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November  2018, 17(6): 2657-2682. doi: 10.3934/cpaa.2018126

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

Hung Luong

Fak. Mathematik, University of Vienna, Oskar MorgensternPlatz 1, A-1090 Wien, Austria

Received  March 2018 Revised  March 2018 Published  June 2018

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.

Keywords: Local well-posedness, Zakharov system, line soliton, transverse stability, Schrödinger equation.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q55.
Citation: Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126
References:
[1]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53 (1984), Springer-Verlag, New York. Google Scholar

[5]

D. Lannes, The Water Waves Problem, Mathematical Surveys and Monographs, 188 (2013), American Mathematical Society, Providence, RI. Google Scholar

[6]

C. Sulem and P. L. Sulem, The Nonlinear SchrÖdinger Equation, Applied Mathematical Sciences, 139 (1999), Springer-Verlag, New York. Google Scholar

[7]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.   Google Scholar

[8]

H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Differential and Integral Equations, 12 (1999), 789-810.   Google Scholar

[9]

J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[12]

F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388.   Google Scholar

[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.   Google Scholar

[14]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDEs,, Math. Pures Appl. (9), 90 (2008), 550-590.   Google Scholar

[15]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. math. Res. Notices, 11 (1996), 515-546.   Google Scholar

[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.   Google Scholar

[17]

H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 79 (1988), 183-210.   Google Scholar

[18]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[21]

I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089.   Google Scholar

[22]

C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal., 127 (1995), 204-234.   Google Scholar

[23]

J. Simon, Compact set in the Space Lp(0, T; B), Annali di Matematica pura ed applicata, (1987), 65-96.   Google Scholar

show all references

References:
[1]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53 (1984), Springer-Verlag, New York. Google Scholar

[5]

D. Lannes, The Water Waves Problem, Mathematical Surveys and Monographs, 188 (2013), American Mathematical Society, Providence, RI. Google Scholar

[6]

C. Sulem and P. L. Sulem, The Nonlinear SchrÖdinger Equation, Applied Mathematical Sciences, 139 (1999), Springer-Verlag, New York. Google Scholar

[7]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.   Google Scholar

[8]

H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Differential and Integral Equations, 12 (1999), 789-810.   Google Scholar

[9]

J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[12]

F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388.   Google Scholar

[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.   Google Scholar

[14]

F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDEs,, Math. Pures Appl. (9), 90 (2008), 550-590.   Google Scholar

[15]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. math. Res. Notices, 11 (1996), 515-546.   Google Scholar

[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.   Google Scholar

[17]

H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 79 (1988), 183-210.   Google Scholar

[18]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[21]

I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089.   Google Scholar

[22]

C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal., 127 (1995), 204-234.   Google Scholar

[23]

J. Simon, Compact set in the Space Lp(0, T; B), Annali di Matematica pura ed applicata, (1987), 65-96.   Google Scholar

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