arXiv subjects

Timur Mashkin

Publications and source records attributed to Timur Mashkin.

3 recordsLinked to original sources

arXiv2018

Invariant Virtual Solitary Manifold of the Perturbed Sine-Gordon Equation

We study the perturbed sine-Gordon equation $\theta_{tt}-\theta_{xx}+\sin \theta= F(\varepsilon,x)$, where we assume that the perturbation $F$ is analytic in $\varepsilon$ and that its derivatives with respect to $\varepsilon$ satisfy certain bounds at $\varepsilon=0$. We construct implicitly an, adjusted to the perturbation $F$, virtual solitary manifold, which is invariant in the following sense: The initial value problem for the perturbed sine-Gordon equation with an appropriate initial state on the constructed manifold has a unique solution, which follows a trajectory on the virtual solitary manifold. The trajectory is precisely described by two parameters, which satisfy a specific system of ODEs. The approach is based on the work of Mashkin (arXiv:1705.05713), where we constructed by an iteration scheme a virtual solitary manifold for the perturbed sine-Gordon equation. In arXiv:1705.05713 we proved a stability result for the perturbed sine-Gordon equation with initial data close to the virtual solitary manifold. The employed iteration scheme produces a sequence of virtual solitary manifolds such that the accuracy of the corresponding stability statements increases after each iteration step, as long as the perturbation $F$ is sufficiently often differentiable. The invariant virtual solitary manifold constructed in this work is generated as a limit of the virtual solitary manifolds produced by the iteration scheme. The method and the kind of result presented in this paper is to our knowledge a novelty in the field of stability of solitons.

math.AP↗

arXiv2017

Solitons in the Presence of a Small, Slowly Varying Electric Field

We consider the perturbed sine-Gordon equation $\theta_{tt}-\theta_{xx}+\sin \theta= \varepsilon^2 f(\varepsilon x)$, where the external perturbation $\varepsilon^2 f(\varepsilon x)$ corresponds to a small, slowly varying electric field. We show that the initial value problem with an appropriate initial state close enough to the solitary manifold has a unique solution, which follows up to time $1/{\varepsilon}$ and errors of order $\varepsilon^{ 3/4}$ a trajectory on the solitary manifold. The trajectory on the solitary manifold is described by ODEs, which agree approximately up to errors of order $\varepsilon^3$ with Hamilton equations for the restricted to the solitary manifold sine-Gordon Hamiltonian.

math-ph↗

arXiv2017

Stability of the Solitary Manifold of the Perturbed Sine-Gordon Equation

We study the perturbed sine-Gordon equation $\theta_{tt}-\theta_{xx}+\sin \theta= F(\varepsilon,x)$, where $F$ is of differentiability class $C^n$ in $\varepsilon$ and the first $k$ derivatives vanish at $0$, i.e., $\partial_\varepsilon^l F(0,\cdot)=0$ for $0\le l\le k $. We construct implicitly a virtual solitary manifold by deformation of the classical solitary manifold in $n$ iteration steps. Our main result establishes that the initial value problem with an appropriate initial state $\varepsilon^n$-close to the virtual solitary manifold has a unique solution which follows up to time $1/(\tilde C\varepsilon^{\frac{k+1}{2}})$ and errors of order $\varepsilon^n$ a trajectory on the virtual solitary manifold. The trajectory on the virtual solitary manifold is described by two parameters which satisfy a system of ODEs. In contrast to previous works our stability result yields arbitrarily high accuracy as long as the perturbation $F$ is sufficiently often differentiable.

math.AP↗