Nonlinear evolution of disturbances in higher time-derivative theories

TL;DR

This paper explores how localized disturbances evolve in higher time-derivative integrable theories, revealing complex behaviors like oscillations and multiple soliton formations, confirmed through numerical solutions of modified KdV equations.

Contribution

It provides the first detailed analysis of the nonlinear evolution of disturbances in higher time-derivative integrable systems, highlighting novel soliton and oscillation phenomena.

Findings

Profiles can settle into two-soliton or multiple N-soliton solutions.

Oscillations spread over time but remain finite.

Singularities may prevent full soliton development or cause standing waves.

Abstract

We investigate the evolution of localized initial value profiles when propagated in integrable versions of higher time-derivative theories. In contrast to the standard cases in nonlinear integrable systems, where these profiles evolve into a specific number of N-soliton solutions as dictated by the conservation laws, in the higher time derivative theories the theoretical prediction is that the initial profiles can settle into either two-soliton solutions or into any number of N-soliton solutions. In the latter case this implies that the solutions exhibit oscillations that spread in time but remain finite. We confirm these analytical predictions by explicitly solving the associated Cauchy problem numerically with multiple initial profiles for various higher time-derivative versions of integrable modified Korteweg-de Vries equations. In the case with the theoretical possibility of a decay into two-soliton solutions, the emergence of underlying singularities may prevent the profiles from fully developing or may be accompanied by oscillatory, chargeless standing waves at the origin.