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Solitons in the presence of a small, slowly varying perturbation. (English) Zbl 1448.35450

Summary: We consider the perturbed sine-Gordon equation \(\theta_{tt} - \theta_{xx} + \sin \theta = \varepsilon^2 f (\varepsilon x)\), where the external perturbation \(\varepsilon^2 f (\varepsilon x)\) is small and slowly varying. 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 up to errors of order \(\varepsilon^3\) with Hamilton equations for the restricted to the solitary manifold sine-Gordon Hamiltonian.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35L70 Second-order nonlinear hyperbolic equations
35C08 Soliton solutions
53D05 Symplectic manifolds (general theory)
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