Exact Solutions of Matrix Generalizations of Some Integrable Systems

Transcription

1 Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part 1, Exact Solutions of Matrix Generalizations of Some Integrable Systems Yuri BERKELA Franko National University of Lviv, Lviv, Ukraine Exact solutions of a matrix generalization of nonlinear Yajima Oikawa model are built in an explicit form. The Melnikov-like system was also integrated. 1 Introduction The hierarchy of Kadomtsev Petviashvili equations can be given as an infinite sequence of the Sato Wilson operator equations [1, 2] α n W tn = W D n W 1) W, n N, α n C, 1) where W =1+w 1 D 1 +w 2 D 2 + is a microdifferential operator MDO) with coefficients w i, i N, depending on the variables t =t 1,t 2,...), t 1 := x and D := x, DD 1 =1. Differential and integral parts of the microdifferential operator W D n W 1 are denoted by W D n W 1) + and W D n W 1) respectively. In the algebra MDO ζ: nl) ζ = a i D i : a i = a i t) A; i, nl) Z, i= the operation of multiplication is induced by the generalized Leibnitz rule ) n D n f := f j) D n j, n Z, D m f) := m f j x m = f m), m Z +, j=0 where D n D m := D m D n := D n+m, n, m Z, andf is the operator of multiplication by a function ft), which belongs to the same functional space A that the coefficients of microdifferential operators L ζ. With the aid of the MDO L is defined by formula L := W DW 1 = D + UD 1 + U 2 D 2 + system 1) can be rewritten in the form of the Lax representation α n L tn =[B n,l]:=b n L LB n, 2) where B n =L n ) + =WD n W 1 ) +, n N. Nonlocal reduced hierarchy of Kadomtsev Petviashvili is the system of operator equations 2) with the additional restriction so-called k-constraint oftheform[3,4,5,6,7]seealso[8]) L k := L r ) k = B k + l q i D 1 ri, where denotes transposition which is in accordance with i=1 dynamics of system 2), if field-variables q i, r i satisfy the system of the following equations: α n q itn = B n q i ), α n r itn = B τ nr i ), the symbol τ denotes the transposition of differential operator.

2 Exact Solutions of Matrix Generalizations of Some Integrable Systems 297 Equations from k reduced hierarchy of Kadomtsev Petviashvili allow the Lax representation [B k + qd 1 r,α n tn B n ] =0, n N. 3) 2 Exact solutions of a matrix generalization of Yajima Oikawa model In the present paper we consider the matrix case of 3): k =2,n = 2 and U 1 := U, U 2,U 3,... Mat N N C), q, r Mat N N C) and obtain the system: α 2 q t2 = q xx +2Uq, α 2 U t2 =qr ) x, α 2 r t2 = r xx 2rU. 4) Introduce the additional reductions of complex conjugation α 2 = i, t 2 = t, U = U := Ū, r = i qm, where M Mat N N C),M = M. System 4) can be represented as: iq t = q xx +2Uq, U t =qmq ) x. 5) System 5) is a matrix generalization of Yajima Oikawa model [9]. Operators of this system in the Lax representation [L, A] = 0) have the form: L = D 2 +2U + iqmd 1 q, A = i t D 2 2U. Proposition 1 [2, 10]). Let B = B + be a differential operator; fd 1 g, fd 1 g ζ. Then the following relations hold: BfD 1 g = BfD 1 g ) + + Bf)D 1 g, ) ), fd 1 g B = fd 1 g B + fd 1 B τ g + ) fd 1 g fd 1 g = f g f D 1 g ) fd 1 g f g. 6) In formulas 6) the symbol ) g f stands for an arbitrary fixed primitive of g f x, t 2 )as a function of x. Let ϕ, ψ be smooth complex matrix N K) functions of real variables x, t 2 R, C = C mn )=const Mat K K C), and also: 1) the improper integral ψ ϕds:= ψ s, t 2 )ϕs, t 2 ) ds converges absolutely x, t 2 ) R R + and admits differentiation by the parameter t 2 R + ; 2) the matrix-function Ωx, t 2 ):=C + ψ ϕds is nondegenerate in x, t 2 ) σ R R +. Define the functions Φ = Φx, t 2 ), Ψ = Ψx, t 2 )andmdow by the following way: Φ=ϕΩ 1, Ψ =Ω 1 ψ, W =1 ΦD 1 ψ. 7) Lemma 1. The components Φ ij, Ψ ij, i = 1,N, j = 1,K, of matrix functions Φ, Ψ 7) can be given as: Φ ij = ϕω 1) Ω j) ϕ i ij = 1)K+j, 8) Ω j) Ψ ij = ψω 1) ψ i ij = 1)K+j. 9) Here Ω j) is obtained from Ω by deletion of j-line; ϕ i, ψ i are i-lines of matrixes ϕ, ψ.

3 298 Yu. Berkela Proof. In order to prove 8), 9) we use a well-known algebraic equality for framed determinant: ) Ω ψ det j := ϕ i α Ω ψ j ϕ i α = α det Ω ϕ iω C ψj, where Ω C is the matrix of cofactors. Φ ij = ϕω 1) Ω j) ij = ϕ iω 1 e j = 1) K+j ϕ i. Here e i =e i1,...,e ik ), e ii =1,e ij =0fori, j = 1,K, i j. By the similar reasoning, formula 9) can be proved. Theorem 1 [10]). MDO W has an inverse operator W 1 and: W 1 =1+ϕD 1 Ψ. Proposition 2. For MDO W 7) the equalities are true: W D 2 W 1 = I ΦD 1 ψ ) D 2 I + ϕd 1 Ψ ) = D 2 +2 ϕω 1 ψ ) x ) ) ΦD ψ 1 xx ψssϕdsψ + ϕ xx Φ ψ ϕ ss ds D 1 Ψ, ϕω 1 ψ ) x W i t D 2) W 1 = i t D 2 2 { ) +ΦD 1 iψt + ψxx + iψ t { iϕ t ϕ xx ) Φ ψ iϕ t ϕ ss ) ds ) } + ψss ϕdsψ } D 1 Ψ. The proof of the Proposition 2 is based on the using of formulas 6) and the generalized Leibnitz rule. Consider operators L 0 = D 2, A 0 = i t D 2, ˆL = WL 0 W 1, Â = WA 0 W 1. Theorem 2. Let: a) ϕ be a solution of the equation iϕ t = ϕ xx ; b) ϕ xx = ϕλ, where Λ=diag λ 2 1,λ2 2,...,λ2 K) =const MatK K C); c) ψ = ϕ; d) C = C. Then 1) Ψ= Φ; 2) ˆL = D 2 +2 ϕω 1 ϕ ) x +ΦJD 1 Φ, where J = CΛ Λ C; 3) Â = i t D 2 2 ϕω 1 ϕ ) x. Proof. 1) From definitions 7) and condition d we have: Φ =ϕ C + ϕ ϕds) = ϕ C + ϕ ϕds) = ψ C + ϕ ψds) =Ψ. 2) From Proposition 2, condition b and properties: Φ ψ ϕds= ϕ ΦC and ψ ϕdsψ = ψ CΨ,

4 Exact Solutions of Matrix Generalizations of Some Integrable Systems 299 it follows that: ˆL = D 2 +2 ϕω 1 ϕ ) x ΦD 1 Λ ψ +ΦD 1 Λ ψ ϕdsψ + ϕλd 1 Ψ Φ ψ ϕdsλd 1 Ψ = D 2 +2 ϕω 1 ϕ ) x +ΦCΛ Λ C) D 1 Ψ. 3) The validity of this item follows from Proposition 2 and conditions a), c). Proposition 3. The matrix J =J mn ), m, n = 1,K has the following properties: 1) J = J ; ) 2) J mn = C mn λ 2 m λ 2 n ; 3) if the matrix C is diagonal, then: J =diag 2ic 1 Im λ 2 1, 2ic 2 Im λ 2 2,...,2ic K Im λ 2 K). Proof. 1) J =CΛ Λ C) =Λ C C Λ=Λ C CΛ = J. The proof of the 2), 3) is based on the using of formulas of operations with matrices. Corollary 1. Let the matrix J be defined by the following condition: ΦJΦ = iqmq, then the functions q =q ij ) and U =u kl ), i, k, l = 1,N, j = 1,K, where Ω j) q ij = 1) j+k ϕ i Ω ϕ, u kl = l ϕ k 0 are solutions of system 5). ) 1 The proof of corollary is based on the using of formula 8), equality for framed determinant and Theorem 2. Consider the simplest case of matrix equation 5): N =2,K = 1. Then ϕ 1 =ĉe λx iλ2t, ϕ 2 =ĉe λx iλ2t, M = µ R and under conditions Re λ>0, µ =4Reλ Im λ C solutions have the form: 4ĉ Re λ Im λe λx iλ2 t q 1 := q 11 = µ +4 ĉ 2 Im λ sinh 2 Re λ x) e4reλim λt, 4ĉ Re λ Im λe λx iλ2 t q 2 := q 21 = µ +4 ĉ 2 Im λ sinh2 Re λ x)e4reλim λt, 4 ĉ 2 Re λ Im λe2reλx+4 Re λ Im λt u 11 = µ +4 ĉ 2 Im λ sinh2 Re λ x)e4reλim λt, 4 ĉ 2 Re λ Im λe4reλim λt+2i Re λx u 12 =ū 21 = µ +4 ĉ 2 Im λ sinh2 Re λ x)e4reλim λt, 4 ĉ 2 Re λ Im λe 2Reλx+4 Re λ Im λt u 22 = µ +4 ĉ 2 Im λ sinh2 Re λ x)e4reλim λt. 3 Exact solutions of higher equation from matrix hierarchy of Yajima Oikawa Let us consider the following operators: L = D 2 +2U + iqmd 1 q, A = t D 3 3UD 3 2 U x 3 2 iqmq. x

5 300 Yu. Berkela The result of equation [L, A] = 0 will be the system: q t = q xxx +3Uq x U xq iµqq q, 10) U t = 1 4 U xxx +3UU x iµq xxq qq xx). 11) This system is a matrix generalization of Melnikov model [7, 11]. Proposition 4. For MDO W the equality is true: W t D 3) W 1 = t D 3 3 ϕω 1 ψ ) D x 3 ϕ xx Ω 1 ψ ϕω 1 ψxx + ϕω 1 ψx ϕω 1 ψ ϕ x Ω 1 ψ ) 2 { ) ) } +ΦD 1 ψt ψxxx ψt ψsss ϕdsψ { } + ϕ t ϕ xxx ) Φ ψ ϕ t ϕ sss ) ds D 1 Ψ. The proof of the proposition is based on the formulas 6). Consider operators L 0 = D 2, A 0 = t D 3, ˆL = WL 0 W 1, Â = WA 0 W 1. Theorem 3. Let: a) ϕ be a solution of the equation ϕ t = ϕ xxx ; b) ϕ xx = ϕλ, where Λ=diag λ 2 1,λ2 2 K),...,λ2 =const MatK K C); c) ψ = ϕ; d) C = C. Then 1) ˆL = D 2 +2 ϕω 1 ϕ ) x +ΦJD 1 Φ, where J = CΛ Λ C; 2) Â = t D 3 3 ϕω 1 ϕ ) x D 3 2 ϕxx Ω 1 ϕ ϕω 1 ϕ xx + ϕω 1 ϕ x ϕω 1 ϕ ϕ x Ω 1 ϕ ). Proof. 2) The validity of this item follows from Proposition 4 and conditions a), c). Remark 1. For system 10) the corollary of the previous part is true see above). Consider the case N =2,K = 1. Then ϕ 1 =ĉe λx+λ3t, ϕ 2 =ĉe λx λ3t, M = µ R and under conditions Re λ>0, µ =4Reλ Im λ C solutions will be of the form: q 1 := q 11 = q 2 := q 21 = u 11 = u 12 =ū 21 = u 22 = 4ĉ Re λ Im λe λx+λ3 t µ +4 ĉ 2 Im λ sinh 2Reλ x +2 Re 3 λ 3Reλ Im 2 λ ) t ), 4ĉ Re λ Im λe λx λ3 t µ +4 ĉ 2 Im λ sinh 2Reλ x +2 Re 3 λ 3Reλ Im 2 λ ) t ), 4 ĉ 2 Re λ Im λe 2Re λ x+re3 λ 3Reλ Im 2 λ)t) µ +4 ĉ 2 Im λ sinh 2Reλ x +2 Re 3 λ x 3Reλ Im 2 λ ) t ), 4 ĉ 2 Re λ Im λe 2iIm λ x+3re2 λ Im λ Im 3 λ)t) µ +4 ĉ 2 Im λ sinh 2Reλ x +2 Re 3 λ x 3Reλ Im 2 λ ) t ), 4 ĉ 2 Re λ Im λe 2Re λ x+re3 λ 3Reλ Im 2 λ)t) µ +4 ĉ 2 Im λ sinh 2Re λ x +2 Re 3 λ x 3Reλ Im 2 λ ) t ).

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