none

First,we use two different methods to get four conservation laws of the nonlinear Schrodinger equation. And then the local existence in time of the classical solutions can be established via an iteration method and the uniqueness of the solution is also proved. At last we prove the existence of the semiclassical limit of the solution.

[1] D. J. Benney, A general theory for interactions between short and long waves, Stud. Appl. Math., 56, 81--94, 1977.
[2] T. Colin and A. Soyeur, Some singular limits for evolutionary Ginzburg Landau equations, Asymptotic Analysis, 13, 361--372, 1996.
[3] T. Colin and D. Lannes, Long-wave short-wave resonance for nonlinear geometric optics, Duke Math. Journal, 107, 351--419, 2001.
[4] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der mathematischen Wissenschaften Vol. 325, Springer, 2000.
[5] B. Desjardins, C. K. Lin and T. C. Tso, Semiclassical limit of the derivative nonlinear Schr"odinger equation, Math. Models Methods Appl. Sci., 10, 261--285, 2000.
[6] B. Desjardins and C. K. Lin, On the semiclassical limit of the general modified NLS equation, J. Math. Anal. Appl., 260, 546--571, 2001.
[7] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79, 703--714, 1977.
[8] I. Gasser, C. K. Lin and P. Markowich, A review of dispersive limit of the (non)linear Schr"odinger-type equation, Taiwanese J. of Mathematics., 4, 501--529, 2000.
[9] E. Grenier, Semiclassical limit of the nonlinear Schr"odinger equation in small time, Proc. Amer. Math. Soc., 126, 523--530, 1998.
[10] C. H. Hsu and C. K. Lin, Convergence of the Godunov scheme for the modified Euler equation, preprint.
[11] J. S. Jiang and C. K. Lin, Homogenization of the Dirac-like system,
Math. Models Methods Appl. Sci., 11, 433--458, 2001.
[12] S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math., 52, 613--654, 1999.
[13] C. Kenig, G.Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40, 33--67, 1991.
[14] P. Laurencot, On a nonlinear Schr"odinger equation arising in the theory of water wave, Nonlinear Analysis, Theory, Methods & Applications, 24, 509--527, 1995.
[15] J. H. Lee and C. K. Lin, The behavior of solutions of NLS equation of derivative type in the semiclassical limit, Chaos, Solitons & Fractals, 13, 1475--1492, 2002.
[16] H. L. Li and C. K. Lin , Semiclassical limit and well-posedness of Schr"odinger-Poisson and quantum hydrodynamics, preprint (2002).
[17] C. K. Lin,
Singular limit of the modified nonlinear Schr"odinger equation,
In: Nonlinear Dynamics and Renormalization Group, Edited by I. M. Sigal and C. Sulem, CRM Proceeding & Lecture Note Vol. 27, pp. 97--109, Amer. Math. Soc. 2001.
[18] Y. C. Ma, The complete solution of the long wave-short wave resonance equations, Stud. Appl. Math., 59, 201--221, 1978.
[19] T. Ogawa, Global well-posedness and conservation laws for the water wave interaction equation, Proceeding of the Royal Society of Edinburgh, 127A, 368--384, 1997.
[20] N. Sepulveda, Solitary waves in the resonant phenomenon between
a surface gravity wave packet and an internal gravity wave, Phys. Fluids, 30, 1984--1992, 1987.
[21] C. Sulem and P.-L. Sulem,
The Nonlinear Schrodinger Equation, Appl. Math. Sci. 139, Springer-Verlag, (1999).
[22] M. Tsutsumi and S. Hatano, Well-posedness of the Cauchy problem for the long wave-short wave resonance equations, Nonlinear Analysis, Theory, Methods & Applications, 24, 155--171, 1994.
[23] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,
Soviet Phys. Jetp, 34, 62--69, 1972.