Linear Instability of Solitary Wave Solutions of the Kawahara Equation and Its Generalizations

The linear stability problem for solitary wave states of the Kawahara—or fifthorder KdV-type—equation and its generalizations is considered. A new formulation of the stability problem in terms of the symplectic Evans matrix is presented. The formulation is based on a multisymplectification of the Kawahara equation, and leads to a new characterization of the basic solitary wave, including changes in the state at infinity represented by embedding the solitary wave in a multiparameter family. The theory is used to give a rigorous geometric sufficient condition for instability. The theory is abstract and applies to a wide range of solitary wave states. For example, the theory is applied to the families of solitary waves found by Kichenassamy–Olver and Levandosky.