scholarly journals The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

2006 ◽  
Author(s):  
Tadashi Kobayashi
Keyword(s):  
Variable Coefficients ◽  
Canonical Forms ◽  
Painlevé Test ◽  
Soliton Equations ◽  
Higher Dimensional ◽  
Dimensional Soliton

scholarly journals Vector Solitons of a Coupled Schrödinger System with Variable Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-19 ◽  
Author(s):  
Juan Carlos Muñoz Grajales
Keyword(s):  
Numerical Simulations ◽  
Finite Energy ◽  
Variable Coefficients ◽  
Soliton Equations ◽  
Nonlinear Schrödinger ◽  
Coupled Schrödinger System ◽  
Nonlinear Schrödinger System ◽  
Schrödinger System

We show the existence of waveforms of finite-energy (vector solitons) for a coupled nonlinear Schrödinger system with inhomogeneous coefficients. Furthermore, some of these solutions are approximated using a Newton-type iteration, combined with a collocation-spectral strategy to discretize the corresponding soliton equations. Some numerical simulations concerned with analysis of a collision of two oncoming vector solitons of the system are also performed.

Painlevé Test, Bäcklund Transformation and Consistent Riccati Expansion Solvability for two Generalised Cylindrical Korteweg-de Vries Equations with Variable Coefficients

2018 ◽  
Vol 73 (3) ◽  
pp. 207-213 ◽  
Author(s):  
Rehab M. El-Shiekh
Keyword(s):  
Vries Equation ◽  
Dusty Plasmas ◽  
Variable Coefficients ◽  
Bäcklund Transformations ◽  
Painlevé Test ◽  
New Exact Solutions ◽  
Backlund Transformations ◽  
Integrability Conditions ◽  
De Vries ◽  
Consistent Riccati Expansion

AbstractIn this paper, the integrability of the (2+1)-dimensional cylindrical modified Korteweg-de Vries equation and the (3+1)-dimensional cylindrical Korteweg-de Vries equation with variable coefficients arising in dusty plasmas in its generalised form was studied by two different techniques: the Painlevé test and the consistent Riccati expansion solvability. The integrability conditions and Bäcklund transformations are constructed. By using Bäcklund transformations and the solutions of the Riccati equation many new exact solutions are found for the two equations in this study. Finally, the application of the obtained solutions in dusty plasmas is investigated.

SOLITON EIGENFUNCTION EQUATIONS: THE IST INTEGRABILITY AND SOME PROPERTIES

1990 ◽  
Vol 02 (04) ◽  
pp. 399-440 ◽  
Author(s):  
B.G. KONOPELCHENKO
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Wide Class ◽  
Eigenvalue Problems ◽  
Nonlinear Differential Equations ◽  
Soliton Equations ◽  
Spectral Transform ◽  
Dimensional Soliton ◽  
Operator Representations ◽  
General Method

Eigenfunctions of the linear eigenvalue problems for the soliton equations obey nonlinear differential equations. It is shown that these eigenfunction equations are integrable by the inverse spectral transform (IST) method. They have triad operator representations. Eigenfunction equations are the generating equations and possess other interesting properties. Eigenfunction equations form a new wide class of nonlinear integrable equations. Eigenfunction equations for several typical, well-known (1+1)-, (2+1)- and multi-dimensional soliton equations are considered. A general method for constructing the auxiliary linear systems for the eigenfunction equations is proposed. It is shown that the vertical hierarchies of the eigenfunction equations contain only finite numbers of different members in the cases considered. The properties of such hierarchies for soliton equations are closely connected with their Painleve properties. Some “linear” properties of the eigenfunction equations are also discussed.

New exact solutions of the reaction–diffusion equation with variable coefficients via the mathematical computation

2018 ◽  
Vol 11 (04) ◽  
pp. 1850051 ◽  
Author(s):  
Jin Hyuk Choi ◽  
Hyunsoo Kim
Keyword(s):  
Diffusion Equation ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Painlevé Test ◽  
New Exact Solutions ◽  
Mathematical Computation

In this paper, we construct new exact solutions of the reaction–diffusion equation with time dependent variable coefficients by employing the mathematical computation via the Painlevé test. We describe the behaviors and their interactions of the obtained solutions under certain constraints and various variable coefficients.

Sign in / Sign up

Export Citation Format

Share Document