Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction

Transcription

1 Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction Kui Chen, Da-jun Zhang Department of Mathematics, Shanghai University, Shanghai , P.R. China June 25, 208 arxiv: v [nlin.si] 25 Apr 207 Abstract In this letter we propose an approach to obtain solutions for the nonlocal nonlinear Schrödinger hierarchy from the known ones of the Ablowitz-Kaup-Newell-Segur hierarchy by reduction. These solutions are presented in terms of double Wronskian and some of them are new. The approach is general and can be used for other systems with double Wronskian solutions which admit local and nonlocal reductions. MSC: 35Q5, 35Q55 PACS: Ik, Ks, Yv Keywords: nonlocal nonlinear Schrödinger hierarchy, Ablowitz-Kaup-Newell-Segur hierarchy, double Wronskian, solutions, reduction Introduction Recently integrable continuous and semidiscrete nonlocal nonlinear Schrödinger NLS equations describing wave propagation in nonlinear PT symmetric media were found [, 2] and have drown much attention. The nonlocal NLS hierarchy are derived from the Ablowitz-Kaup-Newell-Segur AKNS hierarchy through the reduction [] rx,t = q x,t where denotes complex conjugate. As for solutions, these nonlocal models have been solved through Inverse Scattering Transform, Darboux transformation, bilinear approach, etc. cf. [3 2]. In this letter, we propose an approach to construct solutions for the nonlocal NLS hierarchy directly from the known solutions of the AKNS hierarchy. These solutions are presented in terms of double Wronskian. Wronskian technique is an elegant way to construct and verify solutions for soliton equations, which is developed by Freeman and Nimmo [3]. The double Wronskian solutions for the focusing NLS equation was first given in [4]. In 990 Liu [5] showed the whole AKNS hierarchy admitted solutions in double Wronskian form. In this letter we directly apply the reduction to the solutions Corresponding author. djzhang@staff.shu.edu.cn

2 of the AKNS hierarchy obtained in [5] and construct solutions for the nonlocal focusing and defocusing NLS hierarchies. Some of solutions are new, which correspond to multiple poles of transmission coefficient. The letter is organized as follows. We will first in Sec.2 review known results on the AKNS hierarchy and their double Wronskian solutions. Then in Sec.3 we prove the condition under which double Wronskian solutions of the even order AKNS hierarchy are reduced to those of the nonlocal NLS hierarchies. In Sec.4 we present concrete Wronskian elements and some solutions. Finally, conclusion is given in Sec.5. 2 AKNS hierarchy and double Wronskian solutions It is well known that the AKNS spectral problem [6] φ λ q = r λ yields the isospectral AKNS hierarchy qtn q = K n = L n r r tn φ 2 x φ φ 2 2, n 0, 3 where L is a recursion operator x +2q L = 2r x r x r 2q x q x 2r x q, 4 in which x = x and x = x = x x =. An alternative form of 3 is qtn+ r tn+ with setting t = x. By introducing rational transformation = L qtn r tn, n 5 q = 2 g f, r = 2h f, 6 the hierarchy 5 can be written as the following bilinear form [7] 2D tn+ D x D tn g f = 0, 7a 2D tn+ D x D tn f h = 0, 7b Dx 2 f f = 8gh, 7c for n, where Hirota s bilinear operator D is defined as [8] D m x D n yfx,y gx,y = x x m y y n fx,ygx,y x =x,y =y. System 7 admit double Wronskian solutions [5, 9] f = W n+,m+ ϕ,ψ = ϕ n ; ψ m, g = W n+2,m ϕ,ψ = ϕ n+ ; ψ m, h = W n,m+2 ϕ,ψ = ϕ n ; ψ m+. 8a 8b 8c 2

3 Here, ϕ and ψ are respectively n+m+2-th order column vectors defined by with t = x ϕ = ϕ,ϕ 2,,ϕ n+m+2 T, ψ = ψ,ψ 2,,ψ n+m+2 T 9 ϕ = exp A j t j C, ψ = exp where A is a n+m+2 n+m+2 constant matrix A j t j C, 0 A = k ij n+m+2 n+m+2, k ij C, a and C,C are n+m+2-th order constant column vectors C = c,c 2,,c n+m+2 T, C = c,c 2,,c n+m+2 T ; b ϕ n and ψ m respectively denote n+m+2 n+ and n+m+2 m+ Wronski matrices ϕ n = ϕ, x ϕ, 2x ϕ,, nx ϕ, 2a ψ m = ψ, x ψ, 2x ψ,, mx ψ. 2b Return to the AKNS hierarchy 3. For a concrete system in this hierarchy, for example, the second order AKNS system q t2 = q xx +2q 2 r, r t2 = r xx 2r 2 q, 3a 3b one can treat t j j > 2 as dummy variables which are then absorbed into C and C, and present their solutions through 6 and 8 where ϕ and ψ are defined as [20] ϕ = expax+a 2 t 2 C, ψ = exp Ax A 2 t 2 C. 4 With the same idea, for the even order AKNS hierarchy qtn q = K n = L n, n = 2,4,6,, 5 r r tn their solutions can be described through 6 and 8 where ϕ = exp Ax+ A 2j t 2j C, ψ = exp Ax A 2j t 2j C. 6 3 Reduction to the nonlocal NLS hierarchies The nonlocal nonlinear Schrödinger hierarchies [] iqx τ2k = K ± k [qx,q x], k =,2,3,

4 can be reduced from the even order AKNS hierarchy 5 with the reductions rx = ±q x, t 2k = iτ 2k, x,τ 2k R, 8 where the ± in K ± k corresponds to the sign ± in 8. For the case of k =, the nonlocal focusing NLS equation reads and the nonlocal defocusing NLS equation reads For k = 2 we have and iqx τ2 = K = qx xx +2qx 2 q x, 9 iqx τ2 = K + = qx xx 2qx 2 q x. 20 iqx τ4 = K 2 =qx xxxx +8qxqx x q x+6qx x 2 q x +4qxqx x q x x +2qx 2 q x xx +6qx 3 q x 2, iqx τ4 = K + 2 =qx xxxx 8qxqx x q x 6qx x 2 q x 4qxqx x q x x 2qx 2 q x xx +6qx 3 q x 2. In the following, we derive solutions for the nonlocal hierarchies 7 by imposing the reduction relations 8 on the solutions of the even order AKNS hierarchy 5. Theorem. The nonlocal isospectral NLS hierarchies 7 admit the following solutions qx = 2 ϕn+, ψ n ϕ n, ψ, 2 n where ϕ and ψ are 2n+2-th order column vectors i.e. m = n in 9, defined by 6 with t 2k = iτ 2k and satisfy ψx = Tϕ x, C = TC, in which T is a constant matrix determined through AT = TA, TT = δi, δ =. 22a 22b 23a 23b Proof. We are going to prove that with the assumptions 22 and 23, q and r defined by 6 with double Wronskians 8 satisfy the reduction 8. To achieve that, first, under assumptions of 22b and 23a, it is easy to verify that ψx =exp Ax i A 2j τ 2j C =exp TA T x i =Texp Ax+i =Tϕ x. TA T 2j τ 2j TC A 2j τ 2j C 4

5 Next, to examine relation between Wronskians, based on 2, we introduce notation ϕ n ax [bx] = ϕax, bx ϕax, bxϕax, 2, bxϕax n and similar one for ψ m ax bx, where a,b = ±. With such notations, setting n = m and noticing the constraint 22a, f,g,h in 8 are expressed as fx = ϕ n x [x] ; ψ n x [x] = ϕ n x [x] ;T ϕ n x [x], gx = ϕ n+ x [x] ; ψ n x [x] = ϕ n+ x [x] ;T ϕ n x [x], hx = ϕ n x [x] ; ψ n+ x [x] = ϕ n x [x] ;T ϕ n+ x [x]. 24a 24b 24c By calculation we find f x = ϕ n x [ x] ;T ϕ n x [ x] = T δ n+ T ϕ n x [ x] ; ϕ n x [ x] = T δ n+ n+2 ϕ n x [ x] ;T ϕ n x [ x] = T δ n+ n+2 ϕ n x [x] ;T ϕ n x [x] = T δ n+ n+2 fx, where we have made use of x = x and assumed TT = δi where δ is a constant and I is the unit matrix. In a similar way we have g x = nn+2 T δ n+2 hx. With these relations, by making a comparison of q x = 2g x/f x and rx = 2hx/fx one can find that rx = ±q x when δ =. Thus we finish the proof. We note that to reduce 5 to solutions for the local NLS hierarchies, we need same reduction condition as in Theorem except 23a replaced with AT +TA = 0. 4 Solutions of the nonlocal NLS hierarchies In this section, we list Wronskian elements of the solutions for the nonlocal NLS hierarchies Soliton solutions 4.. Focusing case For the nonlocal focusing NLS hierarchy iqx τ2k = K k [qx,q x], k =,2,3,..., 25 according to Theorem, we take δ = and A 0 0 I A = 0 A, T = I 0 5, C = c,c 2,,c 2n+2 T, 26a

6 where A = Diagk,k 2,,k n+, k j C 26b and k j are distinct. The Wronskian elements are taken as Φ ϕx =, ψx = Tϕ x, 27a where and Φ 2 Φ j = ϕ j,,ϕ j,2,,ϕ j,n+ T, j =,2, ϕ,l = e ξ lx,k l, ϕ 2,l = e ξ lx,k l, ξ l x,k l = k l x+i k 2j l τ 2j. 27b 27c 27d The simplest solution to the nonlocal focusing NLS equation9 is obtained by taking n = 0 in double Wronskian and the solution is given as q ss x = 4ic c 2 Imk e 2k x+i2k 2τ 2 c 2 e i4imk x e 8Rek Imk τ 2 c2 2, 28 which is same as the one obtained in [6] and [], up to some scalar transformations. And 28 is a nonsingular periodic solution of the case of Rek = 0, Imk 0 and c c 2. Solution modeled by k and k 2 is q 2ss x = 2 E F where E =c c 2 k k 2 e 2ξ +2ξ 2 [c 2c 3 k kk 2 ke 2k x c c 4 k k2k 2 k2e 2k 2 x ] +c 3 c 4 k2 k e2k +k 2 x [c 2 c 3 k 2 k k 2 k2 e2ξ 2 +c c 4 k k k 2 k e 2ξ ], F = c c 2 k k 2 2 e 2ξ +2ξ 2 + c 3 c 4 k 2 k 2 e 2k +k 2 x c c 4 k 2 k 2 e 2ξ +2k2 x c 2 c 3 k 2 k 2 e 2ξ 2+2k x +4Imk Imk 2 [c c 2 c 3c 4 e2ξ +2k x +c c 2c 3 c 4e 2ξ 2+2k2 x ] Defocusing case For the nonlocal defocusing NLS hierarchy iqx τ2k = K + k [qx,q x], k =,2,3,..., 29 from Theorem, we take δ = and A 0 0 I A = 0 A, T = I 0, C = c,c 2,,c 2n+2 T, 30 where A is the diagonal matrix 26b, ϕ and ψ follow the structure 27 but with T defined as in 30. Solutions to the nonlocal defocusing NLS equation 20 are n = 0 q ss x = 4ic c 2 Imk e 2k x+i2k 2τ 2 c 2 e i4imk x e 8Rek Imk τ 2 + c2 2, 3 6

7 which is a nonsingular solution of the case of Rek = 0, Imk 0 and c c 2, and q 2ss x = 2 G H where G =c c 2 k k 2 e 2ξ +2ξ 2 [c c 4 k k2k 2 k2e 2k 2 x c 2c 3 k kk 2 ke 2k x ] +c 3 c 4 k2 k e2k +k 2 x [c 2 c 3 k 2 k k 2 k2 e2ξ 2 +c c 4 k k k 2 k e 2ξ ], H = c c 2 k k 2 2 e 2ξ +2ξ 2 c 3 c 4 k 2 k 2 e 2k +k 2 x + c c 4 k 2 k 2 e 2ξ +2k2 x c 2 c 3 k 2 k 2 e 2ξ 2+2k x +4Imk Imk 2 [c c 2 c 3c 4 e2ξ +2k x c c 2c 3 c 4e 2ξ 2+2k2 x ]. 4.2 Jordan block solutions Jordan block solutions mean the solutions corresponding to the matrix A taking the form A 0 A = 0 A, 32a where A is a n+ n+ Jordan block 2n+2 2n+2 A = k I +σ, I = δ j,l n+ n+, σ = δ j,l+ n+ n+. 32b 4.2. Focusing case For the nonlocal focusing NLS hierarchy 25, to get their solutions, we take δ =, 0 I T = I 0 33 and C = C,C T, C =,,, n+. 34 The Wronskian elements take the form 27a in which with Φ x,k = [ e ξ x,k + Φ 2 x,k = [e ξ x,k + n l= n l= ξ x,k = k x+i ] l! l k e ξ x,k σ l C T, ] l! l k eξ x,k σ l C T, 35a 35b k 2j τ 2j. 36 For the nonlocal NLS equation 9, when n =, from 36 we have ξ x,k = k x+ik 2 τ 2, 37 and ϕ = e ξ x,k I + k e ξ x,k σc T e ξ x,k I + k e ξ x,k σc T, ψ = e ξ x,k I + k e ξ x,k σc T e ξ x,k I + k e ξ x,k σc T 7. 38

8 The corresponding solution reads with qx = 4k k e 2ξ x,k ak,k e2ξ x,k ak,k e 2ξ x,k 2ξ x,k +e 2ξ x,k 2ξ x,k, 39 b ak,k = k k x+2ik k k τ 2 +, 40a b = 4k k 2 x 2 +8ik k 2 k +k xτ 2 6k k k k 2 τ b Defocusing case For the nonlocal defocusing NLS hierarchy 29, in Theorem we take δ =, and A, C, Φ j same as in Sec.4.2., but T = 0 I I 0. 4 For the nonlocal defocusing NLS equation 20, the simplest solution of Jordan block case is qx = 4k k e 2ξ x,k ak,k+e 2ξ x,k ak,k e 2ξ x,k 2ξ x,k +e 2ξ x,k 2ξ x,k, 42 +b where ak,k and b are denoted by Conclusions In this letter by direct reduction we have derived doubled Wronskian solutions of the nonlocal NLS hierarchies from the known ones of the even-order AKNS hierarchy. Compared with the double Wronskian solution obtained in [7] via Darboux transformation where all {k j } are pure imaginary, here in our solutions all {k j } are complex, which admits more freedom. Besides, we also obtain Jordan block solutions which correspond to multiple-pole case of the transmission coefficient. Since equation 23 admits block diagonal extension, it is easy to extend our results to case where A is a block-diagonal composed by diagonal matrix and Jordan blocks. The reduction procedure developed in this letter is general and can apply to other systems which have double Wronskian solutions and admit local and nonlocal reductions. Acknowledgments This work was supported by the NSF of China [grant numbers 3724, 63007]. References [] M.J. Ablowitz, Z.H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., No pp. [2] M.J. Ablowitz, Z.H. Musslimani, Integrable discrete PT symmetric model, Phys. Rev. E, No pp. 8

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