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Related Concept Videos

Partial Differential Equations01:21

Partial Differential Equations

A stone dropped into a still pond generates waves that propagate outward in circular patterns, creating a dynamic surface whose elevation depends on both position and time. At any given location, the water level oscillates as the wave passes, while at any fixed moment, the surface exhibits smooth, curved structures extending across space. This dual dependence requires a mathematical description that accounts for variation in multiple variables simultaneously.At a fixed point on the water...
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Related Experiment Video

Updated: Jun 26, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Compactification of nonlinear patterns and waves.

Philip Rosenau1, Eugene Kashdan

  • 1School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. rosenau@post.tau.ac.il

Physical Review Letters
|December 31, 2008
PubMed
Summary
This summary is machine-generated.

A novel nonlinear mechanism generates compact patterns with sharp fronts propagating at finite speeds, offering an alternative to models with missing wave speeds. This universal mechanism is mathematically simple and applicable to various physical systems.

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Last Updated: Jun 26, 2026

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Area of Science:

  • Nonlinear dynamics
  • Mathematical physics
  • Wave propagation

Background:

  • Traditional wave models often assume infinite or undefined wave speeds.
  • Developing mechanisms for finite-speed propagation is crucial for accurate physical modeling.

Purpose of the Study:

  • To introduce and characterize a nonlinear mechanism that generates finite-speed wave propagation.
  • To demonstrate the applicability of this mechanism in diverse physical contexts.

Main Methods:

  • Mathematical analysis of nonlinear partial differential equations.
  • Investigation of pattern formation with compact support and sharp fronts.
  • Application to Klein-Gordon and Schrödinger equations with specific nonlinearities.

Main Results:

  • A universal, simple mathematical characterization of the nonlinear mechanism via a sublinear substrate force.
  • Demonstration of pattern formation with compact support and finite propagation speed.
  • Successful application to both Klein-Gordon and Schrödinger equations.

Conclusions:

  • The presented nonlinear mechanism provides a viable alternative to models with missing wave speeds.
  • The mechanism's universality and mathematical simplicity make it broadly applicable in physics.
  • This work advances the understanding of pattern formation and finite-speed propagation in nonlinear systems.