Ring wave solutions of a n+1-dimensional Sine-Gordon model

Area of Science:

• Nonlinear dynamics
• Mathematical physics
• Condensed matter theory

Background:

• The study examines the dynamical properties of ring wave solutions within a specific nonlinear partial differential equation model.
• This model is derived as a continuum approximation of a multidimensional Frenkel-Kontorowa lattice, relevant to solid-state physics and materials science.

Purpose of the Study:

• To analytically and numerically investigate the behavior of ring wave solutions in 2D and 3D spatial dimensions.
• To determine the conditions under which the 'return effect' of ring waves occurs or is absent.
• To analyze the stabilizing influence of dissipative perturbations on ring wave dynamics.

Main Methods:

• Analytical investigation of the model's dynamical properties.
• Numerical simulations to study wave behavior and stability.
• Analysis of the influence of parameters like varepsilon and alpha on wave solutions.

Main Results:

• The 'return effect' of ring waves is shown to be absent only for specific, well-defined values of varepsilon when alpha is zero or positive.
• Numerical results demonstrate that a dissipative perturbation, represented by alphapsi(t) with alpha > 0, effectively stabilizes both the velocity and the wave profile of the ring wave.
• This stabilization occurs particularly in scenarios where the return effect is not present.

Conclusions:

• The presence and absence of the ring wave 'return effect' are critically dependent on the parameter varepsilon.
• Dissipative perturbations play a crucial role in stabilizing ring wave solutions, preventing undesirable return effects and maintaining wave integrity.
• The findings contribute to understanding the complex dynamics of nonlinear wave phenomena in extended systems.