Linear stability of exact solutions for the generalized Kaup-Boussinesq equation and their dynamical evolutions

Abstract

<p style='text-indent:20px;'>The integrability, classification of traveling wave solutions and stability of exact solutions for the generalized Kaup-Boussinesq equation are studied by prolongation structure technique and linear stability analysis. Firstly, it is proved that the generalized Kaup-Boussinesq equation is completely integrable in sense of having Lax pair. Secondly, the complete classification of exact traveling wave solutions of the generalized Kaup-Boussinesq equation are given and a family of exact solutions are proposed. Finally, the stability of these exact solutions are investigated by linear stability analysis and dynamical evolutions, and some stable traveling wave solutions are found. It is shown that the results of linear stability analysis are in excellent agreement with the results from dynamical evolutions.</p>

Prolongation structure and integrability of the modified Sawada-Kotera equation

“…In this section, the Lax pair and conservation laws of the modified SK-KK equation (3) is studied by prolongation technique [20,21].…”

“…We now investigate the prolongation structure and Lax integrability of the modified SK-KK equation (3) by prolongation technique [20,21]. Firstly, define a new set of dependent variables {p, q, r, s} as follows…”

“…Moreover, it is remarked that the modified SK-KK equation (3) has been found through classifying integrable equations by using symmetry approach in [18,19]. However, the prolongation structures [20,21] and Painlevé property of the modified SK-KK equation (3) is still unknown until now. Recent years, there are more and more work on the integrability, exact solutions and the behaviors of solutions for the integrable systems [22][23][24][25][26].…”