − S − 2 ) uδ u u = u 3(1 + S − 1 + S − 2 )( S 2 u − u S − 1 )( S u − u S − 1 ) − 1 ( S u − u S − 2 )(1 + S + S 2 ) uδ u ln u = u (1 + S − 1 + S − 2 )( S 2 u − u S − 1 )( S u − u S − 1 ) − 1 ( u 1 − u ) = u (1 + S − 1 + S − 2 )( u 2 − u ) = u ( u 2 − u + u 1 − u − 1 + u − u − 2 ) 19

Lax representation for Bogoyavlensky hierarchy B ( n ) = ( L ( p +1) n ) ≥ 0 L = S + u S − p , L t n = [ B ( n ) , L ] . Idea to construct a recursion operator : (Tu (’89); G¨ urses, Karasu & Sokolov (’99)) 1. Relate the difference operators B ( n ) : B ( n +1) = LB ( n ) + R with R is the reminder. 2. Find the relation between two flows corresponding to these two difference operators. 20