Homoclinic orbits for soliton equations solvable via the quadratic bundle

Abstract

A method is proposed to generate homoclinic solutions for an integrable nonlinear PDE with periodic boundaries. This approach resembles the dressing method known in the theory of solitons. The pole positions in the dressing factor are given by the complex double points of the Floquet spectrum associated with unstable modes of a seed solution of the nonlinear equation. As an example, we obtain the homoclinic orbit for the modified nonlinear Schrödinger equation solvable by the Wadati–Konno–Ichikawa spectral problem.