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

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Summary

Introduction

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

The model
The unperturbed problem
Main results
Heuristics of the proof
Positive energy for the pendulum
Proof of Theorem A
Proof of Theorem B
Proof of Theorem C
Banach spaces and technical lemmas
The fixed point argument
The difference map
Exponential smallness of ξ1 Consider the function space
Banach spaces and technical results
The operators Lω and Gω
The operator Ph
The fixed point theorem
Full Text
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