Long-time asymptotic behavior and bound state soliton solutions for a generalized derivative nonlinear Schrödinger equation

We obtain the long-time asymptotic behavior and $N$th-order bound state soliton solutions of a generalized derivative nonlinear Schrödinger ($g$-DNLS) equation via the Riemann–Hilbert method. First, in the process of direct scattering, the spectral analysis of the Lax pair is performed, from which a Riemann–Hilbert problem (RHP) is established for the $g$-DNLS equation. Next, in the process of inverse scattering, different from traditional solution finding schemes, we give some Laurent expansions of related functions and use them to obtain solutions of the RHP for the reflection coefficients under different conditions, such as a single pole and multiple poles, where we obtain new $N$th-order bound state soliton solutions. Based on the originally constructed RHP, we use the $\overline{\partial}$-steepest descent method to explicitly find long-time asymptotic behavior of the solutions of the $g$-DNLS equation. With this method, we obtain an accuracy of the asymptotic behavior of the solution that is currently not obtainable by the direct method of partial differential equations.