Solitons for a (2+1)-dimensional variable-coefficient Bogoyavlensky-Konopelchenko equation in a fluid

2017 ◽  
Vol 31 (25) ◽  
pp. 1750216 ◽  
Author(s):  
Ya-Le Wang ◽  
Yi-Tian Gao ◽  
Shu-Liang Jia ◽  
Gao-Fu Deng ◽  
Wen-Qiang Hu
Keyword(s):  
Phase Shift ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Time Dependent ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Long Wave ◽  
Dimensional Variable ◽  
Propagation Properties ◽  
Binary Bell Polynomials

In this letter, a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Bogoyavlensky-Konopelchenko equation is investigated, which describes the interaction of a Riemann wave propagating along the y-axis and a long wave propagating along the x-axis in a fluid. Under two different constraints of the time-dependent coefficients in this equation, two different bilinear forms are derived by virtue of the binary Bell polynomials. Multiple solitary waves are constructed via the Hirota method, whose propagation properties and interaction characteristics are investigated graphically as well. Propagation and interaction of the solitons are illustrated graphically: (i) time-dependent coefficients affect the shape of the solitons; (ii) interaction of the solitons is elastic, i.e., amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.

Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves

2018 ◽  
Vol 32 (08) ◽  
pp. 1750268 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
Yong-Jiang Guo ◽  
Hui-Min Li
Keyword(s):  
Water Waves ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Linear Superposition ◽  
Elastic Interactions ◽  
The One ◽  
Dimensional Variable

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

scholarly journals Bell Polynomials Approach Applied to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wen-guang Cheng ◽  
Biao Li ◽  
Yong Chen
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Bilinear Method ◽  
Bilinear Bäcklund Transformation ◽  
Dimensional Variable ◽  
Binary Bell Polynomials ◽  
Hirota Bilinear

The bilinear form, bilinear Bäcklund transformation, and Lax pair of a (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived through Bell polynomials. The integrable constraint conditions on variable coefficients can be naturally obtained in the procedure of applying the Bell polynomials approach. Moreover, theN-soliton solutions of the equation are constructed with the help of the Hirota bilinear method. Finally, the infinite conservation laws of this equation are obtained by decoupling binary Bell polynomials. All conserved densities and fluxes are illustrated with explicit recursion formulae.

scholarly journals Lump and Interaction Solutions to the ( 3 + 1 )-Dimensional Variable-Coefficient Nonlinear Wave Equation with Multidimensional Binary Bell Polynomials

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Xuejun Zhou ◽  
Onur Alp Ilhan ◽  
Fangyuan Zhou ◽  
Sutarto Sutarto ◽  
Jalil Manafian ◽  
...  
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Regularity Of Solutions ◽  
Bell Polynomials ◽  
Soliton Theory ◽  
Dimensional Variable ◽  
Binary Bell Polynomials

In this paper, we study the ( 3 + 1 )-dimensional variable-coefficient nonlinear wave equation which is taken in soliton theory and generated by utilizing the Hirota bilinear technique. We obtain some new exact analytical solutions, containing interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the generalized ( 3 + 1 )-dimensional nonlinear wave equation in liquid with gas bubbles by the Maple symbolic package. Making use of Hirota’s bilinear scheme, we obtain its general soliton solutions in terms of bilinear form equation to the considered model which can be obtained by multidimensional binary Bell polynomials. Furthermore, we analyze typical dynamics of the high-order soliton solutions to show the regularity of solutions and also illustrate their behavior graphically.

Bilinear forms and soliton solutions for a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation in an optical fiber

2020 ◽  
Vol 34 (30) ◽  
pp. 2050336
Author(s):  
Dong Wang ◽  
Yi-Tian Gao ◽  
Jing-Jing Su ◽  
Cui-Cui Ding
Keyword(s):  
Schrödinger Equation ◽  
Optical Fiber ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Bilinear Forms ◽  
Dimensional Variable

In this paper, under investigation is a (2 + 1)-dimensional variable-coefficient nonlinear Schrödinger equation, which is introduced to the study of an optical fiber, where [Formula: see text] is the temporal variable, variable coefficients [Formula: see text] and [Formula: see text] are related to the group velocity dispersion, [Formula: see text] and [Formula: see text] represent the Kerr nonlinearity and linear term, respectively. Via the Hirota bilinear method, bilinear forms are obtained, and bright one-, two-, three- and N-soliton solutions as well as dark one- and two-soliton solutions are derived, where [Formula: see text] is a positive integer. Velocities and amplitudes of the bright/dark one solitons are obtained via the characteristic-line equations. With the graphical analysis, we investigate the influence of the variable coefficients on the propagation and interaction of the solitons. It is found that [Formula: see text] can only affect the phase shifts of the solitons, while [Formula: see text], [Formula: see text] and [Formula: see text] determine the amplitudes and velocities of the bright/dark solitons.

Solitons, Bäcklund transformations, Lax pair and conservation laws for the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients

2016 ◽  
Vol 30 (03) ◽  
pp. 1650008 ◽  
Author(s):  
Lei Liu ◽  
Bo Tian ◽  
Wen-Rong Sun ◽  
Yu-Feng Wang ◽  
Yun-Po Wang
Keyword(s):  
Conservation Laws ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Time Dependent ◽  
Bäcklund Transformations ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Bell Polynomial ◽  
Backlund Transformations

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg–de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz–Kaup–Newell–Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.

scholarly journals Soliton Molecules and Some Novel Types of Hybrid Solutions to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Shuxin Yang ◽  
Zhao Zhang ◽  
Biao Li
Keyword(s):  
Variable Coefficient ◽  
Resonance Condition ◽  
Soliton Solutions ◽  
Wave Number ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Dimensional Variable

Soliton molecules of the (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived by N-soliton solutions and a new velocity resonance condition. Moreover, soliton molecules can become asymmetric solitons when the distance between two solitons of the molecule is small enough. Finally, we obtained some novel types of hybrid solutions which are components of soliton molecules, lump waves, and breather waves by applying velocity resonance, module resonance of wave number, and long wave limit method. Some figures are presented to demonstrate clearly dynamics features of these solutions.

Solitons and integrability for a (2+1)-dimensional generalized variable-coefficient shallow water wave equation

2017 ◽  
Vol 31 (03) ◽  
pp. 1750012 ◽  
Author(s):  
Ya-Le Wang ◽  
Yi-Tian Gao ◽  
Shu-Liang Jia ◽  
Zhong-Zhou Lan ◽  
Gao-Fu Deng ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Shallow Water Wave ◽  
Binary Bell Polynomials

Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional generalized variable-coefficient shallow water wave equation which can be reduced to several integrable equations, such as the Korteweg–de Vries (KdV) equation and the Calogero–Bogoyavlenskii–Schiff (CBS) equation. Bilinear forms, Bäcklund transformation, Lax pair and infinite conservation laws are derived based on the binary Bell polynomials. N-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the N-soliton interaction in the scaled space and time coordinates; (ii) positions of the solitons depend on the sign of wave numbers after each interaction; (iii) interaction of the solitons is elastic, i.e. the amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.

Soliton interactions and Bäcklund transformation for a (2+1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili equation in fluid dynamics

2018 ◽  
Vol 32 (02) ◽  
pp. 1750170 ◽  
Author(s):  
Zi-Jian Xiao ◽  
Bo Tian ◽  
Yan Sun
Keyword(s):  
Fluid Dynamics ◽  
Shock Waves ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Bilinear Forms ◽  
Backlund Transformation ◽  
Bell Polynomial ◽  
Soliton Interactions ◽  
Petviashvili Equation ◽  
Dimensional Variable

In this paper, we investigate a (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient modified Kadomtsev-Petviashvili (mKP) equation in fluid dynamics. With the binary Bell-polynomial and an auxiliary function, bilinear forms for the equation are constructed. Based on the bilinear forms, multi-soliton solutions and Bell-polynomial-type Bäcklund transformation for such an equation are obtained through the symbolic computation. Soliton interactions are presented. Based on the graphic analysis, Parametric conditions for the existence of the shock waves, elevation solitons and depression solitons are given, and it is shown that under the condition of keeping the wave vectors invariable, the change of [Formula: see text] and [Formula: see text] can lead to the change of the solitonic velocities, but the shape of each soliton remains unchanged, where [Formula: see text] and [Formula: see text] are the variable coefficients in the equation. Oblique elastic interactions can exist between the (i) two shock waves, (ii) two elevation solitons, and (iii) elevation and depression solitons. However, oblique interactions between (i) shock waves and elevation solitons, (ii) shock waves and depression solitons are inelastic.

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