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Sometimes waves do not seem to move; rather, they just vibrate in place. Unmoving waves can be seen on the surface of a glass of milk kept in a refrigerator, which is one example of standing waves. Vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. These waves are formed or created by the superposition of two or more identical moving waves in opposite directions. The waves move through each other, with their...
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Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp
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Three-dimensional rogue waves in nonstationary parabolic potentials.

Zhenya Yan1, V V Konotop, N Akhmediev

  • 1Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100080, China. zyyan_math@yahoo.com

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary

Researchers developed a new method to simplify complex nonlinear Schrödinger (NLS) equations. This technique generates novel 3D rogue wave solutions with potential applications in nonlinear optics and Bose-Einstein condensates.

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Area of Science:

  • Mathematical Physics
  • Nonlinear Dynamics
  • Quantum Mechanics

Background:

  • The nonlinear Schrödinger (NLS) equation is a fundamental model in various scientific fields.
  • Solving higher-dimensional NLS equations with variable coefficients and potentials is challenging.
  • Understanding localized solutions like rogue waves is crucial for applications.

Purpose of the Study:

  • To develop a systematic method for reducing higher-dimensional NLS equations to simpler forms.
  • To investigate the generation of localized exact solutions, specifically rogue waves, in a (3+1)-dimensional context.
  • To explore potential experimental and application avenues for the derived solutions.

Main Methods:

  • Symmetry analysis was employed to identify similarity transformations.
  • A higher-dimensional (3+1)-dimensional inhomogeneous NLS equation was reduced to a (1+1)-dimensional NLS equation with constant coefficients.
  • Lowest-order rational solutions of the (1+1)-dimensional NLS equation were used as seeding functions.

Main Results:

  • A novel similarity transformation was systematically presented.
  • The transformation successfully reduced the complexity of the (3+1)-dimensional NLS equation.
  • Three-dimensional rogue wave solutions exhibiting complex temporal evolution and interactions were obtained.
  • These solutions demonstrated interactions between two time-dependent rogue wave solutions.

Conclusions:

  • The developed technique provides a pathway to analyze complex NLS equations.
  • The generated 3D rogue wave solutions offer new possibilities for experimental research.
  • Potential applications are identified in nonlinear optics and Bose-Einstein condensates.