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Long-wave and short-wave asymptotics in nonlinear dispersive systems
R A Kraenkel1, M A Manna, V Merle
1Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 São Paulo, Brazil.
Summary
This study examines how short and long waves interact in conservative dispersive systems. Long-wave effects are determined by eliminating secularity in nonlinear analysis.
Area of Science:
- Mathematical physics
- Nonlinear dynamics
- Wave phenomena
Background:
- Conservative dispersive systems exhibit complex behaviors across different spatial scales.
- Understanding the interplay between short- and long-wavelengths is crucial for modeling these systems.
- Previous analyses often focused on single-scale phenomena.
Purpose of the Study:
- To investigate the interaction between short- and long-space scales in (1+1) dimensional conservative dispersive systems.
- To develop a nonlinear analysis using multiscale expansions.
- To determine how long-wave effects influence the system's dynamics at leading order.
Main Methods:
- Utilized multiscale expansion techniques for nonlinear analysis.
- Focused on model systems exhibiting both short- and long-wavelength solutions in the linear regime.
- Employed a secularity elimination procedure to analyze long-wave effects.
Main Results:
- The lowest-order governing equations are primarily determined by short-wave properties.
- Long-wave effects are found to be governed by the secularity elimination procedure at the leading order.
- Demonstrated a method to systematically analyze multi-scale interactions.
Conclusions:
- The nonlinear analysis successfully captures the interplay between different spatial scales.
- Secularity elimination is a key procedure for understanding the influence of long waves.
- This approach provides a framework for studying complex wave phenomena in conservative systems.
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