A direct method of solution for the Fokas-Lenells derivative nonlinear Schr"odinger equation: II. Dark soliton solutions

In a previous study (Matsuno Y, J. Phys. A: Math. Theor. 45(2012)23202), we have developed a systematic method for obtaining the bright soliton solutions of the Fokas-Lenells derivative nonlinear Schr"odinger equation (FL equation shortly) under vanishing boundary condition. In this paper, we apply the method to the FL equation with nonvanishing boundary condition. In particular, we deal with a more sophisticated problem on the dark soliton solutions with a plane wave boundary condition. We first derive the novel system of bilinear equations which is reduced from the FL equation through a dependent variable transformation and then construct the general dark $N$-soliton solution of the system, where $N$ is an arbitrary positive integer. In the process, a trilinear equation derived from the system of bilinear equations plays an important role. As a byproduct, this equation gives the dark $N$-soliton solution of the derivative nonlinear Schr"odinger equation on the background of a plane wave. We then investigate the properties of the one-soliton solutions in detail, showing that both the dark and bright solitons appear on the nonzero background which reduce to algebraic solitons in specific limits. Last, we perform the asymptotic analysis of the two- and $N$-soliton solutions for large time and clarify their structure and dynamics.

💡 Research Summary

The paper extends the Hirota bilinear method, previously applied to the Fokas‑Lenells (FL) derivative nonlinear Schrödinger equation under vanishing boundary conditions, to the case of non‑vanishing (plane‑wave) boundary conditions. Starting from the FL equation
(i q_{t}+q_{xx}+2i\kappa (|q|^{2}q){x}=0)
the authors impose a constant‑amplitude background (q\to\rho e^{i(\kappa x-\omega t)}) with (\rho\neq0). By introducing the dependent‑variable transformation
(q=\rho e^{i(\kappa x-\omega t)}\frac{g}{f})
the original nonlinear PDE is reduced to a system consisting of two Hirota bilinear equations together with a crucial trilinear equation. The bilinear part reads
((D{t}+D_{x}^{2}),g\cdot f=0,\qquad (D_{x}^{2}+2i\kappa D_{x}),f\cdot f=2\rho^{2}(|g|^{2}-|f|^{2}))
where (D) denotes Hirota’s differential operator. The trilinear relation, which does not appear in the bright‑soliton case, is
(D_{x}g\cdot f\cdot \bar g=0)
and encodes the phase‑locking between the complex conjugate fields required by the non‑zero background.

To construct explicit solutions, the authors propose τ‑functions expressed as Gram‑type determinants. For an arbitrary positive integer (N) they define
\