Stability of the solitary manifold of the perturbed sine-Gordon equation

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:486

Article

Mashkin, Timur¹ 

[1] Univ Cologne, Math Inst, Weyertal 86-90, D-50931 Cologne, Germany

关键词: Soliton;    Symplectic decomposition;    Modulation equation;    Lyapunov function;    Sine-Gordon equation;   

DOI  :  10.1016/j.jmaa.2020.123904

来源: Elsevier

【 摘 要 】

We study the perturbed sine-Gordon equation theta(tt) - theta(xx) + sin theta = F(epsilon, x), where F is of differentiability class C-n in epsilon and the first k derivatives vanish at epsilon = 0, i.e., partial derivative F-l(epsilon)(0, .) = 0 for 0 <= l <= 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 epsilon(n)-close to the virtual solitary manifold has a unique solution, which follows up to time 1/((C) over tilde epsilon(k+1/2)) and errors of order epsilon(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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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