Multi-Dark Soliton Solutions of the Two-Dimensional Multi-Component Yajima–Oikawa Systems

Abstract

We present a general form of multi-dark soliton solutions of two-dimensional multi-component soliton systems. Multi-dark soliton solutions of the two-dimensional (2D) and one-dimensional (1D) multi-component Yajima-Oikawa (YO) systems, which are often called the 2D and 1D multicomponent long wave-short wave resonance interaction systems, are studied in detail. Taking the 2D coupled YO system with two short wave and one long wave components as an example, we derive the general N-dark-dark soliton solution in both the Gram type and Wronski type determinant forms for the 2D coupled YO system via the KP hierarchy reduction method. By imposing certain constraint conditions, the general N-dark-dark soliton solution of the 1D coupled YO system is further obtained. The dynamics of one dark-dark and two dark-dark solitons are analyzed in detail. In contrast with bright-bright soliton collisions, it is shown that dark-dark soliton collisions are elastic and there is no energy exchange among solitons in different components. Moreover, the dark-dark soliton bound states including the stationary and moving ones are discussed. For the stationary case, the bound states exist up to arbitrary order, whereas, for the moving case, only the two-soliton bound state is possible under the condition that the coefficients of nonlinear terms have opposite signs. * The one-dimensional (1D) YO system was proposed as a model equation for the interaction of a Langmuir wave with an ion-acoustic wave in a plasma by Yajima and Oikawa, 4 which was also derived from several other physical contexts. 3, 5-7 The 1D YO system was solved exactly by the inverse scattering transform method 4, 8 and the (classical) Hirota's bilinear method (which uses the perturbation expansion). 9, 10 It admits both bright and dark soliton solutions. The 2D YO system for the resonant interaction between a long surface wave and a short internal wave in a two-layer fluid was presented and the bright and dark soltion solutions are provided by using the Hirota's bilinear method. 2, 3 The Painlevé property for the 2D YO system was investigated 11 and some special solutions such as positons, dromions, instantons and periodic wave solutions J. Phys. Soc. Jpn. DRAFTwere constructed. 11,12 For the 2D coupled case, the multi-bright soliton solutions expressed by the Wronskian to Eqs.(1)-(3) were provided. 1 Later, the bright N-soliton solutions in the Gram type determinant for the multi-component YO system were obtained. 13,14 Similar to the single component case, the Painlevé property and dromion solutions to Eqs.(1)-(3) were discussed. 15 In a recent paper by Kanna, Vijayajayanthi and Lakshmanan, one and two mixed soliton solutions for the multi-component YO system were constructed. 16 Very recently, the rogue wave solutions for the single YO system in 1D case were derived. 17,18 However, to the best of our knowledge, general multi-dark soliton solutions for the multicomponent 2D and 1D YO system have not been reported yet. Moreover, general multi-dark soliton solutions for the ...

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

“…Finally, it should be pointed out that similar to the multicomponent YO system , we can extend the present study to obtain multisoliton and breather solutions of the multicomponent DYO system which is composed of multi‐SWs and one LW. In addition, in parallel to the investigation of the integrable YO system , the integrable semidiscrete analogue of the DYO system is worth to be expected.…”

From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

“…The KP hierarchy reduction method was first developed by the Kyoto school [49], and later was used to obtain soliton solutions for the NLS equation, the modified Korteweg-de Vries (KdV) equation, the Davey-Stewartson equation and the coupled higher order NLS equations [51,52]. Recently, this method has been applied to derive dark-dark soliton solution in two-coupled NLS equation of the mixed type [53], the general bright-dark soliton solution in an M-coupled NLS equation of the mixed type [54], the general dark-dark and bright-dark soliton solutions to both the two-dimensional and one-dimensional multicomponent Yajima-Oikawa system [55,56]. In compared to the IST method [46], and Hirota's bilinear method [26], the KP hierarchy reduction method starts with the general KP hierarchy including the two-dimensional Toda-hierarchy [57] and derives the general soliton solution in either determinant or pfaffian form reduced directly from the tau functions of the KP hierarchy.…”

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

“…As one of the hot topics in the nonlinear science, dark solitons have been reported in many documents [15,16,17,18]. There have been many different methods for constructing the dark soliton, such as the binary Darboux transformation [15,19,20], the algebraic-geometry reduction method [17], the dressing-Hirota method [18] and the KP hierarchy reduction method [16,21,22]. The main goal of this work is to construct multi-dark soliton solutions for the multi-component coupled Maccari system through the KP hierarchy reduction method.…”