# New equilibrium solution branches of plane Couette flow discovered using a project-then-search method

## TL;DR

This paper introduces a novel project-then-search method to discover 16 new equilibrium solutions in plane Couette flow, leveraging resolvent mode projections to efficiently explore the flow's solution space.

## Contribution

The study presents a new approach combining resolvent mode projections with Newton-Krylov searches to find previously unknown equilibrium branches in plane Couette flow.

## Key findings

- Discovered 16 new equilibrium solutions.
- Achieved a 96% convergence rate with the new method.
- Identified bifurcations and disconnected solution branches.

## Abstract

New equilibrium solution branches for plane Couette flow are reported which add to the inventory of known solution branches. The exact solutions are found by projecting known equilibria onto the resolvent modes of McKeon & Sharma (J. Fl. Mech., vol. 658, 2010, pp. 336-382) to generate approximate solutions that are subsequently used as seeds in a Newton-Krylov-hookstep search. Searches initialised with these projections have a convergence rate of 96%, leading to the discovery of 16 new equilibria. The low-dimensional nature of the resulting equilibria is attributed to the projection generated by the resolvent model; resolvent modes for a given equilibrium solution span a low-dimensional space which the solution approximately inhabits. This property is exploited to generate new branches of equilibria in plane Couette flow, and to jump to known branches. The new branches include bifurcations from previously known bifurcation curves and a disconnected bifurcation curve which displays interesting behaviour when continued in Reynolds number. The unstable manifolds of the new exact coherent states are also computed to expand the known state space structure of plane Couette flow.