Numerical study of transverse (in-)stability of solitary waves in the cubic-quintic nonlinear Schrödinger equation

TL;DR

This work analyzes the transverse stability of line solitary waves for the cubic-quintic nonlinear Schrödinger equation on the waveguide domain $\mathbb{R}_x \times \mathbb{T}_{L_y}$, using a Fourier-based 2D solver to explore both localized and periodic perturbations. A key finding is the existence of a critical torus length $L_y^{\rm crit}(\omega)$ beyond which line solitons become transversely unstable, with the threshold controlled by the 2D mass comparison $M_2D(\phi_\omega)=2\pi L_y M_1D(\phi_\omega)$ against the 2D ground-state mass $M(Q_\omega)$, and without a simple $\omega$-rescaling. The study reveals stable and unstable regimes across frequencies near $\omega\in(0,\tfrac{3}{16})$, showing lump formation $Q_\omega$ as the primary unstable outcome and highlighting how the instability can persist as oscillations or converge to a moving lump depending on $L_y$ and perturbation type. These results provide quantitative benchmarks for transverse instabilities in non-scaling NLS models relevant to nonlinear optics and Bose–Einstein condensates, and they clarify how cubic-quintic nonlinearities modify stability thresholds in waveguide geometries.

Abstract

We study the nonlinear Schrödinger equation with a competing cubic-quintic power law nonlinearity on the waveguide domain $\mathbb R_x \times \mathbb T_{L_y}$. This model is globally well-posed and admits line solitary wave solutions, whose transverse (in-)stability is numerically investigated. We consider both spatially localized perturbations and periodic deformations of the line solitary wave and numerically confirm that there exists a critical torus length $L_y>0$ above which instability appears.

Figures (23)

• Figure 1: The mass (left) and the energy (right) of the 1D solitary wave (\ref{['phi']}) as a function of $\omega$.
• Figure 2: Left: Numerically constructed 2D ground state solutions to \ref{['nls']} for several values of $\omega$. Right: The $L^{\infty}$-norm of these states as a function of $\omega$.
• Figure 3: $M(Q_\omega)$ and $E(Q_\omega)$ as functions of $\omega$ in 2D.
• Figure 4: Time-evolution of the $L^{\infty}$ norm of the solution $u$ to (\ref{['cubic']}) for initial data \ref{['pertcub']} with $\omega=0.04$ (on the left for $u_{0,+}$ and on the right for $u_{0,-}$).
• Figure 5: Left: Solution $|u|$ to the cubic NLS at time $t=2.316$ corresponding to initial data $u_{0, +}$ with $\omega=1$. Right: Time-evolution of the $L^{\infty}$ norm of the solution $u$ on the $y$-axis.

Theorems & Definitions (4)

• Definition 1.1
• Remark 1.2
• Remark 2.1
• Remark 7.1