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Related Experiment Videos

Instabilities in the two-dimensional cubic nonlinear Schrödinger equation.

John D Carter1, Harvey Segur

  • 1Mathematics Department, Seattle University, Seattle, Washington 98122, USA. carterj1@seattleu.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
Summary

One-dimensional traveling wave solutions of the nonlinear Schrödinger equation (NLS) are unstable. Perturbations with two-dimensional structures destabilize these waves, with instability conditions depending on dispersion term signs.

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Wave Phenomena

Background:

  • The two-dimensional cubic nonlinear Schrödinger equation (NLS) models diverse physical systems, including deep water waves and optical fiber pulses.
  • One-dimensional traveling wave solutions are fundamental to understanding NLS dynamics.

Purpose of the Study:

  • To investigate the stability of one-dimensional traveling wave solutions of the NLS equation.
  • To analyze the impact of two-dimensional perturbations on these solutions.

Main Methods:

  • Linear stability analysis of one-dimensional traveling wave solutions.
  • Perturbation analysis considering infinitesimal disturbances with two-dimensional structure.
  • Case analysis based on the signs of coefficients in the linear dispersion terms (elliptic vs. hyperbolic).

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Main Results:

  • All one-dimensional traveling wave solutions of the NLS with linear phase are unstable to certain two-dimensional perturbations.
  • In the elliptic case (dispersion coefficients same sign), instability is restricted to perturbations with wavelengths longer than a cutoff.
  • In the hyperbolic case (dispersion coefficients opposite signs), instability occurs for all wavelengths, with a bounded maximum growth rate as wavelength decreases.

Conclusions:

  • One-dimensional traveling waves in NLS models are inherently unstable when subjected to two-dimensional perturbations.
  • The nature of instability, particularly the wavelength dependence, is critically determined by the system's dispersion characteristics.
  • These findings have implications for the stability of wave phenomena modeled by the NLS equation in various physical contexts.