Mapping Between Nonlinear Sch\"odinger Equations with Real and Complex Potentials

TL;DR

This paper constructs a mapping between solutions of nonlinear Schrödinger equations with real and complex potentials, providing exact solutions with real energies for various complex potentials.

Contribution

It introduces a novel mapping method and derives exact solutions with real energies for complex potentials in nonlinear Schrödinger equations.

Findings

Derived a set of exact solutions with real energies for complex potentials.

Applied the mapping to damped quantum harmonic oscillators.

Obtained dissipative periodic soliton solutions.

Abstract

A mapping between stationary solutions of nonlinear Sch\"odinger equations with real and complex potentials is constructed and a set of exact solutions with real energies are obtained for a large class of complex potentials. As specific examples we consider the case of the damped dynamics of a quantum harmonic oscillator and the case of dissipative periodic soliton solutions of the nonlinear Schr\"odinger equation with complex potential.