Abstract:

In this work, we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schrödinger (DNLS) equation. By establishing a matrix Riemann–Hilbert problem and reconstructing potential function q(x, t) from eigenfunctions ${\{{G}_{j}(x,t,\eta )\}}_{1}^{3}$ in the inverse problem, the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed. Moreover, we also obtain that the spectral functions f(η), s(η), F(η), S(η) are not independent of each other, but meet an important global relation. As applications, the generalized DNLS equation can be reduced to the Kaup–Newell equation and Chen–Lee–Liu equation on the half-line.

Key words: Riemann–Hilbert problem, generalized derivative nonlinear Schrödinger equation, initial-boundary value problems, unified transformation method