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

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.

Figure 1

Sketch of a metric graph $\Gamma$ and a fat graph ${\Omega }_{\varepsilon }=V_{\varepsilon }\cup B_1\cup B_2\cup B_3$, with bonds of width $w_j$, where $w_j=O\left(\varepsilon \right)$. Ideally, the lengths $l_1,l_2,l_3$ of the bonds are infinite, but for numerical simulations of the NLSE we use finite lengths with Dirichlet boundary conditions (DBC) at the ends, and homogeneous Neumann boundary conditions (NBC) elsewhere.

Figure 2

Amplitudes $A_k={max}_{x\in {\Gamma }_k}\left|{\psi }_k(t,x)\right|$ for (1) on the metric graph $\Gamma$ with bond lengths 15. Initial soliton of the form (4) with $\eta ,c=1,10$ and ${\xi }_0=-7.5$; see also (19). (a) Nonballistic case, $\alpha =(1,1,1),\beta =(1,1,1)$; (b) ballistic case, $\alpha =(1,1.73,1.22),\beta =(1,3,1.5)$. In (a), the blue line is hidden by the red line.

Figure 3

Numerical solution of $\left({\mathrm{P}}_{\varepsilon }\right)$ for ${\delta }_{2,3}=1$ and $\varepsilon =0.5$, i.e., $w=(0.5,0.5,0.5)$; $\tilde{\beta }\equiv 1,l=(15,15,15),\theta =(\pi /3,\pi /3)$. Initial condition (20) with $x_0=-l/2$ and $\eta ,c=1,10$. (a) Geometry and mesh near the vertex. (b) $\mathrm{Re}\varphi (0.5,·)$ real part of incoming soliton at $t=0.5$; (c),(d) $\left|\varphi \right(·,x\left)\right|$ during and after transmission and reflection through and at the vertex.

Figure 4

Norms and amplitudes corresponding to the solutions presented in Fig. 3. Dashed lines present respective quantities from $\left({\mathrm{P}}_0\right)$; the blue lines are all hidden by the red lines.

Figure 5

Norms and amplitudes for fat and metric graphs with $1/{\alpha }_2^2+1/{\alpha }_3^2=1$ and ${\beta }_k={\alpha }_k^2$, hence $\tilde{\beta }=1$, and plots of the amplitude distances $m_{k,\varepsilon }$; cf. (18). Here ${\theta }_1={\theta }_2=\pi /3,{\delta }_2=1/3,{\delta }_3=2/3$, and $\varepsilon =0.5$; hence $w=(0.5,0.33,0.17)$ (see Fig. 6). In (a1) we also plot the geometry and mesh near the vertex. For the lengths of the bonds we again have $l_1=l_2=l_3=15$. In (a2),(a3) the full lines are $N_{\varepsilon ,k}$ and $\frac{1}{{\alpha }_k}{A}_{\varepsilon ,k}$, respectively, and the dashed lines are $N_k$ and $A_k$, cf. Fig. 2, and similarly in (b1),(b2) and (c1),(c2).

Figure 6

Continuation of Fig. 5; $\varepsilon =0.2$ in (b) and $\varepsilon =0.1$ in (c).

Figure 7

(a) $r_{k,\varepsilon }^N$ and $r_{k,\varepsilon }^A$ as functions of the (relative) thickness ${\delta }_3$ of the third bond, with fixed $\varepsilon =0.25,{\theta }_{1,2}=\pi /4,{\delta }_1=1,{\delta }_2=1/3,l=(15,15,15)$, and $(\eta ,c)=(1,10)$. (b) $r_{k,\varepsilon }^N$ and $r_{k,\varepsilon }^A$ as functions of $\varepsilon$, with ${\delta }_3=2/3$ (ballistic case) and the remaining parameters fixed as in (a). (c) $r_{1,\varepsilon }^A$ as a function of $\varepsilon$ and angles $\theta :={\theta }_1={\theta }_2$; remaining parameters as in (b).

Figure 8

Reflection coefficients as functions of thickness $\varepsilon$, with fixed ${\theta }_{1,2}=\pi /4,l=(15,15,15)$ for differrent values of (a) speeds (with fixed $\eta =1)$ and (b) amplitudes (with fixed $c=10)$ of soliton in the ballistic case.

Figure 9

Numerical error of the scheme (A4) over a straight bond. (a) Error dependence on $\tau$; (b) $|\tilde{\varphi }(1,·)|-|\varphi (1,·)|,\tau ={10}^{-4}$.