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

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2018, Volume 17, Issue 4: 1633-1650. Doi: 10.3934/cpaa.2018078

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

Tomoyuki Miyaji^1, and
Yoshio Tsutsumi^2, ,

1.
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Tokyo 164-8525, Japan
2.
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

^* Corresponding author
^* Corresponding author

Received: November 30, 2016

Revised: June 30, 2017

Published: April 2018

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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.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B36.

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Figure 1. Bifurcation diagrams for (18) near $(0, 2)$-$(0, 3)$ mode interactions at $\theta_d = 1.2$. (A) one-parameter bifurcation diagrams at $\nu_2 = 0.05$. (B) two-parameter bifurcation diagram

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Figure 2. Limit cycles for (18) near $(0, 2)$-$(0, 3)$ mode interactions at $\theta_d = 1.2$. (A) limit cycles for several values of $\nu_1$ at $\nu_2 = 0.05$. (B) plot of $|\nu_1-\nu_c|$ versus the period of limit cycles, where $\nu_c$ is the homoclinic bifurcation point. The horizontal axis is plotted on a log scale. The dashed line is the graph of a function proportional to $-\mathrm{ln}|\nu_1 - \nu_c|/\lambda_{m}$

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Figure 3. Summary of numerical experiments for $(0, n)$-$(0, n+1)$ mode interactions

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Figure 4. Bifurcation diagrams for (18) near $(0, 10)$-$(0, 11)$ mode interactions at $\theta_d = 1.7$. (A) one-parameter bifurcation diagrams at $\nu_2 = 0.2$. (B) two-parameter bifurcation diagram

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Figure 5. Bifurcation diagrams for (18) near $(0, 1)$-$(0, 2)$ mode interactions at $\theta_d = 1.2$. (A) one-parameter bifurcation diagrams at $\nu_2 = 0.02$. (B) two-parameter bifurcation diagram

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Figure 6. Bifurcation diagrams for (18) near $(0, 1)$-$(0, 2)$ mode interactions at $\theta_d = 0.8$. (A) one-parameter bifurcation diagrams at $\nu_2 = 0.1$. (B) two-parameter bifurcation diagram

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Figure 7. Bifurcation diagrams for second order FD approximation of (19) for $b^2 = 0.0117837$. (A) one-parameter bifurcation diagram at $\theta = 1.25$. (B) two-parameter bifurcation diagram on $(\alpha, \theta)$-plane

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Figure 8. (A) one-parameter bifurcation diagrams for second order FD approximation for (19) at $\theta_d = 1.205, b^2_d = 0.0117837$. (B) close-up of the right figure of Fig. 7. CP and GH mean the cusp bifurcation point and the generalized Hopf bifurcation point, respectively

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Table 1. Numerical approximation of $\int_{\Omega}\varphi_{0n}^3$

$n$	value	$n$	value	$n$	value
1	4.934760E-01	6	1.820328E-01	11	1.382262E-01
2	2.922274E-01	7	1.732584E-01	12	1.310071E-01
3	2.668396E-01	8	1.590709E-01	13	1.271857E-01
4	2.188215E-01	9	1.527833E-01	14	1.215772E-01
5	2.052654E-01	10	1.430242E-01	15	1.184378E-01

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