The Derivative Yajima–Oikawa System: Bright, Dark Soliton and Breather Solutions

Abstract

In this paper, we study the derivative Yajima–Oikawa (YO) system which describes the interaction between long and short waves (SWs). It is shown that the derivative YO system is classified into three types which are similar to the ones of the derivative nonlinear Schrödinger equation. The general N‐bright and N‐dark soliton solutions in terms of Gram determinants are derived by the combination of the Hirota's bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method. Particularly, it is found that for the dark soliton solution of the SW component, the magnitude of soliton can be larger than the nonzero background for some parameters, which is usually called anti‐dark soliton. The asymptotic analysis of two‐soliton solutions shows that for both kinds of soliton only elastic collision exists and each soliton results in phase shifts in the long and SWs. In addition, we derive two types of breather solutions from the different reduction, which contain the homoclinic orbit and Kuznetsov–Ma breather solutions as special cases. Moreover, we propose a new (2+1)‐dimensional derivative Yajima–Oikawa system and present its soliton and breather solutions.

On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

“…It should be pointed out that the two-component mYOLS Equation (10) is different from the long-wave-short-wave resonance Equation (1), and the two-component mYOLS Equation (10) is equivalent to the long-wave-short-wave model (2) under some transformation [21]. Then, with the help of Riccati equations for the Lax pair associated with the vmYOLS Equation (8) [22][23][24], Bäcklund transformation [25,26], Darboux transformation [27][28][29][30][31][32][33][34][35][36][37][38][39], and others [40][41][42][43][44][45][46][47][48][49][50][51].…”

Multi-dark soliton solutions for the (2+1)-dimensional multi-component Maccari system

“…From the KP theory, the nonlinear models under investigation can be seen as some special reductions of the integrable equations in the KP hierarchy [7,16]. Then the solutions that bright, dark and rational solutions for some relative nonlinear systems can be derived from the τ function in the KP hierarchy under the appropriate reductions.…”

“…Additionally, a variety of complex systems usually involve more than one component, such as nonlinear optical fibers and Bose-Einstein condensates, etc.. It is greatly necessary to extend the corresponding researches to multi-component systems [6,7,8]. There always exist coupled effects that cross-phase modulation in the multi-component ones, and the corresponding various solutions can not be correlated by Galileo transformation [9].…”

A study on resonant collision in the two-dimensional multi-component long-wave–short-wave resonance system

“…In this regard, we will construct a family of general semi-rational solutions to the two-dimensional M-LSRI by using the bilinear KP hierarchy reduction method [51–54]. In recent years, this KP reduction method has been successfully employed to construct interesting dark–dark and bright-dark type solitons [55–59], rational rogue waves [60–63] in various nonlinear evolution equations.…”