2 research outputs found
Orbital stability of periodic waves for the nonlinear Schroedinger equation
The nonlinear Schroedinger equation has several families of quasi-periodic
travelling waves, each of which can be parametrized up to symmetries by two
real numbers: the period of the modulus of the wave profile, and the variation
of its phase over a period (Floquet exponent). In the defocusing case, we show
that these travelling waves are orbitally stable within the class of solutions
having the same period and the same Floquet exponent. This generalizes a
previous work where only small amplitude solutions were considered. A similar
result is obtained in the focusing case, under a non-degeneracy condition which
can be checked numerically. The proof relies on the general approach to orbital
stability as developed by Grillakis, Shatah, and Strauss, and requires a
detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure