Bilinear form, bilinear auto-Bäcklund transformation, breather and lump solutions for a (3$$+$$1)-dimensional generalised Yu–Toda–Sasa–Fukuyama equation in a two-layer liquid or a lattice

Pramana ◽  
2021 ◽  
Vol 95 (3) ◽  
Author(s):  
Yuan Shen ◽  
Bo Tian ◽  
Xin Zhao ◽  
Wen-Rui Shan ◽  
Yan Jiang
Keyword(s):  
Bilinear Form ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Lump Solutions

scholarly journals Integrability and lump-type solutions to the 3-D Kadomtsev-Petviashvili-Boussinesq-like equation

2019 ◽  
Vol 23 (4) ◽  
pp. 2373-2380
Author(s):  
Pin-Xia Wu ◽  
Yu-Feng Zhang ◽  
Qi-Qi Yin ◽  
Yan Wang
Keyword(s):  
Bilinear Form ◽  
Bäcklund Transformation ◽  
Dynamic Evolution ◽  
Wave Method ◽  
Backlund Transformation ◽  
Lax Pairs ◽  
Ansatz Method ◽  
Lump Solutions

The (3+1)-D Kadomtsev-Petviashvili-Boussinesq-like equation is studied, and its bilinear form, Backlund transformation and Lax pairs are elucidated. Lump-type solutions are obtained, which include periodic lump and interaction lump solutions, through the three-wave method and the ansatz method. The dynamic evolution mechanisms of solutions are illustrated graphically.

INTEGRABLE PROPERTIES FOR A GENERALIZED NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG–DE VRIES MODEL

2010 ◽  
Vol 24 (10) ◽  
pp. 1023-1032 ◽  
Author(s):  
XIAO-GE XU ◽  
XIANG-HUA MENG ◽  
FU-WEI SUN ◽  
YI-TIAN GAO
Keyword(s):  
Bilinear Form ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Solitonic Solution ◽  
Bilinear Bäcklund Transformation ◽  
De Vries ◽  
The One ◽  
Hirota Bilinear

Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg–de Vries (vcKdV) equation is investigated analytically employing the Hirota bilinear method in this paper. The bilinear form for such a model is derived through a dependent variable transformation. Based on the bilinear form, the integrable properties such as the N-solitonic solution, the Bäcklund transformation and the Lax pair for the vcKdV equation are obtained. Additionally, it is shown that the bilinear Bäcklund transformation can turn into the one denoted in the original variables.

A note on the exact solutions for non-conservative three-wave resonance

1987 ◽  
Vol 106 (3-4) ◽  
pp. 205-207 ◽  
Author(s):  
A. D. D. Craik
Keyword(s):  
Exact Solutions ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Lump Solutions ◽  
Wave Resonance ◽  
The One

SynopsisExact solutions recently discovered for non-conservative three-wave resonance are here related to the ‘one-lump’ solutions obtained by Backlund transformation in conservative cases.

Bilinear form, bilinear Bäcklund transformation and dynamic features of the soliton solutions for a variable-coefficient (3+1)-dimensional generalized shallow water wave equation

2017 ◽  
Vol 31 (22) ◽  
pp. 1750126 ◽  
Author(s):  
Qian-Min Huang ◽  
Yi-Tian Gao
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Bilinear Form ◽  
Water Wave ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Backlund Transformation ◽  
Spectral Parameters ◽  
Shallow Water Wave

Under investigation in this letter is a variable-coefficient (3[Formula: see text]+[Formula: see text]1)-dimensional generalized shallow water wave equation. Bilinear form and Bäcklund transformation are obtained. One-, two- and three-soliton solutions are derived via the Hirota bilinear method. Interaction and propagation of the solitons are discussed graphically. Stability of the solitons is studied numerically. Soliton amplitude is determined by the spectral parameters. Soliton velocity is not only related to the spectral parameters, but also to the variable coefficients. Phase shifts are the only difference between the two-soliton solutions and the superposition of the two relevant one-soliton solutions. Numerical investigation on the stability of the solitons indicates that the solitons could resist the disturbance of small perturbations and propagate steadily.

Bäcklund transformation and soliton solutions for two (3+1)-dimensional nonlinear evolution equations

2016 ◽  
Vol 30 (24) ◽  
pp. 1650309
Author(s):  
Lin Wang ◽  
Qixing Qu ◽  
Liangjuan Qin
Keyword(s):  
Bilinear Form ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Backlund Transformation ◽  
Wronskian Determinant ◽  
Bilinear Bäcklund Transformation ◽  
Bilinear Equations

In this paper, two (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equations (NLEEs) are under investigation by employing the Hirota’s method and symbolic computation. We derive the bilinear form and bilinear Bäcklund transformation (BT) for the two NLEEs. Based on the bilinear form, we obtain the multi-soliton solutions for them. Furthermore, multi-soliton solutions in terms of Wronskian determinant for the first NLEE are constructed, whose validity is verified through direct substitution into the bilinear equations.

Symbolic Computation Study of a Generalized Variable-Coefficient Two-Dimensional Korteweg-de Vries Model with Various External-Force Terms from Shallow Water Waves, Plasma Physics, and Fluid Dynamics

2009 ◽  
Vol 64 (3-4) ◽  
pp. 222-228 ◽  
Author(s):  
Xing Lü ◽  
Li-Li Li ◽  
Zhen-Zhi Yao ◽  
Tao Geng ◽  
Ke-Jie Cai ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
External Force ◽  
Bilinear Form ◽  
Water Waves ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Two Dimensional ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
De Vries

Abstract The variable-coefficient two-dimensional Korteweg-de Vries (KdV) model is of considerable significance in describing many physical situations such as in canonical and cylindrical cases, and in the propagation of surface waves in large channels of varying width and depth with nonvanishing vorticity. Under investigation hereby is a generalized variable-coefficient two-dimensional KdV model with various external-force terms. With the extended bilinear method, this model is transformed into a variable-coefficient bilinear form, and then a Bäcklund transformation is constructed in bilinear form. Via symbolic computation, the associated inverse scattering scheme is simultaneously derived on the basis of the aforementioned bilinear Bäcklund transformation. Certain constraints on coefficient functions are also analyzed and finally some possible cases of the external-force terms are discussed

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