# Retracting fronts for the nonlinear complex heat equation

## TL;DR

This paper investigates the existence of retracting fronts in the nonlinear complex heat equation, revealing their family structure and analogies with front propagation into unstable states.

## Contribution

It establishes the existence of one-parameter families of retracting fronts for the nonlinear complex heat equation, highlighting their properties and analogies with unstable front propagation.

## Key findings

- Existence of one-parameter families of retracting fronts.
- Boundaries of these families include the slowest and steepest fronts.
- Analogies with front propagation into unstable states.

## Abstract

The "nonlinear complex heat equation" $A_t=i|A|^2A+A_{xx}$ was introduced by P. Coullet and L. Kramer as a model equation exhibiting travelling fronts induced by non-variational effects, called "retracting fronts". In this paper we study the existence of such fronts. They go by one-parameter families, bounded at one end by the slowest and "steepest" front among the family, a situation presenting striking analogies with front propagation into unstable states.