Vector shock soliton and the Hirota bilinear method

Abstract

The Hirota bilinear method is applied to find an exact shock soliton solution of the system reaction-diffusion equations for n-component vector order parameter, with the reaction part in form of the third order polynomial, determined by three distinct constant vectors. The bilinear representation is derived by extracting one of the vector roots (unstable in general), which allows us reduce the cubic nonlinearity to a quadratic one. The vector shock soliton solution, implementing transition between other two roots, as a fixed points of the potential from continuum set of the values, is constructed in a simple way. In our approach, the velocity of soliton is fixed by truncating the Hirota perturbation expansion and it is found in terms of all three roots. Shock solitons for extensions of the model, by including the second order time derivative term and the nonlinear transport term are derived. Numerical solutions illustrating generation of solitary wave from initial step function, depending of the polynomial roots are given.

Approximate explicit solution of Camassa-Holm equation by He’s homotopy perturbation method

“…Fortunately, in recent years, a lot of new methods for solving nonlinear ordinary or partial differential equations are developed. For instance, Bäcklund transformation method [1], inverse scattering method [2], Darboux transformation method [3], Adomian's decomposition method [4], Hirota's bilinear method [5], homotopy analysis method [6,7], tanh-function method [8,9], sine-cosine method [10], the bifurcations method of planar dynamical systems [11,12], homotopy perturbation method (HPM) [13][14][15][16][17][18] etc. Generally speaking, all of the above methods have their own advantages and limitations respectively.…”

Variational iteration method for modified Camassa–Holm and Degasperis–Procesi equations

“…Many powerful methods have been presented, for instance, inverse scattering method [1], Adomian's decomposition method [2][3][4], Hirota's bilinear method [5], exp-function method [6], homotopy analysis method [7], homotopy perturbation method [4,[8][9][10][11][12][13][14][15][16][17][18], variational iteration method (VIM) [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] and other methods [38][39][40][41][42][43].…”

Solitary wave solution of nonlinear multi-dimensional wave equation by bilinear transformation method

“…We extended our previous work [16] for this aim. In addition, to our knowledge, the solitary wave solution for the system of multi-dimensional nonlinear wave equation for n-component vector case has not been formulated explicitly by using Hirota method.…”