Bifurcation and stability of travelling waves in self-focusing planar wave guides

A mathematical problem arising in the modelling of travelling waves in self-focusing waveguides is studied. It involves a nonlinear Schrödinger equation in R. In the paraxial approximation, the travelling 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, for very large and very small positive frequencies. In the case of even symmetry, a global bifurcation result is proved, yielding a branch of positive even solutions parametrized by frequencies in the whole half-line (0,∞). The stability of standing waves of the NLS relies on the monotonicity of the L-norm with respect to the frequency. This criterion, precisely going back to formal arguments in the physical literature on self-focusing waveguides, is proved to hold along the whole global branch of solutions in the even case. The stability of standing waves with any positive frequencies follows. The mathematical results are applied to the study of a non-homogeneous planar self-focusing waveguide with a Kerr-type nonlinear response, yielding existence of TE travelling waves and their stability among the set of TE modes. The bifurcation analysis gives informations on the power of the beam. In particular, it provides the possibility of low power cut-off.