# Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with PT-symmetry

## TL;DR

This paper investigates bifurcations of nonlinear bound states in the one-dimensional PT-symmetric Gross-Pitaevskii equation with complex periodic potentials, revealing conditions for spectral band formation and approximating states via an effective cubic nonlinear Schrödinger equation.

## Contribution

It proves bifurcations of PT-symmetric nonlinear bound states from the spectrum edges under general conditions and provides criteria for complex spectral band emergence in PT-symmetric potentials.

## Key findings

- Bifurcation of nonlinear bound states from spectrum endpoints.
- Approximation of bound states by an effective cubic nonlinear Schrödinger equation.
- Conditions for the appearance of complex spectral bands with small imaginary potential.

## Abstract

The stationary Gross-Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the PT (parity-time reversal) symmetry. Under rather general assumptions on the potentials we prove bifurcations of PT-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrodinger operator with a complex PT-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrodinger equation. In addition we provide sufficient conditions for the appearance of complex spectral bands when the complex $\PT$-symmetric potential has an asymptotically small imaginary part.