Compactons and chaos in strongly nonlinear lattices

Area of Science:

• Nonlinear dynamics
• Condensed matter physics
• Mathematical physics

Background:

• Hamiltonian lattices exhibit complex dynamics.
• Localized traveling waves, or solitary waves, are key features in nonlinear systems.
• Understanding chaotic states is crucial for predicting system behavior.

Purpose of the Study:

• To investigate localized traveling waves and chaotic states in strongly nonlinear one-dimensional Hamiltonian lattices.
• To demonstrate that solitary waves are superexponentially localized.
• To present an accurate numerical method for finding solitary waves with arbitrary nonlinearity.

Main Methods:

• Numerical simulations of one-dimensional Hamiltonian lattices.
• Development of a specialized numerical method for identifying superexponentially localized solitary waves.
• Analysis of wave collisions and long-term dynamics.

Main Results:

• Solitary waves in these lattices are superexponentially localized.
• A robust numerical method was developed to find these solitary waves for any nonlinearity index.
• Compactons evolve from general localized perturbations and exhibit nearly elastic collisions.
• For finite lattices, extensive chaotic states are generally observed over long timescales.
• The observed dynamical properties are energy-independent due to system scaling.

Conclusions:

• Superexponential localization of solitary waves is a significant finding in nonlinear lattice dynamics.
• The developed numerical method provides a powerful tool for studying such waves.
• While compactons interact elastically, the long-term behavior of finite lattices tends towards chaos.
• The energy independence of these properties simplifies their applicability across different system regimes.