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

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

* 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.

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. CazenaveJ. 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. CazenaveF. 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. CazenaveF. 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. PoppO. StillerE. 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. CazenaveJ. 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. CazenaveF. 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. CazenaveF. 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. PoppO. StillerE. 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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