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On one dimensional quantum Zakharov system

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  • In this paper, we discuss the properties of one dimensional quantum Zakharov system which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The system (1a)-(1b) with initial data $(E(0),n(0),\partial_t n(0))\in H^k\bigoplus H^l\bigoplus H^{l-2}$ is local well-posedness in low regularity spaces (see Theorem 1.1 and Figure 1). Especially, the low regularity result for $k$ satisfies $-3/4 < k \leq -1/4$ is obtained by using the key observation that the convoluted phase function is convex and careful bilinear analysis. The result can not be obtained by using only Strichartz inequalities for ``Schrödinger" waves.
    Mathematics Subject Classification: Primary: 35Q40, 35L56; Secondary: 81Q99.


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  • [1]

    H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equations: Smoothness and approximation, Journal of Functional Analysis, 79 (1988), 183-210.doi: 10.1016/0022-1236(88)90036-5.


    I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., 261 (2011), 478-506.doi: 10.1016/j.jfa.2011.03.015.


    I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with $L^2$-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089.doi: 10.1088/0951-7715/22/5/007.


    M. Ben-Artzi, H. Koch and J. C. Saut, Disperion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris, Série I, 330 (2000), 87-92.doi: 10.1016/S0764-4442(00)00120-8.


    J. Bourgain, Fourier transform restriction phenomena for certain lattic subsets and application to nonlinear evolution equations I,II, Geom. Funct. Anal., 3 (1993), 107-156, 209-262.doi: 10.1007/BF01895688.


    J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices, 11 (1996), 515-546.doi: 10.1155/S1073792896000359.


    J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Transactions AMS, 360 (2008), 4619-4638.doi: 10.1090/S0002-9947-08-04295-5.


    J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.doi: 10.1006/jfan.1997.3148.


    Y. Guo, J. Zhang and B. Guo, Global well-posedness and the classical limit of the solution for the quantum Zakharov system, Z. Angew. Math. Phys., 64 (2013), 53-68.doi: 10.1007/s00033-012-0215-y.


    F. Haas, Quantum Plasma: An Hydrodynamic Approach, Springer-Verlag, New York, 2011.doi: 10.1007/978-1-4419-8201-8.


    F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets, Physical Review E., 79 (2009), 066402-066501.doi: 10.1103/PhysRevE.79.066402.


    J.-C. Jiang, Bilinear Strichartz estimates for Schrödinger operators in 2 dimensional compact manifolds and cubic NLS, Differential and Integral Equations, 24 (2011), 83-108.


    C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.doi: 10.1512/iumj.1991.40.40003.


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


    N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Inventiones Mathematicae, 172 (2008), 535-583.doi: 10.1007/s00222-008-0110-5.


    T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. RIMS, Kyoto Univ., 28 (1992), 329-361.doi: 10.2977/prims/1195168430.


    T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Differential and Integral Equations, 5 (1992), 721-745.


    B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225.doi: 10.4310/DPDE.2007.v4.n3.a1.


    H. Schochet and M. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580.doi: 10.1007/BF01463396.


    C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse, Appl. Math. Sci., Vol.139, Springer-Verlag, New York, 1999.


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

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