Mariana Haragus and Dmitry E. Pelinovsky
Linear instability of breathers for the focusing nonlinear Schrodinger equation
Journal of Nonlinear Science 32 (2022) 66 (40 pages)
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
Schrodinger 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.
Keywords:
nonlinear Schrodinger equation; breathers; linear instability;
Darboux transformations;