scholarly journals M -Breather, Lumps, and Soliton Molecules for the 2 + 1 -Dimensional Elliptic Toda Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yuechen Jia ◽  
Yu Lu ◽  
Miao Yu ◽  
Hasi Gegen
Keyword(s):  
Soliton Solution ◽  
Toda Lattice ◽  
Discrete Version ◽  
Toda Equation ◽  
Nonlinear Superposition ◽  
Lump Solutions ◽  
Breather Solution ◽  
Higher Dimensional ◽  
Hybrid Solutions ◽  
Line Soliton

The 2 + 1 -dimensional elliptic Toda equation is a higher dimensional generalization of the Toda lattice and also a discrete version of the Kadomtsev-Petviashvili-1 (KP1) equation. In this paper, we derive the M -breather solution in the determinant form for the 2 + 1 -dimensional elliptic Toda equation via Bäcklund transformation and nonlinear superposition formulae. The lump solutions of the 2 + 1 -dimensional elliptic Toda equation are derived from the breather solutions through the degeneration process. Hybrid solutions composed of two line solitons and one breather/lump are constructed. By introducing the velocity resonance to the N -soliton solution, it is found that the 2 + 1 -dimensional elliptic Toda equation possesses line soliton molecules, breather-soliton molecules, and breather molecules. Based on the N -soliton solution, we also demonstrate the interactions between a soliton/breather-soliton molecule and a lump and the interaction between a soliton molecule and a breather. It is interesting to find that the KP1 equation does not possess a line soliton molecule, but its discrete version—the 2 + 1 -dimensional elliptic Toda equation—exhibits line soliton molecules.

N-soliton solutions and localized wave interaction solutions of a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyamaf equation

2021 ◽  
pp. 2150277
Author(s):  
Hongcai Ma ◽  
Qiaoxin Cheng ◽  
Aiping Deng
Keyword(s):  
Dynamic Behavior ◽  
Soliton Solution ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Bilinear Transformation ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Ytsf Equation ◽  
Line Soliton

[Formula: see text]-soliton solutions are derived for a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by using bilinear transformation. Some local waves such as period soliton, line soliton, lump soliton and their interaction are constructed by selecting specific parameters on the multi-soliton solutions. By selecting special constraints on the two soliton solutions, period and lump soliton solution can be obtained; three solitons can reduce to the interaction solution between period soliton and line soliton or lump soliton and line soliton under special parameters; the interaction solution among period soliton and two line solitons, or the interaction solution for two period solitons or two lump solitons via taking specific constraints from four soliton solutions. Finally, some images of the results are drawn, and their dynamic behavior is analyzed.

scholarly journals Five-Dimensional Gauge Theories and Whitham–Toda Equation

1997 ◽  
Vol 12 (29) ◽  
pp. 2183-2191 ◽  
Author(s):  
Masato Hisakado
Keyword(s):  
Gauge Theory ◽  
Toda Lattice ◽  
Gauge Theories ◽  
Toda Chain ◽  
Charge Conjugation ◽  
Two Dimensional ◽  
Toda Equation ◽  
Toda Lattice Hierarchy

The five-dimensional supersymmetric SU (N) gauge theory is studied in the framework of the relativistic Toda chain. This equation can be embedded in two-dimensional Toda lattice hierarchy. This system has the conjugate structure which corresponds to the charge conjugation.

scholarly journals A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2315
Author(s):  
Meng-Li Qin ◽  
Xiao-Yong Wen ◽  
Manwai Yuen
Keyword(s):  
Exact Solutions ◽  
Darboux Transformation ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Lattice System ◽  
Arbitrary Parameter ◽  
Rational Solutions ◽  
Limit States ◽  
Hybrid Solutions ◽  
Relativistic Toda Lattice

This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m,2N−m)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.

scholarly journals High-Order Breather Solutions, Lump Solutions, and Hybrid Solutions of a Reduced Generalized (3 + 1)-Dimensional Shallow Water Wave Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jing Wang ◽  
Biao Li
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
High Order ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Moving Path

We investigate a reduced generalized (3 + 1)-dimensional shallow water wave equation, which can be used to describe the nonlinear dynamic behavior in physics. By employing Bell’s polynomials, the bilinear form of the equation is derived in a very natural way. Based on Hirota’s bilinear method, the expression of N-soliton wave solutions is derived. By using the resulting N-soliton expression and reasonable constraining parameters, we concisely construct the high-order breather solutions, which have periodicity in x,y-plane. By taking a long-wave limit of the breather solutions, we have obtained the high-order lump solutions and derived the moving path of lumps. Moreover, we provide the hybrid solutions which mean different types of combinations in lump(s) and line wave. In order to better understand these solutions, the dynamic phenomena of the above breather solutions, lump solutions, and hybrid solutions are demonstrated by some figures.

Soliton molecules in the (2+1)-dimensional Nizhnik–Novikov–Veselov equation

2021 ◽  
pp. 2150367
Author(s):  
Huiling Wu ◽  
Jinxi Fei ◽  
Zhengyi Ma
Keyword(s):  
Soliton Solution ◽  
Elastic Interaction ◽  
Wave Number ◽  
Explicit Solutions ◽  
Variable Formula ◽  
Structure Interaction ◽  
Molecule Structure ◽  
Long Wave ◽  
Wave Limit ◽  
Line Soliton

The [Formula: see text]-soliton solution of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation is constructed. The line soliton molecule, the breather and the lump soliton are presented successively for [Formula: see text]. The three-soliton molecule structure, interaction of one-soliton molecule and a line soliton, the soliton molecules consisting of a line soliton and the breather/lump soliton of the solution [Formula: see text] are constructed for [Formula: see text]. Moreover, the four-soliton molecule structure, interaction of the soliton molecule and a line soliton, the soliton molecule consisting of the line soliton molecule and a lump soliton, the elastic interaction between the line soliton molecule and a lump soliton, the soliton molecules consisting of the line soliton molecule and the breather, two breather solitons, the breather soliton and a lump of the variable [Formula: see text] for this equation are also derived for [Formula: see text] by applying the velocity resonance, the module resonance of wave number and the long-wave limit ideas. To illustrate these phenomena, the analysis explicit solutions are all given and their dynamics features are all displayed through figures.

The Structures and the Interactions of Soliton in two (2+1)-dimensional KdV-Type Equations

2004 ◽  
Vol 59 (1-2) ◽  
pp. 14-22
Author(s):  
Hang-yu Ruan
Keyword(s):  
Exact Solutions ◽  
Boussinesq Equations ◽  
Repulsive Interaction ◽  
Bilinear Method ◽  
Breather Solution ◽  
Kdv Type Equations ◽  
Line Soliton

Exact solutions in two (2+1)-dimensional KdV-type (Sawada-Kodera and Boussinesq) equations are presented by using the bilinear method. The N-breather solution, the solution to describe the interaction between a line soliton and a y-periodic soliton, and the solution to express the interaction between two y-periodic solitons are included in our results. Detailed behavior of interactions between a line soliton and a y-periodic soliton for the SK equation and between two y-periodic solitons for the BS equation are illustrated both analytically and graphically. For these two equations, we only discuss the repulsive interaction keeping the shapes of the soliton unchanged.

DARBOUX TRANSFORMATION OF THE MODIFIED TODA LATTICE EQUATION

2006 ◽  
Vol 20 (11) ◽  
pp. 641-648 ◽  
Author(s):  
XI-XIANG XU ◽  
HONG-XIANG YANG ◽  
YE-PENG SUN
Keyword(s):  
Spectral Problem ◽  
Soliton Solution ◽  
Darboux Transformation ◽  
Toda Lattice ◽  
Lattice Equation ◽  
Matrix Spectral Problem ◽  
Toda Lattice Equation

A modified Toda lattice equation associated with a properly discrete matrix spectral problem is introduced. Darboux transformation for the resulting lattice equation is constructed. As an application, the soliton solution for the Toda lattice equation is explicitly given.

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