Nonlocal Yajima–Oikawa system: binary Darboux transformation, exact solutions and dynamic properties

Abstract

The Yajima–Oikawa (YO) system is an important long-wave–short-wave resonant interaction model, which can be used to describe a fascinating resonance phenomena in diverse areas, such as hydrodynamics, nonlinear optics and biophysics. In this paper, we propose a new type integrable nonlocal YO system, which can be derived from the special reduction in the two-component YO system. We show that the binary Darboux transformation is an effective method to construct not only multi-soliton solutions, but also other types of solutions for this type nonlocal integrable systems. Additionally, some novel solutions of the nonlocal YO system are obtained, and further are analyzed in detail to reveal several interesting dynamic features, such as the moving bright soliton with sudden position shift, the collision of two-breather waves.

Appendix

In this appendix, the multi-fold binary Darboux transformation for the nonlocal NLS equation (1.2) is presented. We show the Manakov equation

$$\begin{aligned}&iu_t + u_{xx} + 2\sigma (\left| u \right| ^2 + \left| v \right| ^2 )u = 0, \nonumber \\&iv_t + v_{xx} + 2\sigma (\left| u \right| ^2 + \left| v \right| ^2 )v = 0, \end{aligned}$$

(4.1)

is reduced to the nonlocal NLS equation (1.2) by taking the constraint condition $v(x,t) = u(-x,t)$. The Lax pair of (1.2) can also get from the reduction of Lax pair of Manakov equation, which can be presented as

$$\begin{aligned}{} & {} \Phi _x=U\Phi =(-i\lambda \Lambda +Q)\Phi ,\nonumber \\{} & {} \Phi _t=V\Phi =-2i\lambda ^2\Lambda +2\lambda Q+i\Lambda (Q_x -Q^2)\Phi , \end{aligned}$$

(4.2)

where

$$\begin{aligned} Q=\left( \begin{array}{ccc} 0 &{} 0 &{} u\\ 0 &{} 0 &{} u(-x)\\ - u^* &{} - u^*(-x)&{} 0 \\ \end{array} \right) , \Lambda =\left( \begin{array}{ccc} 1&{} 0 &{} 0 \\ 0 &{}1&{}0 \\ 0&{} 0 &{} -1 \\ \end{array} \right) . \end{aligned}$$

As in Refs. [4, 31, 32], we suppose that $\Phi _j=(\varphi _1^j,\varphi _2^j,\varphi _3^j)^T$ is the solution of spectral problem and the time evolution equation (4.2) at $\lambda =\lambda _j$. Then, we see that $\Psi _j=(\varphi _2^j(-x),\varphi _1^j(-x),-\varphi _3^j(-x))^T$ is the eigenfunction at $\lambda =-\lambda _j$. Letting $\eta _j = (\Phi _j,\Psi _j)$ and defining the function

$$\begin{aligned}&\Omega (\eta _j ,\eta _k )= \left( \begin{array}{cc} {\frac{{\Phi _j^\dag \Phi _k }}{{ \lambda _j^* \mathrm{{ - }}\lambda _k }}} &{} {\frac{{\Phi _j^\dag \Psi _k }}{{ \lambda _j^* \mathrm{{ + }}\lambda _k }}} \\ {\frac{{\Psi _j^\dag \Phi _k }}{{-\lambda _j^* \mathrm{{ - }}\lambda _k }}} &{} {\frac{{\Psi _j^\dag \Psi _k }}{{-\lambda _j^* \mathrm{{ + }}\lambda _k }}} \end{array} \right) , \qquad \Omega (\eta _j ,\Phi ) =\left( \begin{array}{c} {\frac{{\Phi _j^\dag \Phi }}{{ \lambda _j^* \mathrm{{ - }}\lambda }}} \\ {\frac{{\Psi _j^\dag \Phi }}{{-\lambda _j^* \mathrm{{ - }}\lambda }}} \end{array} \right) ,j,k=1,2,3,\ldots , \end{aligned}$$

(4.3)

we can present the following form

Theorem 4.1

The N-fold binary Darboux transformation of the nonlocal NLS equation (1.2) can be given by

$$\begin{aligned} \Phi [N] = \Phi - RW^{ - 1} \Omega , \end{aligned}$$

(4.4)

where

$$\begin{aligned} R&= \big (\begin{array}{*{20}c} \eta _1 &{} \eta _2 &{} \cdots &{} \eta _N \\ \end{array}\big )=\Bigg ( {\underbrace{ \begin{array}{cccc} \varphi _1^1 &{} \varphi _2^1(-x) \\ \varphi _2^1 &{} \varphi _1^1(-x) \\ \varphi _3^1 &{} -\varphi _3^1(-x) \end{array}}_{\eta _1}}~ {\underbrace{ \begin{array}{cccc} \varphi _1^2 &{} \varphi _2^2(-x) \\ \varphi _2^2 &{} \varphi _1^2(-x) \\ \varphi _3^2 &{} -\varphi _3^2(-x) \end{array}}_{\eta _2}} \begin{array}{c} \cdots \\ \cdots \\ \cdots \end{array} \underbrace{ \begin{array}{cccc} \varphi _1^N &{} \varphi _2^N(-x) \\ \varphi _2^N &{} \varphi _1^N(-x) \\ \varphi _3^N &{} -\varphi _3^N(-x) \end{array}}_{\eta _N} \Bigg ) \end{aligned}$$

and

$$\begin{aligned} W = \left( {\begin{array}{*{20}c} {\Omega (\eta _1 ,\eta _1 )} &{} {\Omega (\eta _1 ,\eta _2 )} &{} \cdots &{} {\Omega (\eta _1 ,\eta _N )} \\ {\Omega (\eta _2 ,\eta _1 )} &{} {\Omega (\eta _2 ,\eta _2 )} &{} \cdots &{} {\Omega (\eta _2 ,\eta _N )} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {\Omega (\eta _N ,\eta _1 )} &{} {\Omega (\eta _N ,\eta _2 )} &{} \cdots &{} {\Omega (\eta _N ,\eta _N )} \\ \end{array}} \right) , \qquad \Omega (\eta ,\Phi ) = \left( {\begin{array}{*{20}c} {\Omega (\eta _1 ,\Phi )} \\ {\Omega (\eta _2 ,\Phi )} \\ \vdots \\ {\Omega (\eta _N ,\Phi )} \\ \end{array}} \right) . \end{aligned}$$

The relationship between the new potential and the original potential function is

$$\begin{aligned} u[N]=u+2 i \frac{\left|\begin{array}{cc} W &{} {{\textbf {a}}}_3^{\dag } \\ {{\textbf {a}}}_1 &{} 0 \end{array} \right|}{\left|W \right|}, \end{aligned}$$

(4.5)

where

$$\begin{aligned} {{\textbf {a}}}_1&=\big ( \begin{array}{ccccccc} \varphi _1^1&\varphi _2^1(-x)&\varphi _1^2&\varphi _2^2(-x)&\cdots&\varphi _1^N&\varphi _2^N(-x) \end{array} \big ),\\ {{\textbf {a}}}_3&=\big (\begin{array}{ccccccc} \varphi _3^1&-\varphi _3^1(-x)&\varphi _3^2&-\varphi _3^2(-x)&\cdots&\varphi _3^N&\varphi _3^N(-x) \end{array} \big ). \end{aligned}$$