# American Institute of Mathematical Sciences

July 2018, 17(4): 1633-1650. doi: 10.3934/cpaa.2018078

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

 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

Received  December 2016 Revised  July 2017 Published  April 2018

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.

Citation: Tomoyuki Miyaji, Yoshio Tsutsumi. Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1633-1650. doi: 10.3934/cpaa.2018078
##### References:
 [1] M. Abounouh, Asymptotic behaviour for a weakly damped Schrödinger equation in dimension two, Appl. Math. Lett., 6 (1993), 29-32. [2] T. Ackemann and W. J. Firth, Dissipative solitons in pattern-forming nonlinear optical systems, Lecture Notes in Phys., 661 (2005), 55-100. [3] P. Colet, D. Gomila, A. Jacobo and M. A. Matía, Excitability mediated by dissipative solitons in nonlinear optical cavities, Lecture Notes in Phys., 751 (2008), 113-135. [4] E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, Canada, January 2012. Available from: http://cmvl.cs.concordia.ca/auto/. [5] P. Gaspard, Measurement of the instability rate of a far-from-equilibrium steady state at an infinite period bifurcation, J. Phys. Chem., 94 (1990), 1-3. [6] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. [7] J.-M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405. [8] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966. [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Springer-Verlag, New York, 2004. [10] P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schrödinger equations in $\mathbf{R}^N, N≤ 3$, NoDEA, 2 (1995), 357-369. [11] L. A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett., 58 (1987), 2209-2211. [12] T. Miyaji, I. Ohnishi and Y. Tsutsumi, Bifurcation analysis to the Lugiato-Lefever equation in one space dimension, Phys. D, 239 (2010), 2066-2083. [13] T. Miyaji, I. Ohnishi and Y. Tsutsumi, Stability of stationary solution for the Lugiato-Lefever equation, Tohoku Math. J., 63 (2011), 651-663. [14] T. Ooura, Ooura's mathematical software packages, 2006. Available from: http://www.kurims.kyoto-u.ac.jp/ooura/index.html. [15] J. Prüss, On the spectrum of $C_0$ -semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. [16] A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever and L. A. Lugiato, Pattern formation in a passive Kerr cavity, Chaos Solitons Fractals, 4 (1994), 1323-1354. [17] N. Tzvetkov, Invariant measures for the nonlinear Schrodinger equation on the disc, Dynamics of PDE, 3 (2006), 111-160. [18] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported New Series, 1 (1992), 125-163. [19] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Phys. D, 88 (1995), 167-175. [20] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.

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##### References:
 [1] M. Abounouh, Asymptotic behaviour for a weakly damped Schrödinger equation in dimension two, Appl. Math. Lett., 6 (1993), 29-32. [2] T. Ackemann and W. J. Firth, Dissipative solitons in pattern-forming nonlinear optical systems, Lecture Notes in Phys., 661 (2005), 55-100. [3] P. Colet, D. Gomila, A. Jacobo and M. A. Matía, Excitability mediated by dissipative solitons in nonlinear optical cavities, Lecture Notes in Phys., 751 (2008), 113-135. [4] E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, Canada, January 2012. Available from: http://cmvl.cs.concordia.ca/auto/. [5] P. Gaspard, Measurement of the instability rate of a far-from-equilibrium steady state at an infinite period bifurcation, J. Phys. Chem., 94 (1990), 1-3. [6] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. [7] J.-M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405. [8] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966. [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Springer-Verlag, New York, 2004. [10] P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schrödinger equations in $\mathbf{R}^N, N≤ 3$, NoDEA, 2 (1995), 357-369. [11] L. A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett., 58 (1987), 2209-2211. [12] T. Miyaji, I. Ohnishi and Y. Tsutsumi, Bifurcation analysis to the Lugiato-Lefever equation in one space dimension, Phys. D, 239 (2010), 2066-2083. [13] T. Miyaji, I. Ohnishi and Y. Tsutsumi, Stability of stationary solution for the Lugiato-Lefever equation, Tohoku Math. J., 63 (2011), 651-663. [14] T. Ooura, Ooura's mathematical software packages, 2006. Available from: http://www.kurims.kyoto-u.ac.jp/ooura/index.html. [15] J. Prüss, On the spectrum of $C_0$ -semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. [16] A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever and L. A. Lugiato, Pattern formation in a passive Kerr cavity, Chaos Solitons Fractals, 4 (1994), 1323-1354. [17] N. Tzvetkov, Invariant measures for the nonlinear Schrodinger equation on the disc, Dynamics of PDE, 3 (2006), 111-160. [18] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported New Series, 1 (1992), 125-163. [19] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Phys. D, 88 (1995), 167-175. [20] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.
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
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 $-\mathrm{ln}|\nu_1 - \nu_c|/\lambda_{m}$
Summary of numerical experiments for $(0, n)$-$(0, n+1)$ mode interactions
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
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
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
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
(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
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
 $n$ 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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