Nonlinear stability of periodic traveling wave solutions to the Schrödinger and the modified Korteweg-de Vries equations
The paper revisits the previously studied problem of the stability of periodic solutions to the nonlinear Schrödinger (NLS) equation in the form of Jacobi's elliptic functions (dn and cn, hence called ``cnoidal'' and ``dnoidal'' waves). It is proved that some families of these solutions are stable against perturbations that have the same periodicity as the unperturbed solutions, and (nonlinearly) unstable against perturbations with double period (a period-doubling instability). In fact, the instability of all periodic solutions to the NLS solution was known before. It is stated that the stability of other families of periodic solutions is unknown, and approaches developed in earlier well-known papers on this issue are criticized. The author also considers periodic cnoidal-type solutions to the modified Korteweg-de Vries equations, and states that stability remains unknown. In addition, there are considered some recently published particular exact solutions to the NLS equation with an external potential which are also expressed in terms of elliptic functions. It is stated that some of them are stable, and some are not.
scientific article; zbMATH DE number 5144839
scientific article