NONLINEAR EVOLUTION EQUATIONS AND THE PAINLEVÉ TEST

1992 ◽  
Vol 07 (08) ◽  
pp. 1669-1683 ◽  
Author(s):  
W.-H. STEEB ◽  
N. EULER
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Nonlinear Evolution ◽  
Energy Eigenvalue ◽  
Nonlinear Evolution Equations ◽  
The Self ◽  
Painlevé Test ◽  
Yang Mills ◽  
Petviashvili Equation ◽  
Klein Gordon

A survey is given of new results of the Painlevé test and nonlinear evolution equations where ordinary- and partial-differential equations are considered. We study the semiclassical Jaynes-Cumming model, the energy-eigenvalue-level-motion equation, the Kadomtsev-Petviashvili equation, the nonlinear Klein-Gordon equation and the self-dual Yang-Mills equation.

ON THE GENERALIZED TANH METHOD FOR THE (2+1)-DIMENSIONAL BREAKING SOLITON EQUATION

1995 ◽  
Vol 10 (38) ◽  
pp. 2937-2941 ◽  
Author(s):  
BO TIAN ◽  
YI-TIAN GAO
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
The Self ◽  
Solitary Wave Solutions ◽  
Soliton Equation ◽  
Yang Mills ◽  
Wave Solutions ◽  
Tanh Method ◽  
Open Question

There is an open question as to whether or not the recently-proposed tanh method can be modified in order to proceed beyond the traveling or solitary wave solutions for nonlinear evolution equations. On the other hand, the class of the breaking soliton equations, which the self-dual Yang-Mills equation is found to belong to, is of current interest. In this letter, we propose a generalized tanh method, with symbolic computation, to construct a family of soliton-like solutions for a (2+1)-dimensional breaking soliton equation.

A METHOD TO CONSTRUCT WEIERSTRASS ELLIPTIC FUNCTION SOLUTION FOR NONLINEAR EQUATIONS

2011 ◽  
Vol 25 (14) ◽  
pp. 1931-1939 ◽  
Author(s):  
LIANG-MA SHI ◽  
LING-FENG ZHANG ◽  
HAO MENG ◽  
HONG-WEI ZHAO ◽  
SHI-PING ZHOU
Keyword(s):  
Nonlinear Equations ◽  
Elliptic Function ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Order Derivative ◽  
First Order ◽  
Function Solution ◽  
Weierstrass Elliptic Function ◽  
Klein Gordon

A method for constructing the solutions of nonlinear evolution equations by using the Weierstrass elliptic function and its first-order derivative was presented. This technique was then applied to Burgers and Klein–Gordon equations which showed its efficiency and validality for exactly some solving nonlinear evolution equations.

scholarly journals The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Yun-Mei Zhao ◽  
Ying-Hui He ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Equation Method ◽  
Simplest Equation Method ◽  
Klein Gordon ◽  
Schrödinger's Equation ◽  
Wave Reduction

A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

scholarly journals On a direct construction of inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension

2004 ◽  
Vol 2004 (58) ◽  
pp. 3117-3128
Author(s):  
H. H. Chen ◽  
J. E. Lin
Keyword(s):  
Inverse Scattering ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Spatial Dimension ◽  
Nonlinear Evolution Equations ◽  
Scattering Problems ◽  
Direct Construction ◽  
Inverse Scattering Problems ◽  
Petviashvili Equation ◽  
The One

We present a method to construct inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension. The temporal component is the adjoint of the linearized equation and the spatial component is a partial differential equation with respect to the spatial variables. Although this idea has been known for the one-spatial dimension for some time, it is the first time that this method is presented for the case of the higher-spatial dimension. We present this method in detail for the Veselov-Novikov equation and the Kadomtsev-Petviashvili equation.

Application of Extended Transformed Rational Function Method to Some (3+1) Dimensional Nonlinear Evolution Equations

2018 ◽  
pp. 433-437
Keyword(s):  
Rational Function ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Mapping Method ◽  
Nonlinear Evolution Equations ◽  
Petviashvili Equation ◽  
Complexiton Solutions ◽  
Transformed Rational Function Method

The transformed rational function method can be considered as unification of the tanh type methods, the homogeneous balance method, the mapping method, the exp-function method and the F-expansion type methods. In this paper, we present complexiton solutions of (3+1) dimensional Korteweg-de Vries (KdV) equation and a new (3+1) dimensional generalized Kadomtsev-Petviashvili equation by using extended transformed rational function method which provides very useful and effective way to obtain complexiton solutions of nonlinear evolution equations.

NEW TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR EVOLUTION EQUATIONS

2011 ◽  
Vol 25 (02) ◽  
pp. 319-327 ◽  
Author(s):  
CHENG-JIE BAI ◽  
HONG ZHAO ◽  
HENG-YING XU ◽  
XIA ZHANG
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mapping Method ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions ◽  
Trigonometric Function Solutions ◽  
Klein Gordon ◽  
Transformation Relation

The deformation mapping method is extended to solve a class of nonlinear evolution equations (NLEEs). Many types of explicit and exact traveling wave solutions, which contain solitary wave solutions, trigonometric function solutions, and Jacobian elliptic function solutions, are obtained by a simple algebraic transformation relation between the solutions of the NLEEs and those of the cubic nonlinear Klein–Gordon (NKG) equation.

Conservation laws and Hamiltonian structures of the generalized sine-Gordon hierarchy

2014 ◽  
Vol 28 (32) ◽  
pp. 1450248
Author(s):  
Fang Li ◽  
Bo Xue ◽  
Yan Li
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Nonlinear Evolution Equations ◽  
Conserved Quantities ◽  
Trace Identity ◽  
Sine Gordon Equation ◽  
Hamiltonian Structures ◽  
Matrix Spectral Problem

By introducing a 2 × 2 matrix spectral problem, a new hierarchy of nonlinear evolution equations is proposed. A typical equation in this hierarchy is the generalization of sine-Gordon equation. With the aid of trace identity, the Hamiltonian structures of the hierarchy are constructed. In addition, the infinite sequence of conserved quantities of the generalized sine-Gordon equation are obtained.

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