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

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  • 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.

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

    Citation:

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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

    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 $-\mathrm{ln}|\nu_1 - \nu_c|/\lambda_{m}$

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

    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

    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

    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

    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

    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

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

    $n$value $n$value $n$value
    14.934760E-0161.820328E-01111.382262E-01
    22.922274E-0171.732584E-01121.310071E-01
    32.668396E-0181.590709E-01131.271857E-01
    42.188215E-0191.527833E-01141.215772E-01
    52.052654E-01101.430242E-01151.184378E-01
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