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Parametric resonance in the non-autonomous sine-Gordon model
Tomasz Dobrowolski1, Jacek Gatlik2, Zofia Bryłowska2
1Department of Physics and Applied Mathematics, University of the National Education Commission in Krakow, Podchora̧żych 2, 30-084 Cracow, Poland.
Chaos (Woodbury, N.Y.)
|November 4, 2025
Summary
This study simplifies the sine-Gordon model for time- and space-dependent parameters. An effective model accurately predicts kink movement and stability regions, though complex dynamics arise in specific areas.
Area of Science:
- Theoretical Physics
- Nonlinear Dynamics
- Condensed Matter Physics
Background:
- The sine-Gordon model describes various physical phenomena, including soliton propagation.
- Investigating models with space- and time-dependent parameters is crucial for realistic applications.
- Understanding kink dynamics in non-autonomous and inhomogeneous systems presents significant challenges.
Purpose of the Study:
- To develop and validate a simplified effective model for the sine-Gordon equation with space- and time-dependent parameters.
- To analyze kink movement and stability boundaries in temporally non-autonomous and spatially inhomogeneous settings.
- To compare the dynamics of the simplified model with the full field model, particularly concerning parametric instability.
Main Methods:
- Construction of a reduced effective model with one degree of freedom.
- Analysis of kink dynamics under temporal drive, leading to parametric instability.
- Examination of Arnold tongues to map regions of stability and instability.
- Comparison of results between the effective model and the full field model.
Main Results:
- The effective model successfully describes kink movement in complex settings.
- Good agreement was found between the effective and full field models regarding stability regions.
- Parametric instability regions, visualized as Arnold tongues, were accurately characterized.
- More complex field dynamics were observed in the lower portions of Arnold's tongues compared to the effective model.
Conclusions:
- The simplified effective model provides a valuable tool for studying the sine-Gordon model with varying parameters.
- The model accurately captures essential dynamics, especially stability boundaries.
- Discrepancies in complex dynamics highlight the limitations of the approximation in specific regimes.
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