Linear instability of breathers for the focusing nonlinear Schr{ö}dinger equation

Abstract

Relying upon tools from the theory of integrable systems, we discuss the linear instability of the Kuznetsov-Ma breathers and the Akhmediev breathers of the focusing nonlinear Schrödinger equation. We use the Darboux transformation to construct simultaneously the breathers and the exact solutions of the Lax system associated with the breathers. We obtain a full description of the Lax spectra for the two breathers, including multiplicities of eigenvalues. Solutions of the linearized NLS equations are then obtained from the eigenfunctions and generalized eigenfunctions of the Lax system. While we do not attempt to prove completeness of eigenfunctions, we aim to determine the entire set of solutions of the linearized NLS equations generated by the Lax system in appropriate function spaces.

Breather Solutions to the Cubic Whitham Equation

“…As suggested by the short summary above, the study of breather solutions is an active field of research with a wide range of open problems concerning existence, stability and relation to extreme events. For the NLS equation, new breather solutions are found in closed form on a regular basis (see for example [19]), and there are several results showing existence [4], stability [20,21], or nonexistence or instability [22][23][24]. In the present work we present numerical evidence for the existence of breather solutions in a non-integrable model.…”