Abstract
This paper uses the Ansatz method to solve for exact topological soliton solutions to sine-Gordon type equations. Single, double, and triple sine-Gordon and sine-cosine-Gordon equations are investigated along with dispersive and highly dispersive variations. After these solutions are found, strong perturbations are added to each equation and the new solutions are found. In solving both the perturbed and unperturbed sine-Gordon type equations, constraints are imposed on the parameters. The novel contributions of the authors are the soliton solutions to the strongly perturbed sine-Gordon equation and its variations. These results are important to the study of Josephson junctions, crystal dislocations, ultra-short optical pulses, relativistic field theory, and elementary particles.
Highlights
The sine-Gordon equation (SGE) has applications to Josephson junctions, crystal dislocations, ultra-short optical pulses, relativistic field theory, and elementary particles [1].A Josephson junction is a pair of superconductors separated by a thin material that is not superconducting
This paper uses the Ansatz method to solve for exact topological soliton solutions to sine-Gordon type equations
The novel contributions of the authors are the soliton solutions to the strongly perturbed sine-Gordon equation and its variations. These results are important to the study of Josephson junctions, crystal dislocations, ultra-short optical pulses, relativistic field theory, and elementary particles
Summary
The sine-Gordon equation (SGE) has applications to Josephson junctions, crystal dislocations, ultra-short optical pulses, relativistic field theory, and elementary particles [1]. An edge dislocation occurs when one plane of atoms only extends half-way through the crystal. This causes the planes to bend around it. The goal of this paper is to find exact solutions to strongly perturbed sine-Gordon (SG) type equations. The research done in those papers was primarily to find solutions to the sine-Gordon equation and its variations under small, adiabatic perturbations. We conclude this work with a summary of our methods, the applicability of our results, and possible avenues of future work
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