Global solvability of the derivative nonlinear Schrödinger equation

Abstract:

The derivative nonlinear Schrödinger equation $(\text {DNLS})$ \[ \begin {array}{*{20}{c}} {i{q_t} = {q_{xx}} \pm {{({q^\ast }{q^2})}_x},} & {q = q(x,t), i = \sqrt { - 1} ,{q^\ast }(z) = \overline {q(z)} ,} \\ \end {array} \] 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 $\text {DNLS}$ in 1978 [5]. The author obtained the local solvability of $\text {DNLS}$ in his dissertation [6]. In this paper we obtain global existence (in time $t$) of Schwartz class solutions of $\text {DNLS}$ if the ${L^2}$-norm of the generic initial data $q(x,0)$ is bounded.