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arXiv · 2104.04619

A collective coordinate framework to study solitary waves in stochastically perturbed Korteweg-de Vries equations

Madeleine Cartwright·Georg A. Gottwald
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Abstract

Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to describe the effect of random perturbations on coherent solitary waves. We present a collective coordinate approach to describe the effect on coherent solitary waves in stochastically perturbed KdV equations. The collective coordinate approach allows one to reduce the infinite-dimensional stochastic partial differential equation (SPDE) to a finite-dimensional stochastic differential equation for the amplitude, width and location of the solitary wave. The reduction provides a remarkably good quantitative description of the shape of the solitary waves and its location. Moreover, the collective coordinate framework can be used to estimate the time-scale of validity of stochastically perturbed KdV equations for which they can be used to describe coherent solitary waves. We describe loss of coherence by blow-up as well as by radiation into linear waves. We corroborate our analytical results with numerical simulations of the full SPDE.

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Madeleine Cartwright, Georg A. Gottwald. 2021-04-09. A collective coordinate framework to study solitary waves in stochastically perturbed Korteweg-de Vries equations. https://doi.org/10.1103/physreve.104.024201

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