A Novel Dynamic of Localized Solitary Waves for the Hirota-Maccari System

This paper investigates a particular family of semi-rational solutions in determinant form by using the KP hierarchy reduction method, which describe resonant collisions among lumps or resemble line rogue waves and dark solitons in the Hirota-Maccari system. Due to the resonant collisions, the line resemble rogue waves are generated and attenuated in the background of dark solitons with line profiles of finite length, it takes a short time for the lumps to appear from and disappear into the dark solitons background. These novel dynamic of localized solitary waves may be help to understand some physical phenomena of nonlinear localized waves propagation in many physical settings.

Bright--Dark Solitons in the Space-Shifted Nonlocal Coupled Nonlinear Schrodinger Equation

Multiple bright-dark soliton solutions in terms of determinants for the space-shifted nonlocal coupled nonlinear Schro¨dinger (CNLS) equation are constructed by using the bilinear (Kadomtsev-Petviashvili) KP hierarchy reduction method. It is found that the bright-dark two-soliton only occur elastic collisions. Upon their amplitudes, the bright two solitons only admit one pattern whose amplitude are equal, and the dark two solitons have three different non-degenerated patterns and two different degenerated patterns. The bright-dark four-soliton is the superposition of the two-soliton pairs and can generated bound-state solitons. The multiple double-pole bright-dark soliton solutions are generated through the long wave limit of the obtained bright-dark soliton solutions, and their collision dynamics are also investigated.PACS 02.30.Jr · 03.75.Lm · 04.20.Jb · 05.45.Yv

General soliton and (semi-)rational solutions of the partial reverse space y-non-local Mel’nikov equation with non-zero boundary conditions

General soliton and (semi-)rational solutions to the y-non-local Mel’nikov equation with non-zero boundary conditions are derived by the Kadomtsev–Petviashvili (KP) hierarchy reduction method. The solutions are expressed in N × N Gram-type determinants with an arbitrary positive integer N . A possible new feature of our results compared to previous studies of non-local equations using the KP reduction method is that there are two families of constraints among the parameters appearing in the solutions, which display significant discrepancies. For even N , one of them only generates pairs of solitons or lumps while the other one can give rise to odd numbers of solitons or lumps; the interactions between lumps and solitons are always inelastic for one family whereas the other family may lead to semi-rational solutions with elastic collisions between lumps and solitons. These differences are illustrated by a thorough study of the solution dynamics for N = 1, 2, 3. Besides, regularities of solutions are discussed under proper choices of parameters.

Soliton solutions to a (2+1)-dimensional nonlocal NLS equation

In this paper, applying the Hirota’s bilinear method and the KP hierarchy reduction method, we obtain the general soliton solutions in the forms of N × N Gram-type determinants to a (2+1)-dimensional non-local nonlinear Schrodinger equation with time reversal under zero and nonzero boundary conditions. The general bright soliton solutions with zero boundary condition are derived via the tau functions of two-component KP hierarchy. Under nonzero boundary condition, we first construct general soliton solutions on periodic back-ground, when N is odd. Furthermore, we discuss typical dynamics of solutions analytically, and graphically.

Soliton solutions to the nonlocal Davey–Stewartson III equation

The soliton solutions on both constant and periodic backgrounds of the nonlocal Davey–Stewartson III equation are derived by using the bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. These solutions are presented as [Formula: see text] Gram-type determinants, with [Formula: see text] a positive integer. Typical dynamics of these soliton solutions are investigated in analytical and graphical aspects. Two types of soliton solutions are generated with different [Formula: see text]. When [Formula: see text] is even, solitons on the constant background can be constructed, whereas solitons appear on the periodic background for odd [Formula: see text]. Under suitable parameter restrictions, we show the regularity of solutions and display all patterns of two- and four-soliton solutions.

N-soliton and rogue wave solutions of (2+1)-dimensional integrable system with Lax pair

A new generalized (2[Formula: see text]+[Formula: see text]1)-dimensional model, obtained from the Kadomtsev–Petviashvili (KP) equation considered as a system of nonlinear evolution partial differential equations (PDEs), is introduced. With the usage of the Hirota bilinear method and the KP-hierarchy reduction method, N-soliton solutions of the integrable system are constructed. Considering the case of s[Formula: see text]=[Formula: see text]−1 in the linear differential operators, L1 and L2, and a specific set of parameters, two dark solitons, mixed solutions consisting of soliton-type and periodic waves solution are obtained. Based on the particular definition of the matrix elements, one and two rogue waves solutions expressed in terms of rational functions are derived. It is shown that the fundamental rogue waves are line rogue waves, which is different from the property of the moving line solitons of the soliton equations.

Bright-Dark Mixed N-Soliton Solution of Two-Dimensional Multicomponent Maccari System

Based on the KP hierarchy reduction method, we construct the general bright-dark mixed N-soliton solution of the two-dimensional (2D) (M+1)-component Maccari system comprised of M-component short waves (SWs) and one-component long wave (LW) with all possible combinations of nonlinearities. We firstly consider two types of mixed N-soliton solutions (two-bright-one-dark and one-bright-two-dark solitons in SW components) to the (3+1)-component Maccari system in detail. Then by extending our analysis to the (M+1)-component Maccari system, its general m-bright-(M–m)-dark mixed N-soliton solution is obtained. The formula obtained also contains the general all-bright and all-dark N-soliton solutions as special cases. For the two-bright-one-dark mixed soliton solution of the (3+1)-component Maccari system, it can be shown that solioff excitation and solioff interaction take place in the two SW components supporting bright solitons, whereas the SW component supporting dark solitons and the LW component possess V-type solitary and interaction.