Global solvability of the derivative nonlinear Schrödinger equation

Source abstract

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