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

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    * Corresponding author 
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
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  • 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.

    Mathematics Subject Classification: Primary: 35Q56, 35B10, 35B35.

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  • [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. 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.
    [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.
    [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.
    [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. 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.
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