# Defect-like structures and localized patterns in SH357

## TL;DR

This paper investigates the complex pattern formation in the SH357 equation, revealing bistability, localized structures, and diverse steady states through numerical analysis and bifurcation studies.

## Contribution

It provides a detailed numerical study of the SH357 equation, uncovering new localized defect-like structures and complex bifurcation phenomena.

## Key findings

- Bistability between small and large amplitude stripes.
- Existence of heteroclinic connections and localized defect-like structures.
- Rich variety of stable steady states with different stripe configurations.

## Abstract

We study numerically the cubic-quintic-septic Swift-Hohenberg (SH357) equation on bounded one-dimensional domains. Under appropriate conditions stripes with wave number $k\approx 1$ bifurcate supercritically from the zero state and form S-shaped branches resulting in bistability between small and large amplitude stripes. Within this bistability range we find stationary heteroclinic connections or fronts between small and large amplitude stripes, and demonstrate that the associated spatially localized defect-like structures either snake or fall on isolas. In other parameter regimes we also find heteroclinic connections to spatially homogeneous states, and a multitude of dynamically stable steady states consisting of patches of small and large amplitude stripes with different wave numbers or of spatially homogeneous patches. The SH357 equation is thus extremely rich in the types of patterns it exhibits. Some of the features of the bifurcation diagrams obtained by numerical continuation can be understood using a conserved quantity, the spatial Hamiltonian of the system.