Many-body interaction in fast soliton collisions

We study $n$-pulse interaction in fast collisions of $N$ solitons of the cubic nonlinear Schrödinger (NLS) equation in the presence of generic weak nonlinear loss. We develop a generalized reduced model that yields the contribution of the $n$-pulse interaction to the amplitude shift for collisions in the presence of weak $(2m+1)$-order loss, for any $n$ and $m$. We first employ the reduced model and numerical solution of the perturbed NLS equation to analyze soliton collisions in the presence of septic loss $(m=3)$. Our calculations show that the three-pulse interaction gives the dominant contribution to the collision-induced amplitude shift already in a full-overlap four-soliton collision, and that the amplitude shift strongly depends on the initial soliton positions. We then extend these results for a generic weak nonlinear loss of the form $G\left(\right|\psi {|}^2)\psi$, where $\psi$ is the physical field and $G$ is a Taylor polynomial of degree $m_c$. Considering $m_c=3$, as an example, we show that three-pulse interaction gives the dominant contribution to the amplitude shift in a six-soliton collision, despite the presence of low-order loss. Our study quantitatively demonstrates that $n$-pulse interaction with high $n$ values plays a key role in fast collisions of NLS solitons in the presence of generic nonlinear loss. Moreover, the scalings of $n$-pulse interaction effects with $n$ and $m$ and the strong dependence on initial soliton positions lead to complex collision dynamics, which is very different from that observed in fast NLS soliton collisions in the presence of cubic loss.

Figure 1

The total collision-induced amplitude shift of the $j=1$ soliton $\Delta {\eta }_1$ vs frequency difference $\Delta \beta$ in a full-overlap four-soliton collision (a) and in a full-overlap six-soliton collision (b) in the presence of septic loss with coefficient ${\epsilon }_7=0.002$. The solid black line is the analytic prediction of Eqs. (11, 12, 13, 14) and the squares represent the result of numerical simulations with Eq. (5). The dotted red, dashed blue, and dash-dotted green lines correspond to the contributions of two-, three-, and four-soliton interactions to the amplitude shift, $\Delta {\eta }_1^{\left(2\right)}$, $\Delta {\eta }_1^{\left(3\right)}$, and $\Delta {\eta }_1^{\left(4\right)}$, respectively.

Figure 2

The collision-induced amplitude shift of the $j=1$ soliton $\Delta {\eta }_1$ vs the initial phase of the soliton with frequency $\beta =30$ in full-overlap $N$-soliton collisions in the presence of septic loss with ${\epsilon }_7=0.002$. The blue (upper) and red (lower) circles represent the results of numerical simulations with Eq. (5) for four-soliton and six-soliton collisions, respectively. The solid blue and dashed red lines correspond to the analytic predictions of Eqs. (11, 12, 13, 14) for four-soliton and six-soliton collisions. The initial phase of the $\beta =30$ soliton is denoted by ${\alpha }_3\left(0\right)$ in the four-soliton collision and by ${\alpha }_4\left(0\right)$ in the six-soliton collision.

Figure 3

The final soliton amplitudes ${\eta }_j\left(z_f\right)$ vs the initial position of the $j=3$ soliton $y_3\left(0\right)$ in a four-soliton collision in the presence of septic loss with ${\epsilon }_7=0.02$. The solid black curve, dashed red curve, short-dashed blue curve, and dash-dotted green curve represent the analytic predictions of Eqs. (7)–(10) for ${\eta }_j\left(z_f\right)$ with $j=1,2,3,4$, respectively. The black up triangles, red down triangles, blue squares, and green circles correspond to the results obtained by numerical solution of Eq. (5) for ${\eta }_j\left(z_f\right)$ with $j=1,2,3,4$, respectively.

Figure 4

The total collision-induced amplitude shift of the $j=1$ soliton $\Delta {\eta }_1$ vs frequency difference $\Delta \beta$ in a full-overlap four-soliton collision (a) and in a full-overlap six-soliton collision (b) in the presence of generic nonlinear loss of the form (1) with $m_c=3$ and loss coefficients ${\epsilon }_1=0.002$, ${\epsilon }_3=0.004$, ${\epsilon }_5=0.006$, and ${\epsilon }_7=0.001$. The solid black line is the analytic prediction of Eqs. (11, 12, 13, 14) and the squares correspond to the result of numerical simulations with Eq. (2). The dotted red, dashed blue, and dash-dotted green lines represent the contributions of two-, three-, and four-soliton interactions to the amplitude shift, $\Delta {\eta }_1^{\left(2\right)}$, $\Delta {\eta }_1^{\left(3\right)}$, and $\Delta {\eta }_1^{\left(4\right)}$, respectively.