Amplitude equations for systems with long-range interactions

TL;DR

This paper derives amplitude equations for interface dynamics in systems with long-range interactions, revealing nonlocal effects that influence universal behavior in pattern-forming systems.

Contribution

It introduces a novel method to derive amplitude equations incorporating long-range interactions, applicable to systems with linear bulk equations solvable by integral transforms.

Findings

Long-range interactions appear as nonlocal nonlinear terms in the amplitude equations.

The derived equations apply to the Grinfeld and Saffman-Taylor instabilities.

Nonlocal effects persist in the long-wave limit, affecting universal dynamics.

Abstract

We derive amplitude equations for interface dynamics in pattern forming systems with long-range interactions. The basic condition for the applicability of the method developed here is that the bulk equations are linear and solvable by integral transforms. We arrive at the interface equation via long-wave asymptotics. As an example, we treat the Grinfeld instability, and we also give a result for the Saffman-Taylor instability. It turns out that the long-range interaction survives the long-wave limit and shows up in the final equation as a nonlocal and nonlinear term, a feature that to our knowledge is not shared by any other known long-wave equation. The form of this particular equation will then allow us to draw conclusions regarding the universal dynamics of systems in which nonlocal effects persist at the level of the amplitude description.