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Local well-posedness for the Zakharov system on the background of a line soliton

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  • We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove a weak convergence to a nonlinear Schrödinger equation.

    Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q55.

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  • [1] J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. 
    [2] S. H. Schochet and M. I. Weinstein, The nonlinear Schr¨odinger limit of the Zakharov equations governing Langmuir turbulence, Comm. math. Phys., 106 (1986), 569-580. 
    [3] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math, 34 (1981), 481-524. 
    [4] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53 (1984), Springer-Verlag, New York.
    [5] D. Lannes, The Water Waves Problem, Mathematical Surveys and Monographs, 188 (2013), American Mathematical Society, Providence, RI.
    [6] C. Sulem and P. L. Sulem, The Nonlinear SchrÖdinger Equation, Applied Mathematical Sciences, 139 (1999), Springer-Verlag, New York.
    [7] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914. 
    [8] H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Differential and Integral Equations, 12 (1999), 789-810. 
    [9] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202. 
    [10] L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two (Ⅰ), Comm. Math. Phys., 160 (1994), 173-215. 
    [11] L. Glangetas and F. Merle, Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two (Ⅱ), Comm. Math. Phys., 160 (1994), 349-389. 
    [12] F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388. 
    [13] F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 477-496. 
    [14] F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDEs,, Math. Pures Appl. (9), 90 (2008), 550-590. 
    [15] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. math. Res. Notices, 11 (1996), 515-546. 
    [16] T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., 28 (1992), 329-361. 
    [17] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 79 (1988), 183-210. 
    [18] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.
    [19] C. Sulem and P. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C. R. Acad. Sci. Paris Sér. A-B, 289 (1979), A173-A176. 
    [20] H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 551-554. 
    [21] I. BejenaruS. HerrJ. Holmer and D. Tataru, On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. 
    [22] C. KenigG. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal., 127 (1995), 204-234. 
    [23] J. Simon, Compact set in the Space Lp(0, T; B), Annali di Matematica pura ed applicata, (1987), 65-96. 
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