Solitons in the presence of a small, slowly varying perturbation

ABSTRACT

We consider the perturbed sine-Gordon equation ${\theta }_{tt}-{\theta }_{xx}+\sin \theta ={\epsilon }^{2}f\left(\epsilon x\right)$, where the external perturbation ${\epsilon }^{2}f\left(\epsilon x\right)$ 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 $1/{\epsilon }^{}$ and errors of order ${\epsilon }^{3/4}$ a trajectory on the solitary manifold. The trajectory on the solitary manifold is described by ODEs, which agree up to errors of order ${\epsilon }^3$ with Hamilton equations for the restricted to the solitary manifold sine-Gordon Hamiltonian.

Keywords:

• Solitons
• symplectic decomposition
• modulation equations
• Lyapunov function
• sine-Gordon equation

2010 Mathematics Subject Classifications:

• 35Q53
• 35L70
• 35C08

Acknowledgments

My sincere gratitude goes to my PhD advisor Markus Kunze for the continuous support. I would like to thank Justin Holmer for helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the author.