Dispersive hydrodynamics in non-Hermitian nonlinear Schrödinger equation with complex external potential

Abstract

In this paper dispersive hydrodynamics associated with the non-Hermitian nonlinear Schrödinger (NLS) equation with generic complex external potential is studied. In particular, a set of dispersive hydrodynamic equations are obtained. They differ from their classical counterparts (without an external potential), by the presence of additional source terms that alter the density and momentum equations. When restricted to a class of Wadati-type complex potentials, the resulting hydrodynamic system conserves a modified momentum and admits constant intensity/density solutions. This motivates the construction and study of an initial value problem (IVP) comprised of a centred (or non-centred) step-like initial condition that connects two constant intensity/density states. Interestingly, this IVP is shown to be related to a Riemann problem posed for the hydrodynamic system in an appropriate traveling reference frame. The study of such IVPs allows one to interpret the underlying non-Hermitian Riemann problem in terms of an ‘optical flow’ over an obstacle. A broad class of non-Hermitian potentials that lead to modulationally stable constant intensity states are identified. They are subsequently used to numerically solve the associated Riemann problem for various initial conditions. Due to the lack of translation symmetry, the resulting long-time dynamics show a dependence on the location of the step relative to the potential. This is in sharp contrast to the NLS case without potential, where the dynamics are independent of the step location. This fact leads to the formation of diverse nonlinear wave patterns that are otherwise absent. In particular, various gain-loss generated near-field features are present, which in turn drive the optical flow in the far-field which could be comprised of various rich nonlinear wave structures, including DSW-DSW, DSW-rarefaction, and soliton-DSW interactions.