## 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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### References:

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