Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk

Abstract

We study a nonlinear Schrödinger equation with damping, detuning, and spatially homogeneous input terms, which is called the Lugiato-Lefever equation, on the unit disk with the Neumann boundary conditions. We aim at understanding bifurcations of a so-called cavity soliton which is a radially symmetric stationary spot solution. It is known by numerical simulations that a cavity soliton bifurcates from a spatially homogeneous steady state. We prove the existence of the parameter-dependent center manifold and a branch of radially symmetric steady state in a neighborhood of the bifurcation point. In order to capture further bifurcations of the radially symmetric steady state, we study a degenerate bifurcation for which two radially symmetric modes become unstable simultaneously, which is called the two-mode interaction. We derive a vector field on the center manifold in a neighborhood of such a degenerate bifurcation and present numerical simulations to demonstrate the Hopf and homoclinic bifurcations of bifurcating solutions.2000 Mathematics Subject Classification. Primary: 35Q55; Secondary: 35B36.

The Lugiato-Lefever equation with nonlinear damping caused by two photon absorption

“…The theoretical results from [15,16] were accompanied by numerically computed bifurcation diagrams indicating that the most localized and thus soliton-like solutions can be found at those turning points of the branches that are the farthest away from the curve of trivial solutions. We remark that a two-dimensional version of the Lugiato-Lefever equation posed on the unit disk was recently discussed in [22].…”

“…Taking the expressions for αα * and a 2 0 ᾱα * into the transversality condition (22) finally leads to…”

The Lugiato–Lefever Equation with Nonlinear Damping Caused by Two Photon Absorption

“…The theoretical results from [15,16] were accompanied by numerically computed bifurcation diagrams indicating that the most localized and thus soliton-like solutions can be found at those turning points of the branches that are the farthest away from the curve of trivial solutions. We remark that a two-dimensional version of the Lugiato-Lefever equation posed on the unit disk was recently discussed in [22].…”