New Explicit Solutions of (1 + 1)-Dimensional Variable-Coefficient Broer—Kaup System

2010 ◽  
Vol 54 (6) ◽  
pp. 965-970 ◽  
Author(s):  
Yan Zhi-Lian ◽  
Zhou Jian-Ping
Keyword(s):  
Variable Coefficient ◽  
Explicit Solutions ◽  
Dimensional Variable

Explicit Analytical Solutions of Linear and Nonlinear Interior Heat and Mass Transfer Equation Sets for Drying Process

2003 ◽  
Vol 125 (1) ◽  
pp. 175-178 ◽  
Author(s):  
Ruixian Cai ◽  
Na Zhang
Keyword(s):  
Analytical Solutions ◽  
Variable Coefficient ◽  
Infinite Plate ◽  
Explicit Solutions ◽  
Drying Process ◽  
Mass Transfer Equation ◽  
Mathematical Skills ◽  
One Dimensional ◽  
Linear And Nonlinear ◽  
Dimensional Variable

Some algebraic explicit analytical solutions of the unsteady one-dimensional variable coefficient partial differential equation set describing drying process of an infinite plate are derived and given. Such explicit solutions have not yet been published before. Besides their irreplaceable theoretical value, analytical solutions can also serve as benchmark to check the results of recently rapidly developing numerical calculation and study various computational methods. In addition, some mathematical skills used in this paper are special and deserve further attention.

scholarly journals ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

2012 ◽  
Vol 19 (04) ◽  
pp. 1250028
Author(s):  
TING SU ◽  
HUIHUI DAI ◽  
XIAN GUO GENG
Keyword(s):  
Optical Fibers ◽  
Variable Coefficient ◽  
High Order ◽  
Explicit Solutions ◽  
Compatibility Conditions ◽  
Dressing Method ◽  
Integrable Equations ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Hirota Equations

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.

Integrable variable-coefficient derivative Spin-1 Gross–Pitaevskii equations and their explicit solutions

2021 ◽  
Author(s):  
Ting Su ◽  
Junhong Yao ◽  
Yanan Huang
Keyword(s):  
Variable Coefficient ◽  
Separation Of Variables ◽  
Lax Pair ◽  
Explicit Solutions ◽  
Dressing Method

Based on the generalized dressing method, we propose a new integrable variable coefficient Spin-1 Gross–Pitaevskii equations and derive their Lax pair. Using separation of variables, we have derived explicit solutions of the equations. In order to analyze the characteristic of derived solution, the graphical wave of the solutions is plotted with the aid of Matlab.

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.

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.

Dark solitons, Lax pair and infinitely-many conservation laws for a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in the inhomogeneous Heisenberg ferromagnetic spin chain

2017 ◽  
Vol 31 (03) ◽  
pp. 1750013 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
De-Yin Liu ◽  
Xiao-Yu Wu ◽  
Jun Chai ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Spin Chain ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Nonlinear Schrodinger Equation ◽  
Dark Solitons ◽  
Heisenberg Ferromagnetic Spin Chain ◽  
Dimensional Variable

Under investigation in this paper is a generalized (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain. Lax pair and infinitely-many conservation laws are derived, indicating the existence of the multi-soliton solutions for such an equation. Via the Hirota method with an auxiliary function, bilinear forms, dark one-, two- and three-soliton solutions are derived. Propagation and interactions for the dark solitons are illustrated graphically: Velocity of the solitons is linearly related to the coefficients of the second- and fourth-order dispersion terms, while amplitude of the solitons does not depend on them. Interactions between the two solitons are shown to be elastic, while those among the three solitons are pairwise elastic.

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