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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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Keywords: Integrable evolution equation, inverse scattering transform, Riemann-Hilbert problem
Article copyright: © Copyright 1989 American Mathematical Society