Transparent nonlinear networks

Abstract

We consider the reflectionless transport of solitons in networks. The system is modeled in terms of the nonlinear Schrödinger equation on metric graphs, for which transparent boundary conditions at the branching points are imposed. This allows to derive simple constraints, which link equivalent usual Kirchhoff-type vertex conditions to the transparent ones. Our approach is applied to a metric star graph. An extension to more complicated graph topologies is straight forward.

Fokker-Planck equation on metric graphs

“…An evolution equation on graphs is written on each bond and each equation related to other through the vertex boundary conditions, which are imposed at the nodes of a graph. An effective approach for solving evolution equations on graphs was proposed in [49] for nonlinear Schrodinger equation and applied later to sine-Gordon [51] and nonlinear Dirac [53] equations. The method is based on using the solution of an evolution equation on a line to construct solution on a graph by fulfilling vertex boundary conditions.…”

Reflectionless propagation of Manakov solitons on a line:A model based on the concept of transparent boundary conditions

“…Here we will apply the above concept and procedure for the derivation of TBCs for the Manakov system Eq. (1) and their numerical implementation at the artificial boundary points x = 0, x = L. For this purpose, we use the so-called potential approach, which was previously used to derive TBCs for the nonlinear Schrödinger equation [20,39] and the sine-Gordon equation [40]. In [41][42][43] the TBC concept was used to develop transparent quantum graphs model, which was later implemented to describe reflectionless transport of charge carriers in branched conducting polymers [44].…”

Transparent boundary conditions for the sine-Gordon equation:Modeling the reflectionless propagation of kink solitons on a line

“…So far, transparent boundary conditions have been studied for different wave equations having broad applications in physics, such as linear [24][25][26] and nonlinear [27,28] Schrödinger, Dirac [29], diffusion [30] and Bogoliubov de Gennes [31] equations. Recently, the concept of transparent boundary conditions have been extended to linear [32][33][34], nonlinear [35] Schrödinger and Dirac [36] equations on metric graphs. Until today many different numerical schemes like compact schemes [37][38][39], predictor-corrector schemes [37,40], energy-conservative finite difference schemes [41,42], Lattice-Boltzmann methods [43], radial basis functions [44], etc.…”

“…In this paper we address the problem of designing transparent boundary conditions (TBCs) for the 1D sine-Gordon equation using the so-called potential approach previously introduced in [28] (see, also the Refs. [45,46] for further progress) and utilized in [35] for quantum graphs. Here we will adopt this approach for the sine-Gordon equation on a real line.…”