Exact solutions of the modified Gross-Pitaevskii equation in `smart' periodic potentials in the presence of external source

TL;DR

This paper derives a broad set of exact solutions for the modified Gross-Pitaevskii equation with 'smart' Jacobi elliptic potentials, including soliton, singular, and periodic solutions, using a fractional transform method.

Contribution

It introduces a novel fractional transform approach to find exact solutions of the GPE with complex elliptic potentials in the presence of external sources.

Findings

Found exact soliton, singular, and periodic solutions.

Generalized previous results to include external sources.

Extended solution classes for modified GPE in elliptic potentials.

Abstract

We report wide class of exact solutions of the modified Gross-Pitaevskii equation (GPE) in `smart' Jacobi elliptic potentials: $V(\xi)=-V_{0}{\rm sn(\xi,m)}$, $V(\xi)=-V_{0}{\rm cn(\xi,m)}$, and $V(\xi)=-V_{0}{\rm dn(\xi,m)}$ in the presence of external source. Solitonlike solutions, singular solutions, and periodic solutions are found using a recently developed fractional transform: $\rho(\xi)=\frac{A+Bf^2}{1+Df}$, where $f$ is the respective Jacobi elliptic function and the amplitude parameters $A$, $B$, and $D$ {\it nonzero}. These results generalize those contained in (Paul T, Richter K and Schlagheck P 2005 \emph{Phys. Rev. Lett.} {\bf 94}, 020404) for nonzero trapping potential.