scholarly journals Exact Periodic Solutions of the Nonintegrable Kawahara Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Ognyan Yordanov Kamenov ◽  
Anna P. Angova
Keyword(s):  
Evolution Equation ◽  
Periodic Solutions ◽  
Transformation Method ◽  
Periodic Waves ◽  
Kawahara Equation ◽  
Bilinear Transformation ◽  
The Real ◽  
Single Solution ◽  
Spatial Displacements ◽  
Phase Solution

In the present paper, we have obtained an exact biperiodic, one-phase solution of the Kawahara evolution equation. Two classes of real periodic waves generated by the biperiodic solution have been analyzed. A modification of the bilinear-transformation method has been applied allowing to provide a single solution of the residual equation derived from the bidifferential reduction of the considered nonintegrable equation. It is shown that the spatial displacements are individual for each separate harmonic of the real periodic solutions.

scholarly journals Solitary-Wave and Periodic Solutions of the Kuramoto-Velarde Dispersive Equation

2016 ◽  
Vol 46 (3) ◽  
pp. 65-74 ◽  
Author(s):  
Ognyan Y. Kamenov
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Periodic Solutions ◽  
Solitary Waves ◽  
Transformation Method ◽  
Dispersive Equation ◽  
Bilinear Transformation ◽  
Mapping Approach ◽  
Solitary Solutions

Abstract In the present paper, solitary solutions of the Kuramoto- Velarde (K-V) dispersive equation have been found, using the deformation and mapping approach. These exact solutions show the dynamics and the evolution of dispersive solitary waves. In the case α2 = α3, three families of exact periodic solutions have been obtained by employing the bilinear transformation method.

A REDUCTION mKdV METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW DOUBLY-PERIODIC SOLUTIONS FOR NONLINEAR WAVE EQUATIONS

2003 ◽  
Vol 14 (05) ◽  
pp. 661-672 ◽  
Author(s):  
ZHENYA YAN
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Symbolic Computation ◽  
Wave Equations ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Nonlinear Wave Equations ◽  
Cubic Nonlinear Schrödinger Equation

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

New Travelling Solitary Wave and Periodic Solutions of the Generalized Kawahara Equation

2007 ◽  
Author(s):  
Huaitang Chen ◽  
Huicheng Yin ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Solitary Wave ◽  
Periodic Solutions ◽  
Kawahara Equation

scholarly journals Inverse scattering transform for a new non-isospectral integrable non-linear AKNS model

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 153-160 ◽  
Author(s):  
Xudong Gao ◽  
Sheng Zhang
Keyword(s):  
Evolution Equation ◽  
Exact Solutions ◽  
Inverse Scattering ◽  
Spectral Parameter ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Linear Partial Differential Equations ◽  
Scattering Transform ◽  
Non Linear ◽  
Isospectral Problem

Constructing integrable systems and solving non-linear partial differential equations are important and interesting in non-linear science. In this paper, Ablowitz-Kaup-Newell-Segur (AKNS)?s linear isospectral problem and its accompanied time evolution equation are first generalized by embedding a new non-isospectral parameter whose varying with time obeys an arbitrary smooth enough function of the spectral parameter. Based on the generalized AKNS linear problem and its evolution equation, a new non-isospectral Lax integrable non-linear AKNS model is then derived. Furthermore, exact solutions of the derived AKNS model is obtained by extending the inverse scattering transformation method with new time-varying spectral parameter. In the case of reflectinless potentials, explicit n-soliton solutions are finally formulated through the obtained exact solutions.

Sign in / Sign up

Export Citation Format

Share Document