Solitons and diffusive modes in the noiseless Burgers equation: Stability analysis

TL;DR

This paper analyzes the stability and spectral properties of soliton and diffusive modes in the noiseless Burgers equation, revealing how solitons influence diffusive mode spectra and phase shifts.

Contribution

It provides a linear stability analysis of soliton modes and characterizes the spectral and phase shift properties of diffusive modes in the noiseless Burgers equation.

Findings

Diffusive modes develop a spectral gap in the presence of solitons.

Diffusive modes are phase-shifted according to Levinson's theorem.

A zero-frequency Goldstone mode arises from broken translational symmetry.

Abstract

The noiseless Burgers equation in one spatial dimension is analyzed from the point of view of a diffusive evolution equation in terms of nonlinear soliton modes and linear diffusive modes. The transient evolution of the profile is interpreted as a gas of right hand solitons connected by ramp solutions with superposed linear diffusive modes. This picture is supported by a linear stability analysis of the soliton mode. The spectrum and phase shift of the diffusive modes are determined. In the presence of the soliton the diffusive modes develop a gap in the spectrum and are phase-shifted in accordance with Levinson's theorem. The spectrum also exhibits a zero-frequency translation or Goldstone mode associated with the broken translational symmetry.