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

Tomoyuki Miyaji,Yoshio Tsutsumi

doi:10.3934/cpaa.2018078

Abstract

We study a nonlinear Schrodinger 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.

Highlights

We study the Lugiato-Lefever equation (LLE) on the unit disk:
We derive the normal form of bifurcation caused by two-mode interaction in Section 5, that is, we prove Theorem 1.4
In this and the following subsections, we study steadystate mode interactions of two radially symmetric modes by using AUTO07-p [4] that is a software for bifurcation analysis developed by E

Summary

Introduction

We study the Lugiato-Lefever equation (LLE) on the unit disk:. √ where i = −1, ∆ is the two-dimensional Laplace operator, Ω ⊂ R2 is the unit disk, and n is an outward normal unit vector on ∂Ω. We study the Lugiato-Lefever equation (LLE) on the unit disk:. √ where i = −1, ∆ is the two-dimensional Laplace operator, Ω ⊂ R2 is the unit disk, and n is an outward normal unit vector on ∂Ω. The symbols θ and b are real numbers corresponding to detuning and diffraction parameters, respectively. LLE is a nonlinear Schrodinger equation with damping, detuning and driving force. It was proposed by Lugiato and Lefever as a model equation for pattern formation in the ring cavity with the Kerr medium [11]. Scroggie et al have studied pattern formation of the two-dimensional. The Lugiato-Lefever equation, center manifold, normal form, steadystate mode interactions, Hopf bifurcation, homoclinic bifurcation. The Lugiato-Lefever equation, center manifold, normal form, steadystate mode interactions, Hopf bifurcation, homoclinic bifurcation. ∗ Corresponding author

TOMOYUKI MIYAJI AND YOSHIO TSUTSUMI

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