Exciton dynamics in branched conducting polymers: Quantum graphs based approach

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

“…(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]. Within such an approach, the Manakov system (1) is formally reduced to a system of linear PDEs by introducing the following potential…”

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

“…J. For checking the discrete energy conservation (and thus the stability and the suitability to model the long time behavior of the solution) we multiply (33) with the central difference quotient…”

“…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.…”