Soliton transport in tubular networks: Transmission at vertices in the shrinking limit

Abstract

Soliton transport in tubelike networks is studied by solving the nonlinear Schrödinger equation (NLSE) on finite thickness ("fat") graphs. The dependence of the solution and of the reflection at vertices on the graph thickness and on the angle between its bonds is studied and related to a special case considered in our previous work, in the limit when the thickness of the graph goes to zero. It is found that both the wave function and reflection coefficient reproduce the regime of reflectionless vertex transmission studied in our previous work.

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

“…A final note on this regime: We have only given the leading shift of the nonlinear wave numbers in (16). This is consistent as long as the intensity is only growing moderately as φ 2 max = O(k) (at fixed and g).…”

“…Furthermore, several interesting modifications and applications of quantum graphs without nonlinearity have been developed in the past, that call for including effects of nonzero nonlinearity. One example are fat graphs consisting of bonds with finite widths [16]. What is the effect of nonlinear interaction on quantum spectral filters modeled by star graphs [23,24]?…”

“…In the linear setting, these amplitudes are described by a scattering matrix [11,12]. Scattering through nonlinear graphs was studied previously by different methods in [13][14][15][16]. We will use canonical perturbation theory to show how the nonlinear setting connects to the known linear description at low intensities and also discuss how multistability as a proper nonlinear effect can be described in this framework.…”

Stationary nonlinear Schrödinger equation on the graph for the triangle with outgoing bonds

“…The nonlinear evolution equation on metric graphs have attracted much attention over the last decade [1][2][3][4][5][6][7][8][9][10]. Such interest is caused by the possibility of modeling nonlinear waves and soliton transport in networks and branched structures by nonlinear wave equations on metric graphs.…”

Ground States for NLS on Graphs: a Subtle Interplay of Metric and Topology

“…[11,22,27,10,16]), has rapidly become highly popular in a quite spread scientific community, ranging from experts in pointwise potentials ( [12,13,20]) up to specialists of the Nonlinear Schrödinger Equation and its standing waves ( [3,15,19,21,24]). …”