N-soliton solution for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

Abstract

An explicit two-soliton solution for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions is derived, demonstrating details of interactions between two bright solitons, two dark solitons, as well as one bright soliton and one dark soliton. Shifts of soliton positions due to collisions are analytically obtained, which are irrespective of the bright or dark characters of the participating solitons. The derivative nonlinear Schrödinger (DNLS) equation is an integrable model describing various nonlinear waves such as nonlinear Alfvén waves in space plasma(see, e.g., [1,2,3,4,5,6,7]), sub-picosecond pulses in single mode optical fibers(see, e.g., [8,9,10,11,12]), and weak nonlinear electromagnetic waves in ferromagnetic [13], antiferromagnetic [14], and dielectric[15] systems under external magnetic fields. Both of vanishing boundary conditions (VBC) and nonvanishing boundary conditions (NVBC) for the DNLS equation are physically significant. For problems of nonlinear Alfvén waves, weak nonlinear electromagnetic waves in magnetic and dielectric media, waves propagating strictly parallel to the ambient magnetic fields are modelled by VBC while those oblique waves are modelled by NVBC. In optical fibers, pulses under bright background waves are modelled by NVBC.

Freak waves in laboratory and space plasmas

“…However, its applicability to studying the intermediate behaviour of solutions is restricted to the initial conditions for which the corresponding scattering problem can be solved analytically. To our knowledge, the only non-trivial exact solutions to the DNLS equation obtained so far are different kinds of the N-soliton solutions (see, e.,g., [48][49][50][51]). Since, at present, it is not clear if the scattering problem for the DNLS equation for the initial condition (17) can be solved analytically, the nonlinear evolution of this perturbation was studied numerically [43].…”

The derivative nonlinear Schrödinger equation on the half line

“…It also arises in the study of wave propagation in optical fibers [1]. The equation appears in many other contexts and for more information the reader can consult [42], [9], and the references therein.…”

“…The compatibility condition is necessary since the solutions we are interested in are continuous space-time functions for s > 1 2 . This problem bears significant physical importance as described in [9]: The DNLS equation on R is known, [43], to be locally wellposed in H s for any s ≥ 1 2 . This result is sharp since for s < 1 2 the data to solution map fails to be uniformly continuous, see [43] and [3] for the detailed argument.…”

Domain wall arrays, fronts, and bright and dark solitons in a generalized derivative nonlinear Schrödinger equation

“…Investigation of the DNLSE has not only mathematical interest and significance, but also an important physical application background. Solutions of the DNLSE under both the vanishing boundary conditions (VBC) [12] and the nonvanishing boundary conditions (NVBC) [13] are physically and mathematically discussed.…”