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Localized structures in a nonlinear wave equation stabilized by negative global feedback: one-dimensional and
Horacio G Rotstein1, Anatol A Zhabotinsky, Irving R Epstein
1Department of Mathematics and Center for Biodynamics, Boston University, Boston, MA 02215, USA. horacio@math.bu.edu
Summary
Global feedback in nonlinear wave equations stabilizes fronts, creating localized solutions. This study explores rich dynamics in one- and two-dimensional front evolution, generalizing classical models.
Area of Science:
- Nonlinear dynamics
- Mathematical physics
- Wave phenomena
Background:
- The study investigates nonlinear wave equations, generalizing Klein-Gordon and sine-Gordon equations.
- Previous research focused on classical cases without global feedback.
Purpose of the Study:
- To derive and analyze an equation governing front motion in nonlinear wave equations with global feedback.
- To explore the dynamics of one- and two-dimensional fronts under global feedback.
- To understand the conditions leading to localized solutions and phase stabilization.
Main Methods:
- Derivation of a strongly nonlinear equation for front motion.
- Analysis of one- and two-dimensional front dynamics.
- Comparison with classical models lacking global feedback.
Main Results:
- A generalized damped Born-Infeld equation for two-dimensional fronts was derived.
- Global feedback introduces significantly richer dynamics compared to classical models.
- Most cases result in localized solutions, stabilizing one phase within another.
Conclusions:
- Global feedback is crucial for rich front dynamics and the formation of localized solutions.
- The nature of stabilized solutions depends on feedback strength and model parameters.
- This work extends the understanding of front evolution in nonlinear wave systems.
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