Pfaffian solution for coupled discrete nonlinear Schrödinger equation

doi:10.1016/S0960-0779(98)00272-0

Chaos, Solitons & Fractals

Volume 11, Issues 1–3, January–March 2000, Pages 91-95

https://doi.org/10.1016/S0960-0779(98)00272-0 Get rights and content

Abstract

The Pfaffian solution for the coupled discrete nonlinear Schrödinger equation is studied by using the direct method of soliton theory. The bilinear form of the equation contains a new Pfaffian identity. The Pfaffian representation of Toeplitz determinant is also derived.

Introduction

The nonlinear Schrödinger equation (NLS), $i ∂_{t} u= ∂_{x}^{2} u+2uu^{*} u,$ where ^* means complex conjugate, has several types of integrable discrete analogs. One of the simplest way to construct integrable discrete equations is the method of Miwa transformation which makes the correspondence of the independent variables keeping the structure of the solution space [1]. For NLS, this method yields, for instance [2], $i ∂_{t} u_{n} = u_{n−1} −u_{n} +u_{n} u ̄_{n} u_{n−1},$ $−i ∂_{t} u ̄_{n} = u ̄_{n+1} − u ̄_{n} + u ̄_{n} u_{n} u ̄_{n+1},$ or $i ∂_{t} u_{n} = u_{n−1} 1−u_{n−1} u ̄_{n+1} −u_{n},$ $−i ∂_{t} u ̄_{n} = u ̄_{n+1} 1− u ̄_{n+1} u_{n−1} − u ̄_{n} .$ However as is clearly seen, we cannot take ū as the complex conjugate of u in these equations. This inconsistency with the condition of complex conjugation (c.c.) comes from the incompatibility of the flows of time evolution and ^* operation in the solution space. On the other hand, more than two decades ago Ablowitz and Ladik proposed the discrete NLS which satisfies the c.c. condition [3], $i ∂_{t} u_{n} =Δ_{n}^{2} u_{n} +u_{n} u^{*}_{n} (u_{n+1} +u_{n−1}),$ where Δ_n is the central difference operator defined as $Δ_{n} F_{n} =F_{n+12} −F_{n−12}$ .

They constructed the above equation by using the inverse scattering method which automatically keeps the c.c. symmetry in this case. It is known that the solution of this equation is given in terms of the Toeplitz determinant [4], which means that after all the appropriate choice of the independent variable one recovers the c.c. symmetry.

However in the case of coupled equations, the situation is completely different. The coupled NLS, $i ∂_{t} u^{k} = ∂_{x}^{2} u^{k} +2∑_{l=1}^{M} u^{l} u^{l*} u^{k} 1⩽k⩽M$ is the integrable multicomponent generalization of NLS. The usual discretization of this equation again breaks the c.c. symmetry. Even starting from Eq. (1), the natural multicomponent generalization based on the inverse scattering method gives the following coupled equations [5], $i ∂_{t} u_{n}^{k} =Δ_{n}^{2} u_{n}^{k} + ∑ l=1 M u_{n}^{l} u ̄_{n}^{l} (u_{n+1}^{k} +u_{n−1}^{k})− ∑ l=1 M (u_{n−1}^{k} u_{n}^{l} −u_{n−1}^{l} u_{n}^{k}) u ̄_{n}^{l} 1⩽k⩽M$ $−i ∂_{t} u ̄_{n}^{k} =Δ_{n}^{2} u ̄_{n}^{k} + ∑ l=1 M u_{n}^{l} u ̄_{n}^{l} (u ̄_{n+1}^{k} + u ̄_{n−1}^{k})− ∑ l=1 M (u ̄_{n+1}^{k} u ̄_{n}^{l} − u ̄_{n+1}^{l} u ̄_{n}^{k})u_{n}^{l} 1⩽k⩽M$ which is incompatible with the condition $u ̄ =u^{*}$ . Thus in the coupled case, it is not so clear how to construct the discrete analogs.

Recently another coupled discrete NLS (cdNLS) satisfying the c.c. condition was proposed [5], $i ∂_{t} u_{n}^{k} =Δ_{n}^{2} u_{n}^{k} +∑_{l=1}^{M} u_{n}^{l} u_{n}^{l*} (u_{n+1}^{k} +u_{n−1}^{k}) 1⩽k⩽M.$ The purpose of this paper is to investigate the structure of the solution space of this cdNLS system. We show that the solution of Eq. (2) is given in terms of the Pfaffian with new difference structure. It is also shown that the Pfaffian gives a kind of generalization of the Toeplitz determinant. The integrability of Eq. (2) follows from the fact that the Pfaffian solution includes M arbitrary functions satisfying the dispersion relation.

Section snippets

New algebraic identity of Pfaffian

We denote the (i,j) element of Pfaffian simply as (i,j). We have the asymmetric condition: (i,j)=−(j,i) for any i and j. We also use the conventional notation of the linear combination of index, $(α+β,γ)=(α,γ)+(β,γ), (α,β+γ)=(α,β)+(α,γ), (cα,β)=(α,cβ)=c(α,β).$

Theorem 1 New bilinear identity of Pfaffian

Assuming the difference rule for the elements of Pfaffian, $(i^{(n+1)},j^{(m+1)})=(i^{(n)},j^{(m)})+(i^{(n+1)},h_{0})(j^{(m)},h_{0})−(j^{(m+1)},h_{0})(i^{(n)},h_{0})$ then the τ function defined by the Pfaffian, $τ_{N_{1}N_{2}…N_{M}}^{n} = (S^{N_{1}−1} 1^{(n)},…,S1^{(n)},1^{(n)},S^{N_{2}−1} 2^{(n)},…,S2^{(n)},2^{(n)}, …,S^{N_{M}−1} M^{(n)},…,SM^{}$

Pfaffian solution for cdNLS

Now we introduce the time evolution in the above τ function.

Theorem 2 Bilinear equations of cdNLS

Imposing the differential rule for the elements of Pfaffian in Theorem 1, $∂_{t} (i^{(n)},j^{(m)})=(i^{(n+1)},h_{0})(j^{(m−1)},h_{0})−(j^{(m+1)},h_{0})(i^{(n−1)},h_{0}) ∂_{t} (i^{(n)},h_{0})=(i^{(n+1)},h_{0})−(i^{(n−1)},h_{0})$ then the τ function satisfies the bilinear equation, $D_{t} τ_{N_{1}⋯N_{k}+1⋯N_{M}}^{n} ·τ_{N_{1}⋯N_{M}}^{n} =τ_{N_{1}⋯N_{k}+1⋯N_{M}}^{n+1} τ_{N_{1}⋯N_{M}}^{n−1} −τ_{N_{1}⋯N_{k}+1⋯N_{M}}^{n−1} τ_{N_{1}⋯N_{M}}^{n+1} 1⩽k⩽M.$

Proof

[7]

By using the elementary properties of Pfaffian, $(1,2,…,2N,a,b)=(1,2,…,2N)(a,b)− ∑ 1⩽i<j⩽2N Γ_{ij} (i,a)(j,b)−(j,a)(i,b),$ $∂_{x} (1,2,⋯,2N)= ∑ 1⩽i<j⩽2N Γ_{ij} ∂_{x} (i,j),$

Concluding remarks

We have shown that the solution of cdNLS (2) is given by the Pfaffian (4) whose elements satisfy , . The solution of the single component discrete NLS (1) is described by the Toeplitz determinant and the classification of the equation in the KP hierarchy is clear. However to satisfy the condition of complex conjugation in the multicomponent case, we need to change the structure of the solution space completely, switching from Toeplitzs determinant to the Pfaffians by using the algebraic

Acknowledgements

The author would like to express his sincere thanks to Prof. Ablowitz, Prof. Herbst and Dr. Trubatch for their valuable discussions. The present work is based on the results obtained in the collaboration with them. He is also grateful to Prof. Hirota, Prof. Nakamura, Dr. Tsujimoto and Dr. Iwao for their discussions and comments.

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