Nonlinear waves in networks: Model reduction for the sine-Gordon equation

Abstract

To study how nonlinear waves propagate across Y- and T-type junctions, we consider the two-dimensional (2D) sine-Gordon equation as a model and examine the crossing of kinks and breathers. Comparing energies for different geometries reveals that, for small widths, the angle of the fork plays no role. Motivated by this, we introduce a one-dimensional effective model whose solutions agree well with the 2D simulations for kink and breather solutions. These exhibit two different behaviors: a kink crosses if it has sufficient energy; conversely a breather crosses when v>1-ω, where v and ω are, respectively, its velocity and frequency. This methodology can be generalized to more complex nonlinear wave models.

Study of the two‐dimensional sine‐Gordon equation arising in Josephson junctions using meshless finite point method

“…The geometry consists of 2 perpendicular JTLs, main line of width w 0 laying along the x direction and additional line of width w along the y axis, forming a T junction as shown in Figure . The propagation of Josephson vortex at Y‐type and T‐type junctions is discussed in Gulevich and Kusmartsev and Caputo and Dutykh …”

Soliton generation in optical fiber networks

“…Stationary Schödinger equation on metric graphs and standing wave soliton in networks are studied in [28-30, 34, 36]. Integrable sine-Gordon equation on metric graphs is studied in [31,35,38]. Linear and nonlinear systems of PDE on metric graphs are considered in [39][40][41].…”

Transparent quantum graphs

“…The condition is obtained in a simple form of the sum rule for the coefficients, which can characterize physical properties of the network branches. We note that the linear and nonlinear wave equations on metric graphs have been studied earlier in the context of quantum graphs [24][25][26][27][28] and soliton dynamics in networks [29,30,[32][33][34][35][36][37][38][39][40]. It was found in the Refs.…”