Rogue wave, interaction solutions to the KMM system

Under investigation in this paper is the higher-order Broer-Kaup(HBK) system, which describes the bidirectional propagation of long waves in shallow water. Via the standard truncated Painlevé expansion method, the residual symmetry of this system is derived. By introducing an appropriate auxiliary-dependent variable, the residual symmetry is successfully localized to Lie point symmetries. Via solving the initial value problems, the finite symmetry transformations are presented. However, the solution which obtained from the residual symmetry is a special group invariant solutions. In order to find more general solution of HBK system, we further generalize the residual symmetry method to the consistent tanh expansion (CTE) method and prove that the HBK system is CTE solvable, then the resonant soliton solutions and interaction solutions among different nonlinear excitations are obtained by the CET method.

Section: The Residual Symmetry Of the Higher-order Broer-kaup System mentioning

confidence: 99%

Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

Xia

Yao

Xin

2021

“…Lump solutions, which can be considered a kind of rational function solutions, decay polynomially in all directions of space [31][32][33][34][35][36]. One can construct lump solutions by the Hirota bilinear method and the Darboux transformation [37][38][39][40][41][42][43][44][45].…”

Section: Lump Solution Of the ð2 + 1þ-dimensionalmentioning

confidence: 99%

Characteristics of the Soliton Molecule and Lump Solution in the $(2 + 1)$ -Dimensional Higher-Order Boussinesq Equation

Ren

2021

The soliton molecules, as bound states of solitons, have attracted considerable attention in several areas. In this paper, the 2 + 1 -dimensional higher-order Boussinesq equation is constructed by introducing two high-order Hirota operators in the usual 2 + 1 -dimensional Boussinesq equation. By the velocity resonance mechanism, the soliton molecule and the asymmetric soliton of the higher-order Boussinesq equation are constructed. The soliton molecule does not exist for the usual 2 + 1 -dimensional Boussinesq equation. As a special kind of rational solution, the lump wave is localized in all directions and decays algebraically. The lump solution of the higher-order Boussinesq equation is obtained by using a quadratic function. This lump wave is just the bright form by some detail analysis. The graphics in this study are carried out by selecting appropriate parameters. The results in this work may enrich the variety of the dynamics of the high-dimensional nonlinear wave field.

“…In 2019, Lou [11] introduced a velocity resonant mechanism to form soliton molecules and asymmetric solitons for three-fifth order systems. Very recently, soliton molecules and some hybrid solutions involving Lump, breather, and positon have been investigated for some (1 + 1)-dimensional and (2 + 1)-dimensional equations by Hirota bilinear method and Darboux transformation [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Soliton Molecules and Full Symmetry Groups to the KdV-Sawada-Kotera-Ramani Equation

Xiong

2021

By N -soliton solutions and a velocity resonance mechanism, soliton molecules are constructed for the KdV-Sawada-Kotera-Ramani (KSKR) equation, which is used to simulate the resonances of solitons in one-dimensional space. An asymmetric soliton can be formed by adjusting the distance between two solitons of soliton molecule to small enough. The interactions among multiple soliton molecules for the equation are elastic. Then, full symmetry group is derived for the KSKR equation by the symmetry group direct method. From the full symmetry group, a general group invariant solution can be obtained from a known solution.