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May  2021, 20(5): 2021-2038. doi: 10.3934/cpaa.2021056

A generalized complex Ginzburg-Landau equation: Global existence and stability results

Simão Correia 1,, and  Mário Figueira 2,

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

* Corresponding author

Received  October 2020 Revised  February 2021 Published  May 2021 Early access  April 2021

Fund Project: The first author is supported by Fundação para a Ciência e Tecnologia, through the grant UIDB/MAT/04459/2020. The second author is supported by Fundação para a Ciência e Tecnologia, through the grant UIDB/04561/2020

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.

Keywords: complex Ginzburg-Landau, stability, periodic solutions.
Mathematics Subject Classification: Primary: 35Q56, 35B10, 35B35.
Citation: Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2021-2038. doi: 10.3934/cpaa.2021056
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]

P. Hartman, Ordinary Differential Equations, SIAM, 1987.

[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]

P. Hartman, Ordinary Differential Equations, SIAM, 1987.

[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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