Bifurcation analysis to the Lugiato–Lefever equation in one space dimension

We study the stability and bifurcation of steady states for a certain kind of damped driven nonlinear Schrödinger equation with cubic nonlinearity and a detuning term in one space dimension, mathematically in a rigorous sense. It is known by numerical simulations that the system shows lots of coexisting spatially localized structures as a result of subcritical bifurcation. Since the equation does not have a variational structure, unlike the conservative case, we cannot apply a variational method for capturing the ground state. Hence, we analyze the equation from a viewpoint of bifurcation theory. In the case of a finite interval, we prove the fold bifurcation of nontrivial stationary solutions around the codimension two bifurcation point of the trivial equilibrium by exact computation of a fifth-order expansion on a center manifold reduction. In addition, we analyze the steady-state mode interaction and prove the bifurcation of mixed-mode solutions, which will be a germ of localized structures on a finite interval. Finally, we study the corresponding problem on the entire real line by use of spatial dynamics. We obtain a small dissipative soliton bifurcated adequately from the trivial equilibrium.