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Anomalous autoresonance threshold for chirped-driven Korteweg-de-Vries waves
L Friedland1, A G Shagalov2, S V Batalov2
1Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 14, 2015
Summary
Large amplitude waves in the Korteweg-de-Vries (KdV) equation are controlled using chirped frequency drives. A new fully nonlinear theory explains autoresonance and its threshold, applicable to all wavelengths.
Area of Science:
- Nonlinear dynamics
- Fluid mechanics
- Wave propagation
Background:
- The Korteweg-de-Vries (KdV) equation describes nonlinear waves.
- Autoresonance is a phenomenon where a system synchronizes with a driving force.
- Previous studies on autoresonance thresholds used weakly nonlinear theories.
Purpose of the Study:
- To investigate the excitation and control of large amplitude traveling waves in the KdV equation.
- To develop a fully nonlinear theory for autoresonance in the KdV system.
- To analyze the threshold for the transition to autoresonance and its dependence on system parameters.
Main Methods:
- Utilizing a chirped frequency driving perturbation to excite and control KdV waves.
- Analyzing the passage through linear resonance to achieve autoresonance.
- Developing a fully nonlinear theory to describe the phenomenon across all wavelengths.
Main Results:
- Large amplitude traveling waves in the KdV equation can be controlled by chirped frequency drives.
- The transition to autoresonance exhibits a sharp threshold on the driving amplitude.
- The previously established scaling of the autoresonance threshold (α(3/4)) is violated in the long wavelength KdV limit due to strong nonlinearity.
Conclusions:
- A fully nonlinear theory is developed for autoresonance in the KdV equation, applicable to all wavelengths.
- The study reveals a breakdown of weak nonlinearity assumptions in the long wavelength limit.
- This work provides a more comprehensive understanding of nonlinear wave control via autoresonance.
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