Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity

Abstract

The solitary waves to the double dispersion equation with quadratic-cubic nonlinearity are explicitly constructed.Grillakis, Shatah and Strauss’ stability theory is applied for the investigation of the orbital stability or instability of solitary waves to the double dispersion equation. An analytical formula, related to some conservation laws of the problem, is derived. As a consequence, the dependence of orbital stability or instability on the parameters of the problem is demonstrated. A complete characterization of the values of the wave velocity, for which the solitary waves to the generalized Boussinesq equation are orbitally stable or unstable, is given.In the special case of a quadratic nonlinearity our results are reduced to those known in the literature.