## Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity.(English)Zbl 1099.37057

Summary: We investigate the KdV like equation with higher-order nonlinearity $u_t+a(1+ bu^n)u^nu_x+u_{xxx}=0$ with $$n\geq 1$$,$$a,b\in\mathbb{R}$$ and $$a\neq 0$$. The bifurcations and explicit expressions of solitary wave solutions for the equation are discussed by using the bifurcation method and qualitative theory of dynamical systems. The bifurcation diagrams, existence and number of the solitary waves are given.

### MSC:

 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 35Q51 Soliton equations 35Q53 KdV equations (Korteweg-de Vries equations) 37G99 Local and nonlocal bifurcation theory for dynamical systems

### Keywords:

explicit solutions