Sine-Gordon solitons in networks: Scattering and transmission at vertices

¹ Tashkent Financial Institute - 60A, Amir Temur Str., Tashkent 100000, Uzbekistan
² Turin Polytechnic University in Tashkent - 17 Niyazov Str., Tashkent 100095, Uzbekistan
³ Faculty of Mathematics, National University of Uzbekistan - Vuzgorodok, Tashkent 100174, Uzbekistan
⁴ Faculty of Physics, National University of Uzbekistan - Vuzgorodok, Tashkent 100174, Uzbekistan
⁵ Department of Applied Physics, Osaka City University - Osaka 558-8585, Japan
⁶ Institut für Mathematik, Universität Oldenburg - D-26111 Oldenburg, Germany

^(a) sobirovzar@gmail.com
^(b) hannes.uecker@uni-oldenburg.de

Received: 9 April 2016
Accepted: 8 September 2016

Abstract

We consider the sine-Gordon equation on metric graphs with simple topologies and derive vertex boundary conditions from the fundamental conservation laws together with successive space-derivatives of sine-Gordon equation. We analytically obtain traveling-wave solutions in the form of standard sine-Gordon solitons such as kinks and antikinks for star and tree graphs. We show that for this case the sine-Gordon equation becomes completely integrable just as in case of a simple 1D chain. This simple analysis provides a cornerstone for the numerical solution of the general case, including a quantification of the vertex scattering. Applications of the obtained results to Josephson junction networks and DNA double helix are discussed.