Exact explicit rational solutions of two-and one-dimensional multicomponent Yajima-Oikawa (YO) systems, which contain multi-shortwave components and single long-wave one, are presented by using the bilinear method. For two-dimensional system, the fundamental rational solution first describes the localized lumps, which have three different patterns: bright, intermediate and dark states. Then, rogue waves can be obtained under certain parameter conditions and their behaviors are also classified to above three patterns with different definition. It is shown that the simplest (fundamental) rogue waves are line localized waves which arise from the constant background with a line profile and then disappear into the constant background again. In particular, two-dimensional intermediate and dark counterparts of rogue wave are found with the different parameter requirements. We demonstrate that multirogue waves describe the interaction of several fundamental rogue waves, in which interesting curvy wave patterns appear in the intermediate times. Different curvy wave patterns form in the interaction of different types fundamental rogue waves. Higher-order rogue waves exhibit the dynamic behaviors that the wave structures start from lump and then retreats back to it, and this transient wave possesses the patterns such as parabolas. Furthermore, different states of higher-order rogue wave result in completely distinguishing lumps and parabolas. Moreover, one-dimensional rogue wave solutions with three states are constructed through the further reduction. Specifically, higher-order rogue wave in one dimensional case is derived under the parameter constraints.

In this paper, we derive a general mixed (bright-dark) multi-soliton solution to a one-dimensional multicomponent Yajima-Oikawa (YO) system, i.e., the (M + 1)-component YO system comprised of M-component short waves (SWs) and one-component long wave (LW) for all possible combinations of nonlinearity coefficients including positive, negative and mixed types. With the help of the KP-hierarchy reduction method, we firstly construct two types of general mixed N-soliton solution (two-bright-one-dark soliton and one-bright-two-dark one for SW components) to the (3+1)component YO system in detail. Then by extending the corresponding analysis to the (M + 1)-component YO system, a general mixed N-soliton solution in Gram determinant form is obtained. The expression of the mixed soliton solution also contains the general all bright and all dark N-soliton solution as special cases. Besides, the dynamical analysis shows that the inelastic collision can only take place among SW components when at least two SW components have bright solitons in mixed type soliton solution. Whereas, the dark solitons in SW components and the bright soliton in LW component always undergo usual elastic collision.Recently, one of the authors 42) has constructed general brightdark N-soliton solution to the vector NLS equation of all possible combinations of nonlinearities including all focusing, all-defocusing and mixed types. In a previous study, 28) we have obtained both the Gram-type and Wronski-type determinant solutions of N dark-soliton for Eq. (1).The goal of this paper is to construct general bright-dark multi-soliton solution to the multicomponent YO system (1). The rest of the paper is arranged as follows. In Sect. 2, we deduce two types of general mixed soliton solution, which include two-bright-one-dark and one-bright-two-dark soliton for SW components, to the (3+1)-component YO system by using the KP-hierarchy reduction technique. In Sect. 3, general bright-dark soliton solution consisting of m bright solitons and M À m dark solitons to the multicomponent YO system (1) is obtained by generalizing our analysis. We summarize the paper in Sect. 4.2. General Mixed-Type Soliton Solution to the One-Dimensional (3+1)-Component YO System 1 0;0 ðk 1 Þ Á 0;0 ðk 1 Þ ¼ À2 1;0 ðk 1 Þ À1;0 ðk 1 Þ;

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 line-soliton solutions of the Kadomtsev--Petviashvili (KP) equation are investigated in this article using the tau-function formalism. In particular, the Wronskian and the Grammian forms of the tau-function are discussed, and the equivalence of these two forms are established. Furthermore, the interaction properties of two special types of 2-soliton solutions of the KP equation are studied in details.Comment: 16 pages, 6 figures, To appear in Applicable Analysis, Special Issue "Solitons and Integrable Systems

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.