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$L^∞$-energy method for a parabolic system with convection and hysteresis effect
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 |
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.
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. |
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.
|
[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. |






value | | value | | 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 |
value | | value | | 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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