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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 & 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.  Google Scholar

[2]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Arch. Racin. Mech. Anal., 82 (1983), 313-375.  doi: 10.1007/BF00250555.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Coddington and N. Levinson, Theory of ordinary differential equations, New York (McGraw-Hill), (1955)  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.  doi: 10.1016/0022-247X(69)90175-9.  Google Scholar

[17]

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

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Arch. Racin. Mech. Anal., 82 (1983), 313-375.  doi: 10.1007/BF00250555.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. Coddington and N. Levinson, Theory of ordinary differential equations, New York (McGraw-Hill), (1955)  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.  doi: 10.1016/0022-247X(69)90175-9.  Google Scholar

[17]

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

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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