Slow soliton interaction with delta impurities

TL;DR

This paper investigates how a soliton interacts with a small delta potential in the Gross-Pitaevskii equation, showing that the soliton's evolution follows classical dynamics for a significant time before perturbations become significant.

Contribution

It provides a detailed analysis of soliton dynamics in the presence of a delta impurity, deriving effective classical Hamiltonian behavior for the solution over a specific timescale.

Findings

Soliton evolution is governed by an effective Hamiltonian.

The solution remains close to a soliton for a time scale depending on $|q|$ and $v_0$.

The analysis quantifies the impact of a delta impurity on soliton dynamics.

Abstract

We study the Gross-Pitaevskii equation with a delta function potential, $ q \delta_0 $, where $|q|$ is small, and analyze the solutions for which the initial condition is a soliton with initial velocity $v_0$. We show that up to time $ (|q| + v_0^2)^{-\frac12} \log(1/|q|) $ the bulk of the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, $ (\xi^2 + q \sech^2 (x))/2 $.