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

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 $v>1-\omega$, where $v$ and $\omega$ are, respectively, its velocity and frequency. This methodology can be generalized to more complex nonlinear wave models.

Figure 1

Sketch of the computational domain $\Omega$.

Figure 2

Sketch of the tree geometry for the 1D effective model.

Figure 3

The fork region $F$.

Figure 4

Relative energy $|{\mathcal{E}}_n-\langle {\mathcal{E}}_n\rangle |/\langle {\mathcal{E}}_n\rangle$ as a function of time for the 2D finite element solution of a breather propagating in a 2D domain.

Figure 5

Motion in a T junction. Snapshots of a kink starting in branch 1 with the velocity $v_1=0.75$. The times are $t=1350$ (top) and $t=3000$ (bottom).

Figure 6

Motion of a kink in a ${90}^{\circ }$ Y junction. Snapshots of a kink starting in branch 1 with the velocity $v_1=0.75$. The times are $t=900$ (top), $t=1500$ (middle), and $t=3500$ (bottom). The widths are $w_1=1$ and $w_2=w_3=0.7$.

Figure 7

Time evolution of the energy for the kink motion in branches 1 and 2 for the T junction, shown as the solid line (red online), and for the Y junction, shown as the dashed line. The energy for the 1D effective model is plotted with points. The parameters are the same as those in Figs. 5 and 6.

Figure 8

Energy in branch 1 as a function of time for the 1D model and a breather of initial velocity $v_1=0.4$ and frequencies ${\omega }_1=0.5,0.7,0.725$, and $0.75$. Notice the crossing for ${\omega }_1=0.75$. The widths are $w_1=w_2=w_3=1$.

Figure 9

Parameter space $(v,\omega )$ for the crossing of breathers for the 2D and the 1D models. The crosses (red online) indicate crossing and the squares reflection. The widths are $w_1=w_2=w_3=1$.

Figure 10

Time evolution of the energy in branches 1 and 2 for a breather for the 2D partial differential equation, shown as the solid line (red online), and the 1D effective model, shown as the dashed line. The parameters are $w_1=w_2=w_3=1$, $v_1=0.8$, ${\omega }_1=0.3$, and $x_0=10$.

Figure 11

Snapshots of the energy density of a breather at different times in branches 1 (top) and 2 (bottom). The times are indicated on the plots. The parameters are $w_1=w_2=w_3=1$, $v_1=0.8$, and ${\omega }_1=0.3$. The initial position of the breather is $x_0=10$.

Figure 12

Snapshots of the breather analytical solution (dashed line) together with the numerical solution (continuous line), in branch 1 before the collision [panels (a) and (b)], in branch 1 after the collision [panels (c) and (d)], and in branch 2 [panels (e) and (f)]. The corresponding times are $t=20.2$, $40.4$, $80.8$, $90.9$ for panels (a)–(d) and $t=80.8$, $90.9$ for panels (e) and (f).