Related papers: Kinks in the Presence of Rapidly Varying Perturbat…
Dynamics of sine-Gordon kinks in the presence of rapidly varying periodic perturbations of different physical origins is described analytically and numerically. The analytical approach is based on asymptotic expansions, and it allows to…
We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon…
We demonstrate that $\pi$-kinks exist in non-parametrically ac driven sine-Gordon systems if the ac drive is sufficiently fast. It is found that, at a critical value of the drive amplitude, there are two stable and two unstable equilibria…
We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between…
The behavior of the kink in the sine-Gordon (sG) model in the presence of periodic inhomogeneity is studied. An ansatz is proposed that allows for the construction of a reliable effective model with two degrees of freedom. Effective models…
We have examined the dynamical behavior of the kink solutions of the one-dimensional sine-Gordon equation in the presence of a spatially periodic parametric perturbation. Our study clarifies and extends the currently available knowledge on…
Extending a recent effective theory formulation for the dynamics of kinks in the sine-Gordon model [1], we propose an analogous effective description of $\phi^4$ kinks. Three different reduced models based on the kink position, width and…
The sine-Gordon model is studied with model parameters that depend on both space and time. An effective model with one degree of freedom is constructed, allowing the description of the kink movement in both a temporally non-autonomous and…
We establish the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. Our proof consists of a space-time resonances approach based on the distorted Fourier transform to…
We consider static solutions of the sine-Gordon theory defined on a cylinder, which can be either periodic or quasi-periodic in space. They are described by the different modes of a simple pendulum moving in an inverted effective potential…
We study the asymptotic stability of the sine-Gordon kinks under small perturbations in weighted Sobolev norms. Our main tool is the B\"acklund transform which reduces the study of the asymptotic stability of the kinks to the study of the…
Parametrically excited sine-Gordon equation is considered. Excitation is a fast oscillating periodic function with zero mean. Technique of classical method of averaging enables to construct the averaged equations in a variety of assumptions…
In this paper, we study the long-time dynamics and stability properties of the sine-Gordon equation $$f_{tt}-f_{xx}+\sin f=0.$$ Firstly, we use the nonlinear steepest descent for Riemann-Hilbert problems to compute the long-time asymptotics…
We prove the asymptotic stability of the moving kinks for the nonlinear relativistic wave equations in one space dimension with a Ginzburg-Landau potential: starting in a small neighborhood of the kink, the solution, asymptotically in time,…
We consider a dc-driven damped sine-Gordon model with a small nonlinear spatial-disorder term, onto which a sinusoidal modulation is superimposed. It describes, e.g., a weakly disordered system with a regular grain structure. We demonstrate…
In the present work we explore the dynamics of single kinks, kink-anti-kink pairs and bound states in the prototypical fractional Klein-Gordon example of the sine-Gordon equation. In particular, we modify the order $\beta$ of the temporal…
The sine-Gordon model with space- and time-dependent parameters is considered. A highly accurate effective model with two degrees of freedom is constructed, allowing the description of the kink movement in this model even for extremely long…
In the present study the interaction of a sine-Gordon kink with a localized inhomogeneity is considered. In the absence of dissipation, the inhomogeneity considered is found to impose a potential energy barrier. The motion of the kink for…
The sine-Gordon equation on a metric graph with a structure represented by a $\mathcal{Y}$-junction, is considered. The model is endowed with boundary conditions at the graph-vertex of $\delta'$-interaction type, expressing continuity of…
We give an exhaustive, non-perturbative classification of exact travelling-wave solutions of a perturbed sine-Gordon equation (on the real line or on the circle) which is used to describe the Josephson effect in the theory of…