Stability and bifurcation analysis to dissipative cavity soliton of Lugiato-Lefever equation in one dimensional bounded interval

In this paper, a mathematically rigorous analysis for bifurcation structure of a spatially uniform stationary solution in nonlinear Schrödinger equations with a cubic nonlinearity and a dissipation, and with a detuning term is presented. Numerically, it has been reported that the “snake bifurcation” occurs, but this equation does not have the variational (Hamiltonian) structure. Therefore, both a variational technique capturing the ground state of conservative system and a dynamical system technique with reversible system of 1:1 resonance cannot be applied to the problem. We here ensure that the pitchfork type bifurcation happens only under the physically natural conditions by using the bifurcation theory with the symmetry, and moreover, make a much finer analysis at the codimension two bifurcation point to give a proof to the “fold bifurcation” around the singular point. In the consequence, the bending solution branch at least once has been captured in an adequate parameter area near the singularity, which means that a part of the global bifurcation structure is infinitesimally folding into the singularity with codimension two. AMS classification scheme numbers: 34C23, 37L10, 70K20, 70K50 Submitted to: Nonlinearity ‡ Corresponding author Stability and bifurcation analysis to Lugiato-Lefever equation 2