# The origin of localized snakes-and-ladders solutions of plane Couette flow

## TL;DR

This paper explains how localized solutions in plane Couette flow originate from Taylor-Couette system states, revealing a connection between pattern formation and fluid dynamics through a snaking structure.

## Contribution

It demonstrates the mechanism linking localized solutions in plane Couette flow to Taylor-Couette states, highlighting a pattern-formation perspective in fluid dynamics.

## Key findings

- Localized solutions form a snaking structure similar to pattern-forming PDEs.
- These solutions originate from Taylor vortices via a modulational instability.
- The mechanism links pattern formation theory with Navier-Stokes flow.

## Abstract

Spatially localized exact solutions of plane Couette flow are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations. [PRL 104,104501 (2010)]. We demonstrate the mechanism by which these snaking solutions originate from well-known periodic states of the Taylor-Couette system. They are formed by a localized slug of Wavy-Vortex flow that emerges from a background of Taylor vortices via a modulational sideband instability. This mechanism suggests a deep connection between pattern-formation theory and Navier-Stokes flow.