Soliton synchronization in the focusing nonlinear Schrödinger equation

The focusing nonlinear Schrödinger equation (NLSE) describes propagation of quasimonochromatic waves in weakly nonlinear media. The aim of this study is to determine conditions of soliton synchronization in the NLSE in terms of the solitons' position and phase parameters. For this purpose, the concept of asymptotic middle states of solitons in the NLSE is first introduced. With soliton solutions of the NLSE, it is shown that soliton synchronization can be achieved by synchronizing the asymptotic middle states of the solitons, and conditions of soliton synchronization in terms of the solitons' position and phase parameters are given. Although the interaction of the solitons is nonlinear, the conditions are linear equations. Then, aided with the synchronization conditions, simple initial conditions are presented for producing synchronized interaction of solitons without the need to obtain analytic expressions for the synchronized interaction of the solitons. The initial conditions are summations of fundamental solitons with no mutual overlap, so they might be convenient to implement in applicative contexts.

Figure 1

Synchronization of solitons in NLSE (1) may not in general be achieved by synchronizing position and phase, ${\mathrm{\Theta }}_j$ and ${\mathrm{\Phi }}_j$, of the individual single-soliton evolutions. Here $a_j=1(j=1,2,3,4), b_1=-b_4=0.5, b_2=-b_3=0.2$. ${\theta }_j$ and ${\varphi }_j$ are solved from ${\mathrm{\Theta }}_j=0$ and ${\mathrm{\Phi }}_j=0$ with $(x_p,t_p)=(0,0)$. The contrast to the synchronization scenario in Fig. 2 can be seen.

Figure 2

Four-soliton synchronization in NLSE (1) from analytical solutions with $a_j=1(j=1,2,3,4), b_1=-b_4=0.5, b_2=-b_3=0.2$, and $(x_p,t_p)=(0,0)$. ${\theta }_j$ and ${\varphi }_j$ are solved from Eqs. (11) and (12). It can be seen that the wave magnitude at $(x_p,t_p)$ is much higher than the surrounding wave state, and the wave form is localized in both $x$ and $t$ directions.

Figure 3

Numerical simulation of localized high-amplitude wave generation by five-soliton synchronization in NLSE (1) with the initial condition given by Eq. (32), where $a_1=1, a_2=1.1, a_3=1.2, a_4=1.3, a_5=1.4$, and $b_1=1, b_2=0.5, b_3=0, b_4=-0.6, b_5=-1.1$, and $(x_p,t_p)=(0,15)$, and ${\theta }_j$ and ${\varphi }_j$ are solved from the synchronization conditions (11) and (12) with ${\varphi }_c=0$. (a) The initial wave profile at $t=0$ and the generated high-amplitude wave profile at $t=t_p$. (b) Close view of the wave profile at $t=t_p$, from which we can see the expected peak magnitude of ${\sum }_{j=1}^5\left|a_j\right|=6$ is achieved. (c) Density plot of $\left|U\right|$ for the synchronized soliton collision, from which we can see the localization of the high-amplitude wave.