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On different kinds of solutions to simplified modified form of a Camassa-Holm equation

Gündoğdu Hami, Gözükızıl Ömer Faruk

Journal of Applied Mathematics and Computational Mechanics

2019

Vol. 18, nr 2

31--40

In this research, our purpose is to investigate some types of solutions to a simplified modified form of the Camassa-Holm equation. The Jacobi elliptic function expansion method is applied to this equation. Then, a lot of travelling wave solutions are obtained. The derived solutions are in the form of Jacobi elliptic functions, hyperbolic functions, and trigonometric functions. Graphics of solutions are drawn in order to determine the types of the solutions. Furthermore, different kinds of solutions such as the singular kink wave solution, the kink wave solution, and the periodic solution are achieved.

The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation

Shepelsky D., Zielinski L.

Opuscula Mathematica

2017

Vol. 37, no. 1

167--187

The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation [formula] with zero boundary conditions (as [formula]) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented.

Two-Dimensional Chebyshev Wavelet Method for Camassa-Holm Equation with Riesz Fractional Derivative Describing Propagation of Shallow Water Waves

Gupta A. K., Saha Ray S.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

77--89

In this article, the authors present a new wavelet based method viz. Chebyshev wavelet method to compute the numerical solution of Riesz time-fractional Camassa-Holm equation. The approximate solutions of Riesz time-fractional Camassa-Holm equation thus obtained by two-dimensional Chebyshev wavelet method are compared with those obtained by analytical methods such as homotopy analysis method (HAM) and variational iteration method (VIM). The present scheme is quite simple, effective and appropriate for obtaining the numerical solution of the Riesz time-fractional Camassa-Holm equation.