Curve Lengthening Bifurcations in Modally Filtered Nonlinear Schr\"odinger Systems

TL;DR

This paper extends parametric nonlinear Schr"odinger equations to preserve a specific bifurcation behavior, enabling better modeling of interface motion transitions in optical resonance systems.

Contribution

It introduces a class of modally filtered nonlinear Schr"odinger systems that maintain the curve lengthening bifurcation and stability properties of the original system.

Findings

Preserves the curve lengthening bifurcation in extended equations.

Maintains linear stability of the interface front.

Allows sign flip in the normal velocity term while preserving Willmore effects.

Abstract

Extensions of the parametric nonlinear Schr\"odinger equations (PNLS) for phase-sensitive optical resonance are developed that preserve the curve lengthening bifurcation seen in the original system. This bifurcation occurs in sharp interface reductions when the motion of the interface transitions from curvature-driven flow (curve shortening) to motion against curvature regularized by higher order Willmore effects (curve lengthening). We construct a specific class of down-phase self-interaction operators via a spectral transform of the down-up operator. While in the bifurcation regime, the corresponding modally filtered nonlinear Schr\"odinger systems preserve the linear stability of the front, admit the sign flip in the linear term in the normal velocity while preserving the proper sign of the Willmore terms.