Dynamics of dissipative periodic optical waves in an external potential

We study the dynamics of periodic waves in a damped-driven nonlinear Schrödinger (NLS) equation perturbed by a small spatially dependent transport term.

This equation arises in nonlinear optics as a model for light propagation in a passive optical cavity driven by a bichromatic laser source.

Our main result shows that, for initial data close to a stable stationary wave of the translation-invariant unperturbed problem, the solution of the perturbed problem remains close to translated copies of this wave, with the translation parameter evolving according to an effective ordinary differential equation.

If this effective equation admits a stable equilibrium, we prove that the solution of the perturbed NLS converges to a stationary state associated with that equilibrium.

In particular, the asymptotically selected state attracts solutions with initial data that are not necessarily close to it, showing that its basin of attraction extends far beyond a small neighborhood of the state.

The analysis is complicated by the loss of derivatives caused by the heterogeneous transport and combines the renormalization group method with a refined bootstrap argument that exploits parity-induced cancellations of quadratic nonlinear terms.

Numerical simulations illustrate the analytical results.