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A generalized complex Ginzburg-Landau equation: Global existence and stability results
1. | Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal |
2. | CMAF-CIO, Universidade de Lisboa, Edifício C6, Campo Grande, 1749-016 Lisboa, Portugal |
We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable. In particular, we construct bound-states either explicitly in the real line or through a bifurcation argument for a double eigenvalue of the Dirichlet-Laplace operator on bounded domains.
References:
[1] |
I. S. Aronson and L. Kramer,
The world of the complex Ginzburg-Landau equation, Rev. Modern Phys., 74 (2002), 99-143.
doi: 10.1103/RevModPhys.74.99. |
[2] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, Arch. Racin. Mech. Anal., 82 (1983), 313-375.
doi: 10.1007/BF00250555. |
[3] |
M. S. Berger, A bifurcation theory for nonlinear elliptic partial differential equations and related systems, Bifurcation theory and nonlinear eigenvalue problems, Keller, Joseph and Antman, W.A. Benjamin, Inc. (1969) 113–190 |
[4] |
T. Cazenave, J. P. Dias and M. Figueira,
Finite-time blowup for a complex Ginzburg-Landau equation with linear driving, J. Evol. Equ., 14 (2014), 403-415.
doi: 10.1007/s00028-014-0220-z. |
[5] |
T. Cazenave, F. Dickstein and F. Weissler,
Finite time blowup for a complex Ginzburg-Landau equation, SIAM J. Math. Anal., 45 (2013), 244-266.
doi: 10.1137/120878690. |
[6] |
T. Cazenave, F. Dickstein and F. Weissler,
Standing waves of the complex Ginzburg-Landau equation, Nonlinear Anal., 103 (2014), 26-32.
doi: 10.1016/j.na.2014.03.001. |
[7] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.
doi: 10.1137/1.9781611971309. |
[8] |
S. Correia and M. Figueira, Some stability results for the complex Ginzburg-Landau equation, to appear in Comm. Contemp. Math.
doi: 10.1142/S021919971950038X. |
[9] |
R. Cipolatti, F. Dickstein and J. P. Puel, Existence of standing waves for the complex Ginzburg-Landau equation, J. Math. Anal. Appl., 422 (2015), 579–593.
doi: 10.1016/j.jmaa.2014.08.057. |
[10] |
E. Coddington and N. Levinson, Theory of ordinary differential equations, New York (McGraw-Hill), (1955) |
[11] |
R. J. Deissler and H. R. Brand,
Periodic, Quasiperiodic and Cahotic Localized Solutions of the Quintic Complex Ginzburg-Landau Equation, Phys. Rev. Lett., 4 (1994), 478-482.
|
[12] |
P. M. del, J. García-Melián and M. Musso, Local bifurcation from the second eigenvalue of the Laplacian in a square, Proc. Amer. Math. Soc., 131 (2003)
doi: 10.1090/S0002-9939-03-06906-5. |
[13] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., 1969. |
[14] |
J. Ginibre and G. Velo,
The Cauchy problem in local spaces for the complex Ginzburg-Landau equation $ \rm{I :} $ Compactness methods, Phys. D, 95 (1996), 191-228.
doi: 10.1016/0167-2789(96)00055-3. |
[15] |
J. Ginibre and G. Velo,
The Cauchy problem in local spaces for the complex Ginzburg-Landau equation $ \rm{II:} $ Contraction methods, Commun. Math. Phys., 187 (1997), 45-79.
doi: 10.1007/s002200050129. |
[16] |
J. K. Hale,
Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.
doi: 10.1016/0022-247X(69)90175-9. |
[17] | |
[18] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981. |
[19] |
C. D. Levermore and M. Oliver, The complex Ginzburg-Landau Equation as a Model Problem, AMS, Providence, R.I., 1996,141–190.
doi: 10.1080/03605309708821254. |
[20] |
N. Masmoudi and H. Zaag,
Blow-up profile for the complex Ginzburg-Landau equation, J. Funct. Anal., 255 (2008), 1613-1666.
doi: 10.1016/j.jfa.2008.03.008. |
[21] |
N. Okazawa and T. Yokota,
Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation, Discrete Contin. Dyn. Syst., 28 (2010), 311-341.
doi: 10.3934/dcds.2010.28.311. |
[22] |
S. Popp, O. Stiller, E. Kuznetsov and L. Kramer,
The cubic complex Ginzburg-Landau equation for a backward bifurcation, Phys. D, 114 (1998), 81-107.
doi: 10.1016/S0167-2789(97)00170-X. |
show all references
References:
[1] |
I. S. Aronson and L. Kramer,
The world of the complex Ginzburg-Landau equation, Rev. Modern Phys., 74 (2002), 99-143.
doi: 10.1103/RevModPhys.74.99. |
[2] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, Arch. Racin. Mech. Anal., 82 (1983), 313-375.
doi: 10.1007/BF00250555. |
[3] |
M. S. Berger, A bifurcation theory for nonlinear elliptic partial differential equations and related systems, Bifurcation theory and nonlinear eigenvalue problems, Keller, Joseph and Antman, W.A. Benjamin, Inc. (1969) 113–190 |
[4] |
T. Cazenave, J. P. Dias and M. Figueira,
Finite-time blowup for a complex Ginzburg-Landau equation with linear driving, J. Evol. Equ., 14 (2014), 403-415.
doi: 10.1007/s00028-014-0220-z. |
[5] |
T. Cazenave, F. Dickstein and F. Weissler,
Finite time blowup for a complex Ginzburg-Landau equation, SIAM J. Math. Anal., 45 (2013), 244-266.
doi: 10.1137/120878690. |
[6] |
T. Cazenave, F. Dickstein and F. Weissler,
Standing waves of the complex Ginzburg-Landau equation, Nonlinear Anal., 103 (2014), 26-32.
doi: 10.1016/j.na.2014.03.001. |
[7] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.
doi: 10.1137/1.9781611971309. |
[8] |
S. Correia and M. Figueira, Some stability results for the complex Ginzburg-Landau equation, to appear in Comm. Contemp. Math.
doi: 10.1142/S021919971950038X. |
[9] |
R. Cipolatti, F. Dickstein and J. P. Puel, Existence of standing waves for the complex Ginzburg-Landau equation, J. Math. Anal. Appl., 422 (2015), 579–593.
doi: 10.1016/j.jmaa.2014.08.057. |
[10] |
E. Coddington and N. Levinson, Theory of ordinary differential equations, New York (McGraw-Hill), (1955) |
[11] |
R. J. Deissler and H. R. Brand,
Periodic, Quasiperiodic and Cahotic Localized Solutions of the Quintic Complex Ginzburg-Landau Equation, Phys. Rev. Lett., 4 (1994), 478-482.
|
[12] |
P. M. del, J. García-Melián and M. Musso, Local bifurcation from the second eigenvalue of the Laplacian in a square, Proc. Amer. Math. Soc., 131 (2003)
doi: 10.1090/S0002-9939-03-06906-5. |
[13] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., 1969. |
[14] |
J. Ginibre and G. Velo,
The Cauchy problem in local spaces for the complex Ginzburg-Landau equation $ \rm{I :} $ Compactness methods, Phys. D, 95 (1996), 191-228.
doi: 10.1016/0167-2789(96)00055-3. |
[15] |
J. Ginibre and G. Velo,
The Cauchy problem in local spaces for the complex Ginzburg-Landau equation $ \rm{II:} $ Contraction methods, Commun. Math. Phys., 187 (1997), 45-79.
doi: 10.1007/s002200050129. |
[16] |
J. K. Hale,
Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.
doi: 10.1016/0022-247X(69)90175-9. |
[17] | |
[18] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981. |
[19] |
C. D. Levermore and M. Oliver, The complex Ginzburg-Landau Equation as a Model Problem, AMS, Providence, R.I., 1996,141–190.
doi: 10.1080/03605309708821254. |
[20] |
N. Masmoudi and H. Zaag,
Blow-up profile for the complex Ginzburg-Landau equation, J. Funct. Anal., 255 (2008), 1613-1666.
doi: 10.1016/j.jfa.2008.03.008. |
[21] |
N. Okazawa and T. Yokota,
Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation, Discrete Contin. Dyn. Syst., 28 (2010), 311-341.
doi: 10.3934/dcds.2010.28.311. |
[22] |
S. Popp, O. Stiller, E. Kuznetsov and L. Kramer,
The cubic complex Ginzburg-Landau equation for a backward bifurcation, Phys. D, 114 (1998), 81-107.
doi: 10.1016/S0167-2789(97)00170-X. |
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