# 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.   Google Scholar [2] T. Ackemann and W. J. Firth, Dissipative solitons in pattern-forming nonlinear optical systems, Lecture Notes in Phys., 661 (2005), 55-100.   Google Scholar [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.   Google Scholar [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/. Google Scholar [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.   Google Scholar [6] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394.   Google Scholar [7] J.-M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405.   Google Scholar [8] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966. Google Scholar [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Springer-Verlag, New York, 2004. Google Scholar [10] P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schrödinger equations in $\mathbf{R}^N, N≤ 3$, NoDEA, 2 (1995), 357-369.   Google Scholar [11] L. A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett., 58 (1987), 2209-2211.   Google Scholar [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.   Google Scholar [13] T. Miyaji, I. Ohnishi and Y. Tsutsumi, Stability of stationary solution for the Lugiato-Lefever equation, Tohoku Math. J., 63 (2011), 651-663.   Google Scholar [14] T. Ooura, Ooura's mathematical software packages, 2006. Available from: http://www.kurims.kyoto-u.ac.jp/ooura/index.html. Google Scholar [15] J. Prüss, On the spectrum of $C_0$ -semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.   Google Scholar [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.   Google Scholar [17] N. Tzvetkov, Invariant measures for the nonlinear Schrodinger equation on the disc, Dynamics of PDE, 3 (2006), 111-160.   Google Scholar [18] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported New Series, 1 (1992), 125-163.   Google Scholar [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.   Google Scholar [20] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990. Google Scholar

show all references

##### References:
 [1] M. Abounouh, Asymptotic behaviour for a weakly damped Schrödinger equation in dimension two, Appl. Math. Lett., 6 (1993), 29-32.   Google Scholar [2] T. Ackemann and W. J. Firth, Dissipative solitons in pattern-forming nonlinear optical systems, Lecture Notes in Phys., 661 (2005), 55-100.   Google Scholar [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.   Google Scholar [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/. Google Scholar [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.   Google Scholar [6] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394.   Google Scholar [7] J.-M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405.   Google Scholar [8] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966. Google Scholar [9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third edition, Springer-Verlag, New York, 2004. Google Scholar [10] P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schrödinger equations in $\mathbf{R}^N, N≤ 3$, NoDEA, 2 (1995), 357-369.   Google Scholar [11] L. A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett., 58 (1987), 2209-2211.   Google Scholar [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.   Google Scholar [13] T. Miyaji, I. Ohnishi and Y. Tsutsumi, Stability of stationary solution for the Lugiato-Lefever equation, Tohoku Math. J., 63 (2011), 651-663.   Google Scholar [14] T. Ooura, Ooura's mathematical software packages, 2006. Available from: http://www.kurims.kyoto-u.ac.jp/ooura/index.html. Google Scholar [15] J. Prüss, On the spectrum of $C_0$ -semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.   Google Scholar [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.   Google Scholar [17] N. Tzvetkov, Invariant measures for the nonlinear Schrodinger equation on the disc, Dynamics of PDE, 3 (2006), 111-160.   Google Scholar [18] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported New Series, 1 (1992), 125-163.   Google Scholar [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.   Google Scholar [20] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990. Google Scholar
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
. CP and GH mean the cusp bifurcation point and the generalized Hopf bifurcation point, respectively">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
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
 [1] Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 [2] Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021151 [3] Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301 [4] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [5] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [6] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [7] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 [8] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 [9] Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 [10] Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks & Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011 [11] Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147 [12] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [13] Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151 [14] Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71 [15] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 [16] Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197 [17] Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 [18] Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 [19] Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419 [20] Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1

2020 Impact Factor: 1.916