Kink dynamics in a 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.)
|October 17, 2025
Summary
A new two-degree-of-freedom model accurately describes complex kink motion in the sine-Gordon model. This effective model aids in understanding and designing soliton-based devices.
Area of Science:
- Nonlinear dynamics
- Theoretical physics
- Condensed matter physics
Background:
- The sine-Gordon model is a fundamental tool for studying solitons.
- Understanding complex kink dynamics is crucial for soliton-based applications.
- Existing models may struggle with long-term simulations and non-trivial trajectories.
Purpose of the Study:
- To develop a highly accurate, reduced-order model for the sine-Gordon model with space- and time-dependent parameters.
- To enable the description of kink movement over extremely long times.
- To analyze complex kink trajectories and motions induced by temporal drives.
Main Methods:
- Construction of an effective model with two degrees of freedom.
- Analysis of kink movement and coherent structure trajectories.
- Testing the reduced-order model against the full field-theoretic model under temporal drive conditions.
Main Results:
- A two-degree-of-freedom effective model was successfully constructed.
- The model accurately describes kink movement for long times and nontrivial trajectories.
- The approximation faithfully reproduces the behavior of the full field-theoretic model, even for complex motions.
Conclusions:
- The developed two-degree-of-freedom model provides a computationally efficient and accurate approach to studying the sine-Gordon model.
- This work facilitates a deeper understanding of soliton dynamics.
- The findings pave the way for improved design and application of soliton-based devices.
Related Concept Videos
01:21Linear Approximation in Time Domain
340
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
340
01:25Modeling with Differential Equations
4
Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
4
01:26Scaling
554
In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...
554
01:07Damped Oscillations
6.8K
In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...
Although friction and other non-conservative...
6.8K
01:20Relation between Mathematical Equations and Block Diagrams
2.9K
In a spring-mass-damper system, the second-order differential equation describes the dynamic behavior of the system. When transformed into the Laplace domain under zero initial conditions, this equation can be effectively analyzed and manipulated. The transformation into the Laplace domain converts differential equations into algebraic equations, simplifying the process of isolating the output.
2.9K
01:18Second Order systems II
383
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
383