Bifurcation and Stability of Travelling Waves in Self-focusing Planar Waveguides

摘要

A mathematical problem arising in the modelling of travelling waves in selffocusing waveguides is studied. it involves a nonlinear Schrodinger equation in R~(I1.n the p+araxial1 apro)ximation, traveling waves are interpreted as standing waves of the. NLS. Local and global bifurcation results are established for the corresponding "stationary" equation. The combination of variational arguments with an implicit function theorem yields smooth branches of positive solutions parametrized by the "frequency" of the standing wave of NLS. Precise informations on the asymptotic behaviour of various norms of the solutions along the branches are given. In the case of even symmetry, a global bifurcation result is proved, providing a branch of positive even solutions parametrized by frequencies in the whole half-line (0,∞ ). The stability of the standing waves of the NLS relies on the monotonicity of their L~2-norm as a function of the frequency.