scholarly journals Integrable discretization of soliton equations via bilinear method and Bäcklund transformation

2014 ◽  
Vol 58 (2) ◽  
pp. 279-296 ◽  
Author(s):  
YingNan Zhang ◽  
XiangKe Chang ◽  
Juan Hu ◽  
XingBiao Hu ◽  
Hon-Wah Tam
Keyword(s):  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Soliton Equations ◽  
Integrable Discretization

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.

scholarly journals Bäcklund Transformation and Exact Solutions to a Generalized (3 + 1)-Dimensional Nonlinear Evolution Equation

2022 ◽  
Vol 2022 ◽  
pp. 1-12
Author(s):  
Yali Shen ◽  
Ying Yang
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Dynamic Properties ◽  
Traveling Wave Solutions ◽  
Mixed Solution ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Interaction Solutions

In this article, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE), which can be obtained by a multivariate polynomial, is investigated. Based on the Hirota bilinear method, the N-soliton solution and bilinear Bäcklund transformation (BBT) with explicit formulas are successfully constructed. By using BBT, two traveling wave solutions and a mixed solution of the generalized (3 + 1)-dimensional NLEE are obtained. Furthermore, the lump and the interaction solutions for the equation are constructed. Finally, the dynamic properties of the lump and the interaction solutions are described graphically.

New Multi-Soliton Solutions of the (2+1)-Dimensional Breaking Soliton Equations

2003 ◽  
Vol 17 (22n24) ◽  
pp. 4376-4381 ◽  
Author(s):  
Jie-Fang Zhang ◽  
Chun-Long Heng
Keyword(s):  
Bäcklund Transformation ◽  
Direct Method ◽  
Soliton Solutions ◽  
Variable Separation ◽  
Backlund Transformation ◽  
Soliton Equations ◽  
Simplified Form

A simple and direct method is used to solve the (2+1)-dimensional breaking soliton equations: qt=iqxy-2iq∫(qr)ydx, rt=-irxy+2ir∫(qr)ydx. This technique yields a simplified form of the (2+1)-dimensional breaking soliton equations by use of a special Bäcklund transformation and a variable separation solution of this model is derived. Some special types of multi-soliton structure are constructed by selecting the arbitrary functions and arbitrary constants appropriately.

Grammian solutions for a (2+1)-dimensional integrable coupled modified Date–Jimbo–Kashiwara–Miwa equation

2019 ◽  
Vol 33 (10) ◽  
pp. 1950119 ◽  
Author(s):  
Sheng-Nan Wang ◽  
Juan Hu
Keyword(s):  
Bäcklund Transformation ◽  
Perturbation Expansion ◽  
Hirota Bilinear Method ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Bilinear Bäcklund Transformation ◽  
Hirota Bilinear ◽  
Djkm Equation ◽  
Generation Procedure

In this paper, Grammian solutions to a (2[Formula: see text]+[Formula: see text]1)-dimensional modified Date–Jimbo–Kashiwara–Miwa (mDJKM) equation are presented by using Hirota bilinear method and perturbation expansion. Starting from the Grammian solutions, an integrable coupled mDJKM equation is then obtained and the corresponding Grammian solutions are first derived by utilizing the source generation procedure. Besides, we also construct and solve a coupled DJKM equation via source generation procedure. It is interesting that the coupled mDJKM system constitute a bilinear Bäcklund transformation for the coupled DJKM system. This means that the commutativity of source generation procedure and Bäcklund transformation is valid for the (2[Formula: see text]+[Formula: see text]1)-dimensional DJKM equation.

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

Bilinear Bäcklund transformation, breather- and travelling-wave solutions for a (2+1)-dimensional extended Kadomtsev–Petviashvili II equation in fluid mechanics

2021 ◽  
pp. 2150315
Author(s):  
Yong-Xin Ma ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
He-Yuan Tian ◽  
Shao-Hua Liu
Keyword(s):  
Fluid Mechanics ◽  
Bäcklund Transformation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Test Technique ◽  
Wave Solutions ◽  
Bilinear Bäcklund Transformation

Fluid-mechanics studies are applied in mechanical engineering, biomedical engineering, oceanography, meteorology and astrophysics. In this paper, we investigate a (2+1)-dimensional extended Kadomtsev–Petviashvili II equation in fluid mechanics. Based on the Hirota bilinear method, we give a bilinear Bäcklund transformation. Via the extended homoclinic test technique, we construct the breather-wave solutions under certain constraints. We obtain the velocities of the breather waves, which depend on the coefficients in that equation. Besides, we derive the lump solutions with the periods of the breather-wave solutions tending to the infinity. Based on the polynomial-expansion method, travelling-wave solutions are constructed. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. We graphically discuss the effects of those coefficients on the breather wave and lump.

scholarly journals Time-dependent defects in integrable soliton equations

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190652
Author(s):  
Baoqiang Xia ◽  
Ruguang Zhou
Keyword(s):  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Time Dependent ◽  
Full Line ◽  
Backlund Transformation ◽  
Soliton Equations ◽  
Defect System

We study (1 + 1)-dimensional integrable soliton equations with time-dependent defects located at x  =  c ( t ), where c ( t ) is a function of class C 1 . We define the defect condition as a Bäcklund transformation evaluated at x  =  c ( t ) in space rather than over the full line. We show that such a defect condition does not spoil the integrability of the system. We also study soliton solutions that can meet the defect for the system. An interesting discovery is that the defect system admits peaked soliton solutions.

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