Nonlocal nonlinear Schrödinger equation on metric graphs: A model for generation and transport of parity-time-symmetric nonlocal solitons in networks

We consider the parity-time (PT)-symmetric, nonlocal, nonlinear Schrödinger equation on metric graphs. Vertex boundary conditions are derived from the conservation laws. Soliton solutions are obtained for the simplest graph topologies, such as star and tree graphs. The integrability of the problem is shown by proving the existence of an infinite number of conservation laws. A model for soliton generation in such PT-symmetric optical fibers and their networks governed by the nonlocal nonlinear Schrödinger equation is proposed. Exact formulas for the number of generated solitons are derived for the cases when the problem is integrable. Numerical solutions are obtained for the case when integrability is broken.

Figure 1

Star graph with six bonds.

Figure 2

Plot of soliton on a metric star graph obtained from numerical solution of Eq. (8) for the values of ${\beta }_j$ fulfilling the sum rule in Eq. (8) (${\beta }_{-1}=1$, ${\beta }_1=1.15$, ${\beta }_{-2}=2.19$, ${\beta }_2=1.91$, ${\beta }_{-3}=2.42$, ${\beta }_3=2.09$). The initial conditions are given on the bonds $b_{-1}$ and $b_1$.

Figure 3

Plot of soliton on a metric star graph obtained from numerical solution of Eq. (8) for the values of ${\beta }_j$ breaking the sum rule in Eq. (8) (${\beta }_{-1}=0.65$, ${\beta }_1=0.79$, ${\beta }_{-2}=2.7$, ${\beta }_2=2.09$, ${\beta }_{-3}=3.06$, ${\beta }_3=2.87$). The initial conditions are given on the bonds $b_{-1}$ and $b_1$.

Figure 4

A sketch of tree graph adopted for nonlocal nonlinear Schrödinger equation.

Figure 5

Initial pulse profile.

Figure 6

Numerical solution of initial value problem given by Eqs. (1) and (29) for rectangular initial pulse profile, demonstrating generation of two solitons on a line.

Figure 7

Initial pulse profile for the star graph.

Figure 8

Evolution of the solitons profile on time at $t=0$ (red line), $t=0.06$ (green line), and $t=0.12$ (blue line) on the star graph, showing the radiation during the soliton propagation.

Figure 9

Numerical solution of initial value problem given by Eqs. (8) and (13), for the initial rectangular pulse profile (see Fig. 7), demonstrating generation of six solitons on a star graph. The nonlinearity coefficients are chosen as ${\beta }_{-1}=1$, ${\beta }_1=1.15$, ${\beta }_{-2}=2.19$, ${\beta }_2=1.91$, ${\beta }_{-3}=2.42$, ${\beta }_3=2.09$.

Figure 10

Initial pulse profile for the tree graph.