\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation

Abstract Related Papers Cited by
  • The inviscid limit behavior of solution is considered for the multi-dimensional derivative complex Ginzburg-Landau(DCGL) equation. For small initial data, it is proved that for some $T>0$, solution of the DCGL equation converges to the solution of the derivative nonlinear Schrödinger (DNLS) equation in natural space $C([0,T]; H^s)(s\geq \frac{n}{2})$ if some coefficients tend to zero.
    Mathematics Subject Classification: Primary: 35E15, 35K15, 35K55; Secondary: 35Q55.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    J.Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 107-156, 209-262.

    [2]

    H. R. Brand and R. J. Deissler, Interaction of localized solutions for subcritical bifurcations, Phys. Rev. Lett., 63 (1989), 2801-2804.

    [3]

    M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.doi: 10.1103/RevModPhys.65.851.

    [4]

    R. J. Deissler and H. R. Brand, Generation of counterpropagating nonlinear interacting traveling waves by localized noise, Phys. Lett. A, 130 (1988), 293-298.doi: 10.1016/0375-9601(88)90613-5.

    [5]

    A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations, Phys. D, 53 (1991), 249-266.doi: 10.1016/0167-2789(91)90065-H.

    [6]

    J. Q. Duan and P. Holmes, On the Cauchy problem of a generalized Ginzburg-Landau equation, Nonlinear Anal., 22 (1994), 1033-1040.doi: 10.1016/0362-546X(94)90065-5.

    [7]

    H. G. Feichtinger, Modulation spaces on locally compact Abelian group, Proc. Internat. Conf. on Wavelet and Applications, (2003), 99-140.

    [8]

    H. J. Gao and J. Q. Duan, On the initial-value problem for the generalized two-dimensional Ginzburg-Landau equation, J. Math. Anal. Appl., 216 (1997), 536-548.doi: 10.1006/jmaa.1997.5682.

    [9]

    L. Han, B. Wang and B. Guo, Inviscid limit for the derivative Ginzburg-Landau equation with small data in higher spatial dimensions, Print, arXiv:1004.1221.

    [10]

    Z. H. Huo and Y. L. Jia, Well-posedness and inviscid limit behavior of solution for the generalized 1D Ginzburg-Landau equation, J. Math. Pures Appl., 92 (2009), 18-51.doi: 10.1016/j.matpur.2009.04.003.

    [11]

    Z. H. Huo and Y. L. Jia, Global well-posedness for the generalized 2D Ginzburg-Landau equation, J. Differ. Eqns., 247 (2009), 260-276.doi: 10.1016/j.jde.2009.03.015.

    [12]

    A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798.doi: 10.1090/S0894-0347-06-00551-0.

    [13]

    Y. L. Jia, Inviscid limit behavior of generalized 1-D Ginzburg-Landau, submitted.

    [14]

    C. E. Kenig, G. Ponce, C. Rolvent and L. Vega, The genreal quasilinear untrahyperbolic Schrodinger equation, Advances in Mathematics, 206 (2006), 402-433.doi: 10.1016/j.aim.2005.09.005.

    [15]

    C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.

    [16]

    C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.doi: 10.1002/cpa.3160460405.

    [17]

    C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.doi: 10.1090/S0894-0347-96-00200-7.

    [18]

    C. E. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math., 134 (1998), 489-545.doi: 10.1007/s002220050272.

    [19]

    C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for quasi-linear Schrodinger equations, Invent. Math., 158 (2004), 343-388.doi: 10.1007/s00222-004-0373-4.

    [20]

    Y. S. Li and B. L. Guo, Global existence of solutions to the derivative 2D Ginzburg-Landau equation, J. Math. Anal. Appl., 249 (2000), 412-432.doi: 10.1006/jmaa.2000.6880.

    [21]

    T. Ozawa and J. Zhai, Global existence of small classical solutions to nonlinear Schröinger equationsm, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 303-331.doi: 10.1016/j.anihpc.2006.11.010.

    [22]

    H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Eq., 4 (1999), 561-680.

    [23]

    H. Takaoka, Global well-posedness for Schrödinger equation with derivative in a nonlinear term and data in low-order Sobolev Spaces, Elec. J. Diff. Eq., 2001, 23 pp. (electronic).

    [24]

    T. Tao, Multilinear weighted convolution of $ L^2 $ functions, and applications to nonlinear dispersive equation, Amer.J. Math., 123 (2001), 839-908.doi: 10.1353/ajm.2001.0035.

    [25]

    B. X. Wang, The Cauchy problem for critical and subcritial semilinear parabolic equations in $L^r$ (II). Initial data in critial Sobolev spaces $H^{-s,r}$, Nonlinear Anal., 52 (2003), 851-868.doi: 10.1016/S0362-546X(02)00136-0.

    [26]

    B. X. Wang, B. L. Guo and L. F. Zhao, The global well-posedness and spatial decay of solutions for the derivative complex Ginzburg-Landau equation in $H^1$, Nonlinear Anal., 57 (2004), 1059-1076.doi: 10.1016/j.na.2004.03.032.

    [27]

    B. Wang and Y. Wang, The inviscid limit of the derivative complex Ginzburg-Landau equation, J. Math. Pures Appl., 83 (2004), 477-502.doi: 10.1016/j.matpur.2003.11.002.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(69) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return