Abstract
In this work we study a model of interaction of kinks of the sine-Gordon equation with a weak defect. We obtain rigorous results concerning the so-called critical velocity derived in [7] by a geometric approach. More specifically, we prove that a heteroclinic orbit in the energy level 0 of a 2-dof Hamiltonian Hε is destroyed giving rise to heteroclinic connections between certain elements (at infinity) for exponentially small (in ε) energy levels. In this setting Melnikov theory does not apply because there are exponentially small phenomena.
Highlights
Given an evolutionary partial differential equation, a solitary wave is a solution which travels with constant speed and localized in space
Kinks have attracted the focus of researchers due to their significant role in many scientific fields as optical fibers, fluid dynamics, plasma physics and others
We study a model of interaction between kinks of the sine-Gordon equation and a weak defect
Summary
Given an evolutionary partial differential equation, a solitary wave is a solution which travels with constant speed and localized in space. There are several types of solitary waves which are important in modeling physical phenomena. Kinks are solitary waves which travel from one asymptotic state to another. We study a model of interaction between kinks of the sine-Gordon equation and a weak defect. The defect is modeled as a small perturbation given by a Dirac delta function. Such interaction has been studied for the nonlinear Schrödinger equation in [13,14]. We give conditions on the energy of the system to admit “kink-like” solutions in this reduced model
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