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Curve Lengthening Bifurcations in Modally Filtered Nonlinear Schrödinger Systems

Keith Promislow, Abba Ramadan

Abstract

Extensions of the parametric nonlinear Schrödinger 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 Schrödinger 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.

Curve Lengthening Bifurcations in Modally Filtered Nonlinear Schrödinger Systems

Abstract

Extensions of the parametric nonlinear Schrödinger 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 Schrödinger 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.
Paper Structure (12 sections, 7 theorems, 128 equations, 1 figure)

This paper contains 12 sections, 7 theorems, 128 equations, 1 figure.

Table of Contents

  1. Overview
  2. Modally Filtered Schrödinger Equation (MPSE)
  3. Notation
  4. Spectral Analysis in 1D
  5. Essential Spectrum of $\textrm{L}$
  6. Point Spectrum of $\textrm{L}$
  7. Curvature Driven Flow
  8. Sub-Operator Expansions
  9. Residual Expansion
  10. Normal Velocity and Matching
  11. The Wilmore Flow
  12. Discussion

Key Result

Lemma 1

There exists $\mu_*>0$ such that kernel of $\textrm{L}$ is simple for all $|\mu|\leq \mu_*.$ For $\mu\neq0$ the kernel of $\textrm{L}$ and of its adjoint $\textrm{L}^\dag$ are spanned by the vectors respectively. For $\mu=0$ the kernels of $\textrm{L}$ and its adjoint are spanned by

Figures (1)

  • Figure 1: Contour plots of the modulus $|U|$ over $[-2\pi,2\pi]^2$ simulated from the PNLS version of \ref{['e:MFPNLS']} from the same initial data. The top row corresponds to parameters that induce motion by curvature $\mu>0$. In the bottom row the parameters induc motion against curvature ($\mu<0)$. Reprinted with permission from bib:PR24.

Theorems & Definitions (15)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 5 more