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Transactions of the American Mathematical Society

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Global solvability of the derivative nonlinear Schrödinger equation


Author: Jyh-Hao Lee
Journal: Trans. Amer. Math. Soc. 314 (1989), 107-118
MSC: Primary 35Q20; Secondary 34A55, 34B25
MathSciNet review: 951890
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Abstract: The derivative nonlinear Schrödinger equation $ ($DNLS$ )$

\begin{displaymath}\begin{array}{*{20}{c}} {i{q_t} = {q_{xx}} \pm {{({q^\ast }{q... ...sqrt { - 1} ,{q^\ast }(z) = \overline {q(z)} ,} \\ \end{array} \end{displaymath}

was first derived by plasma physicists [9,10]. This equation was used to interpret the propagation of circular polarized nonlinear Alfvén waves in plasma. Kaup and Newell obtained the soliton solutions of DNLS in 1978 [5]. The author obtained the local solvability of DNLS in his dissertation [6]. In this paper we obtain global existence (in time $ t$) of Schwartz class solutions of DNLS if the $ {L^2}$-norm of the generic initial data $ q(x,0)$ is bounded.

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  • [1] R. Beals and R. Coifman, Scattering, transformations spectrales et équations d’évolution non linéaires, Goulaouic-Meyer-Schwartz Seminar, 1980–1981, École Polytech., Palaiseau, 1981, pp. Exp. No. XXII, 10 (French). MR 657992
  • [2] R. Beals and R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984), no. 1, 39–90. MR 728266, 10.1002/cpa.3160370105
  • [3] R. Beals and R. R. Coifman, Inverse scattering and evolution equations, Comm. Pure Appl. Math. 38 (1985), no. 1, 29–42. MR 768103, 10.1002/cpa.3160380103
  • [4] Robin K. Bullough and Philip J. Caudrey (eds.), Solitons, Topics in Current Physics, vol. 17, Springer-Verlag, Berlin-New York, 1980. MR 625877
  • [5] David J. Kaup and Alan C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Mathematical Phys. 19 (1978), no. 4, 798–801. MR 0464963
  • [6] J. H. Lee, Analytic properties of Zakharov-Shabat inverse scattering problem with polynomial spectral dependence of degree $ 1$ in the potential, Ph. D. Dissertation, Yale University, 1983.
  • [7] Jyh-Hao Lee, A Zakharov-Shabat inverse scattering problem and the associated evolution equations, Chinese J. Math. 12 (1984), no. 4, 223–233. MR 774286
  • [8] -, Hamiltonian structure of soliton equations, Proc. CCNAA-AIT Seminar on Differential Equations (Taiwan, June 1985), pp. 205-214.
  • [9] Koji Mio, Tatsuki Ogino, Kazuo Minami, and Susumu Takeda, Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan 41 (1976), no. 1, 265–271. MR 0462141
  • [10] E. MjØlhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys. 16 (1976), 321-334.
  • [11] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118–134 (Russian, with English summary); English transl., Soviet Physics JETP 34 (1972), no. 1, 62–69. MR 0406174

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0951890-5
Keywords: Integrable evolution equation, inverse scattering transform, Riemann-Hilbert problem
Article copyright: © Copyright 1989 American Mathematical Society