Fast two-pulse collisions in linear diffusion-advection systems with weak quadratic loss in spatial dimension 2

TL;DR

This paper develops a two-dimensional perturbation method to analyze fast two-pulse collisions in linear diffusion-advection systems with weak quadratic loss, revealing shape changes and amplitude shifts influenced by initial conditions and orientation.

Contribution

It generalizes the perturbation approach from 1D to 2D, providing new insights into pulse dynamics and shape changes during collisions in higher dimensions.

Findings

Pulse shape changes in the transverse direction during collisions.

Universal form of amplitude shift for separable initial conditions.

Dependence of amplitude shift on initial condition anisotropy and orientation.

Abstract

We investigate the dynamics of fast two-pulse collisions in linear diffusion-advection systems with weak quadratic loss in spatial dimension 2. We introduce a two-dimensional perturbation method, which generalizes the perturbation method used for studying two-pulse collisions in spatial dimension 1. We then use the generalized perturbation method to show that a fast collision in spatial dimension 2 leads to a change in the pulse shape in the direction transverse to the advection velocity vector. Moreover, we show that in the important case of a separable initial condition, the longitudinal part in the expression for the amplitude shift has a simple universal form, while the transverse part does not. Additionally, we show that anisotropy in the initial condition leads to a complex dependence of the amplitude shift on the orientation angle between the pulses. Our perturbation theory predictions are in very good agreement with results of extensive numerical simulations with the weakly perturbed diffusion-advection model. Thus, our study significantly enhances and generalizes the results of previous works on fast collisions in diffusion-advection systems, which were limited to spatial dimension 1.