Orbital stability of solitary waves to the coupled compound KdV and MKdV equations with two components

Abstract

This paper is to study the following coupled version of compound KdV and MKdV equations with two components $ \left\{ \begin{array}{l} {u_t} + \alpha v{v_x} + \beta {u^2}{u_x} + {u_{xxx}} + \lambda u{u_x} = 0,\beta \gt 0,\\ {v_t} + \alpha {(uv)_x} + 2v{v_x} = 0, \end{array} \right. $ which clearly has Hamiltonian form. The orbital stability and instability of solitary waves with nonzero asymptotic value have been few studied. In this paper, we mainly consider the orbital stability and instability of solitary waves with zero or nonzero asymptotic value for this equations. Precisely, we first obtain two explicitly exact solitary waves with zero asymptotic value and four explicitly exact solitary waves with nonzero asymptotic value. Secondly, we conclude some results on the orbital stability of solitary waves with zero or nonzero asymptotic value. To this aim, in order to overcome the difficulty in studying orbital stability of solitary waves with nonzero asymptotic value, we use a translation transformation to transfer this problem into solitary waves with zero asymptotic value for a reduced nonlinear coupled equations. Then by applying the classical orbital stability theory presented by Grillakis et al. and Bona et al., we obtain the orbital stability and instability of solitary waves with zero asymptotic value for the new equations. We finally derive some results on orbital stability of solitary waves with zero or nonzero asymptotic value. In addition, we also obtain the stability results for the coupled compound KdV and MKdV equations with the degenerate condition $v = 0$, called the compound KdV and MKdV equation, which have been studied by Zhang et al.