Matrix generalizations of integrable systems with Lax - PowerPoint PPT Presentation

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Matrix generalizations of integrable systems with Lax integro-differential representations Chvartatskyi O. 1 advisor Yu. M. Sydorenko 1 1 IVAN FRANKO NATIONAL UNIVERSITY OF LVIV XIV-th International Conference Geometry, Integrability and Quantization Varna, Bulgaria, 8–13 June, 2012

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Contents 1 Introduction 2 Symmetry reductions of the KP–hierarchy 3 Exact solutions of some nonlinear models from the KP-hierarchy 4 Integro-differential Lax representations for Davey-Stewartson and Chen-Lee-Liu equations

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Introduction We consider linear space ζ of micro-differential operators over the field С of the following form:   n ( L )  a i D i : n ( L ) ∈ Z  � L ∈ ζ = (1)  ,  i = −∞ where coefficients a i are functions dependent on spatial variable x = t 1 and evolution parameters t 2 , t 3 . . . . Coefficients a i ( t ) , t = ( t 1 , t 2 , . . . ) , are supposed to be smooth functions of vector variable t that has a finite number of elements that belong to some functional space A . This space is a differential algebra under arithmetic operations. An operator of differentiation is ∂ denoted in the following way: D := ∂ x .

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Introduction Addition and multiplication of operators by scalars (elements of the field C) are introduced in the following way: N 1 N 2 λ 1 a 1 i D i ± λ 2 a 2 i D i = � � λ 1 L 1 ± λ 2 L 2 = i = −∞ i = −∞ max < N 1 , N 2 > � ( λ 1 a 1 i ± λ 2 a 2 i ) D i , λ 1 , λ 2 ∈ C . i = −∞ The structure of Lie algebra on a linear space ζ (1) is defined by the commutator [ · , · ] : ζ × ζ → ζ , [ L 1 , L 2 ] = L 1 L 2 − L 2 L 1 , where the composition of micro-differential operators L 1 and L 2 is induced by general Leibniz rule:

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Introduction ∞ � n � D n f := � f ( j ) D n − j , (2) j j = 0 n ∈ Z , f ∈ A ⊂ ζ , f ( j ) := ∂ j f ∂ x j ∈ A ⊂ ζ, D n D m = D m D n = D n + m , � n � � := n ( n − 1 ) ... ( n − j + 1 ) n � n , m ∈ Z , where := 1 , . 0 j j ! Formula (2) defines the composition of the operator D n ∈ ζ and the operator of multiplication by function f ∈ A ⊂ ζ in contradistinction to the denotation D k { f } := ∂ k f ∂ x k ∈ A , k ∈ Z + .

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Introduction Consider a microdifferential Lax operator: ∞ L := WDW − 1 = D + � U i D − i , (3) i = 1 which is parametrized by the infinite number of dynamic variables U i = U i ( t 1 , t 2 , t 3 , ... ) , i ∈ N , which depend on an arbitrary (finite) number of independent variables t 1 := x , t 2 , t 3 , ... All dynamic variables U i can be expressed in terms of functional coefficients of formal dressing Zakharov-Shabat operator: ∞ � w i D − i , W = I + (4) i = 1 The inverse of formal operator W has the form: ∞ W − 1 = I + � a i D − i . (5) i = 1

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Introduction In scalar case, Kadomtsev -Petviashvili hierarchy is a commuting family of evolution Lax equations for the operator L (3) α i L t i = [ B i , L ] := B i L − LB i , (6) where α i ∈ C , i ∈ N , the operator B i := ( L i ) + is a differential part of the i -th power of microdifferential symbol L . By symbol L t i we will denote the following operator: ∞ L t i := ( WDW − 1 ) t i = � ( U j ) t i D − j . (7) j = 1 Formally transposed and conjugated operators L τ , L ∗ have the form:

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Introduction ∞ L τ := − D + ( − 1 ) j D − j U j , L ∗ := ¯ � L τ . (8) j = 1 Zakharov-Shabat equations are consequences of the commutativity of two arbitrary flows in (6) with i = m and i = n L t m t n = L t n t m ⇒ ⇒ [ α n ∂ t n − B n , α m ∂ t m − B m ] = α m B nt m − α n B mt n +[ B n , B m ] = 0 . (9)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Symmetry reductions of the KP–hierarchy References Sidorenko Yu., Strampp W. Symmetry constraints of the KP–hierarchy // Inverse Problems. – 1991. – V. 7. – P. L37-L43. Konopelchenko B., Sidorenko Yu., Strampp W. (1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems // Phys. Lett. A. – 1991. – V. 157. – P. 17-21. Cheng Yi. Constrained of the Kadomtsev – Petviashvili hierarchy // J. Math. Phys. – 1992. – Vol.33. – P. 3774-3787. Sidorenko Yu., Strampp W. Multicomponent integrable reductions in Kadomtsev-Petviashvilli hierarchy // J. Math. Phys. – 1993. – V. 34, №4. – P. 1429-1446.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Symmetry reductions of the KP–hierarchy References A. M. Samoilenko, V. G. Samoilenko and Yu. M. Sidorenko Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system // Ukr. Math. Journ., 1999, Vol. 51, № 1, p. 86-106

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Symmetry reductions of the KP–hierarchy Consider a symmetry reduction of the KP-hierarchy, which is a generalization of the Gelfand-Dickey k-reduction: ( L k ) − := ( L k ) < 0 = q M 0 D − 1 r ⊤ = � x q ( x , t 2 , t 3 , . . . ) M 0 r ⊤ ( s , t 2 , t 3 , . . . ) · ds , = (10) where Mat l × l ( C ) ∋ M 0 is a constant matrix, and functions q = ( q 1 , ..., q l ) , r = ( r 1 , ..., r l ) are fixed solutions of the following system of differential equations: � α n q t n = B n { q } , (11) α n r t n = − B τ n { r } , where n ∈ N .

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Symmetry reductions of the KP–hierarchy Reduced flows (6), (10), (11) admit Lax representation [ L k , M n ] = 0 , L k = B k + q M 0 D − 1 r ⊤ , M n = α n ∂ t n − B n . (12) Equation (12) is equivalent to the ( 1 + 1 )-dimensional integrable systems for functional coefficients U i , i = 1 , k − 1 and functions q , r : � U it n = P in [ U 1 , U 2 , ..., U k − 1 , q , r ] , (13) q t n = B n [ U i , q , r ] { q } , r t n = − B τ n [ U i , q , r ] { r } , where i = 1 , k − 1 , P in and B n are differential polynomials with respect to dynamic variables that are indicated in square brackets.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Symmetry reductions of the KP–hierarchy (2+1)-dimensional generalizations of Lax representations (12) have the form: [ L k , M n ] = 0 , (14) where L k is (2+1)-dimensional integro-differential operator: L k = α∂ y − B k − q M 0 D − 1 r ⊤ , (15) and M n in (14) is evolutional differential operator of n -th order with respect to spatial variable x : n � v j D j M n = α n ∂ t n − (16) j = 1

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Symmetry reductions of the KP–hierarchy Consider examples of equations (12)-(13) and their generalizations (14)-(16) for some k and n : 1. k = 1 , n = 2 : L 1 = D + q M 0 D − 1 q ∗ , M 2 = α 2 ∂ t 2 − D 2 − 2 q M 0 q ∗ , where α 2 ∈ i R , M ∗ 0 = M 0 . Equation [ L 1 , M 2 ] = 0 is equivalent to nonlinear Schrodinger equation (NLS): α 2 q t 2 = q xx + 2 ( q M 0 q ∗ ) q . (17)

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Symmetry reductions of the KP–hierarchy Now let us consider spatially two-dimesional generalizations of the operators L 1 , M 2 : L 1 = ∂ y − q M 0 D − 1 q ∗ , (18) M 2 = α 2 ∂ t 2 − c 1 D 2 − 2 c 1 S 1 , where α 2 ∈ i R , S 1 = S 1 ( x , y , t 2 ) = ¯ S 1 ( x , y , t 2 ) , c 1 ∈ R Lax equation [ L 1 , M 2 ] = 0 is equivalent to Davey-Stewartson system DS-III: � α 2 q t 2 = c 1 q xx − 2 c 1 S 1 q (19) S 1 y = ( q M 0 q ∗ ) x System (19) is spatially two-dimensional l –component generalization of NLS.

Introduction Symmetry reductions of the KP–hierarchy Exact solutions of some nonlinear models from the KP-hi Symmetry reductions of the KP–hierarchy 2. k = 2 , n = 2 : L 2 = D 2 + 2 u + q M 0 D − 1 q ∗ , M 2 = α 2 ∂ t 2 − D 2 − 2 u , where M ∗ 0 = −M 0 , u = ¯ u , α 2 ∈ i R . Operator equation [ L 2 , M 2 ] = 0 is equivalent to Yajima-Oikawa system: � α 2 q t 2 = q xx + 2 u q , (20) α 2 u t 2 = ( q M 0 q ∗ ) x .