Modeling of Ultra-Short Soliton Propagation in Deterministic and Stochastic Nonlinear Cubic Media

TL;DR

This paper investigates the dynamics of ultra-short soliton pulses in deterministic and stochastic nonlinear media, demonstrating their particle-like behavior and persistence under stochastic perturbations through numerical analysis.

Contribution

It introduces a higher-order short pulse equation and a stochastic version, providing numerical evidence of soliton stability and behavior in noisy environments.

Findings

Short pulse solitons approximate Maxwell solutions.

Solitons persist despite stochastic perturbations.

Numerical schemes show noise statistics align across scales.

Abstract

We study the short pulse dynamics in the deterministic and stochastic environment in this thesis. The integrable short pulse equation is a modelling equation for ultra-short pulse propagation in the infrared range in the optical fibers. We investigate the numerical proof for the exact solitary solution of the short pulse equation. Moreover, we demonstrate that the short pulse solitons approximate the solution of the Maxwell equation numerically. Our numerical experiments prove the particle-like behaviour of the short pulse solitons. Furthermore, we derive a short pulse equation in the higher order. A stochastic counterpart of the short pulse equation is also derived through the use of the multiple scale expansion method for more realistic situations where stochastic perturbations in the dispersion are present. We numerically show that the short pulse solitary waves persist even in the presence of the randomness. The numerical schemes developed demonstrate that the statistics of the coarse-graining noise of the short pulse equation over the slow scale, and the microscopic noise of the nonlinear wave equation over the fast scale, agree to fairly good accuracy.