High-speed soliton ejection generated from the scattering of bright solitons by modulated reflectionless potential wells

We investigate numerically and theoretically the conditions leading to soliton ejection stimulated through the scattering of bright solitons by modulated reflectionless potential wells. Such potential wells allow for the possibility of controlled ejection of solitons with significantly high speeds. At the outset, we describe the scattering setup and characterize the soliton ejection in terms of the different parameters of the system. Then, we formulate a theoretical model revealing the underlying physics of soliton ejection. The model is based on energy and norm exchange between the incident soliton and a stable trapped mode corresponding to an exact solution of the governing nonlinear Schrödinger equation. Remarkably, stationary solitons can lead to high-speed soliton ejection where part of the nonlinear interaction energy transforms to translational kinetic energy of the ejected soliton. Our investigation shows that soliton ejection always occurs whenever the incident soliton norm is greater than that of the trapped mode whereas their energy is almost the same. Once the incident soliton is trapped, the excess in norm turns to an ejected soliton in addition to a small amount of radiation that share translational kinetic energy. We found that higher ejection speeds are obtained with multinode trapped modes that have higher binding energy. Simultaneous two-soliton ejection has been also induced by two solitons scattering with the potential from both of its sides. An ejection speed almost twice as that of single soliton ejection was obtained. Ejection outcome and ejection speed turn out to be sensitive to the relative phase between the two incoming solitons, which suggests a tool for soliton phase interferometry.

Figure 1

Spatiotemporal plot showing soliton ejection from the potential well with $V_0=3.9$ and $\alpha =0.69\sqrt{V_0}$ with an ejection speed $v_e=0.32$. Other parameters: $u_0=1,x_0=-10,v_i=0.15,g=1$.

Figure 2

Scattering coefficients of the soliton with initial amplitude $u_0=6.3$ using the considered potential well in terms of incidence speed. Other parameters: $g=1,V_0=100,x_0=-15,\alpha =0.69\sqrt{V_0}$.

Figure 3

Speed of the ejected soliton propagated with different initial velocities. Spatiotemporal inset figures describe the dynamics of the soliton propagated with certain specific initial velocities. Constant ejection speed $v_e=0.32$ is obtained for incident velocities up to the vertical dotted line at incident soliton speed $v_i=0.285$. Parameters: $x_0=-10,u_0=g=1,V_0=2,\alpha =0.69\sqrt{V_0}$.

Figure 4

Speed of ejected soliton versus the position of stationary input soliton. Spatiotemporal inset figures describe the dynamics of the soliton propagated from certain specific initial positions. Vertical dotted lines show the boarders at which soliton ejection disappears ($x_0<-15$ and $x_0>-0.3$) and the boarders within which the ejection speed is constant ($-12.1<x_0<-8$). Parameters: $u_0=g=1,V_0=2,\alpha =0.69\sqrt{V_0}$.

Figure 5

Speed of the ejected soliton for soliton propagating from different initial positions with $v_i=0.15$. Spatiotemporal inset figures describe the dynamics of the soliton propagated from certain specific initial positions. Other parameters: $u_0=g=1,V_0=2,\alpha =0.69\sqrt{V_0}$.

Figure 6

Speed of the ejected soliton for different input amplitudes. Spatiotemporal inset figures describe the dynamics of the soliton propagated with certain specific initial amplitudes. Trajectory of the ejected soliton appears dotted due to the high soliton speed which misses some snapshot frames. Parameters: $x_0=-15,v_i=0.1,g=1,V_0=100,\alpha =3.5\sqrt{V_0}$.

Figure 7

Speed of the ejected soliton for different potential depths. Spatiotemporal inset figures describe the dynamics of the soliton propagated with certain potential depths. Trajectory of the ejected soliton appears dotted due to the high soliton speed which misses some snapshot frames. Parameters: $x_0=-15,u_0=5,v_i=0.1,g=1,\alpha =3.5\sqrt{V_0}$.

Figure 8

Speed gain for the different cases considered during the characterization in terms of: (a) initial position of incident soliton, (b) initial speed of incident soliton, (c) amplitude of incident soliton, and (d) potential depth. Parameters used are the same of those in (a) Fig. 5, (b) Fig. 3, (c) Fig. 6, and (d) Fig. 7.

Figure 9

A snapshot of ejected and trapped soliton profiles at $t=111.49$. Solid red curve corresponds to the numerical solution of Eq. (1). Dashed black curve corresponds to the analytical profile of a bright soliton Eq. (11). Dotted-dashed green curve corresponds to the exact analytical expression representing the trapped mode Eq. (8). Dotted blue curve corresponds to the potential well divided by $V_0$. Parameters: $x_0=-10,v_i=0.1,g=1,V_0=2,u_0=\alpha =\sqrt{2V_0}$.

Figure 10

Ejection speed of the soliton as a function of the potential depth. Solid black curve corresponds to the numerical solution of Eq. (1). Points correspond to the ejection speed calculated using: Eq. (13) (red triangles), Eq. (16) (green squares), Eq. (18) (purple circles), and Eq. (22) (blue diamonds). Parameters: $g=1,v_i=0.1,x_0=-10,\alpha =u_0=\sqrt{2V_0}$.

Figure 11

Time evolution of energy components for a bright soliton scattered by the potential well. (a) Total energy and (b) trapped mode energy. The curves correspond to kinetic energy ($\mathrm{KE},{\mathrm{KE}}_{\mathrm{trap}}$), interaction energy ($\mathrm{IE},{\mathrm{IE}}_{\mathrm{trap}}$), potential energy ($\mathrm{PE},{\mathrm{PE}}_{\mathrm{trap}}$), radiated energy (RE), and total energy ($\mathrm{TE},E_{\mathrm{trap}}$). The point of interaction is indicated by the vertical dotted gray line. Parameters: $g=1,v_i=0.1,x_0=-10,V_0=2,\alpha =u_0=\sqrt{2V_0}$.

Figure 12

(a) Spatiotemporal plot showing the four-node trapped mode and (b) pulse profile of the ejected soliton at $t=63$. Solid red curve corresponds to the numerical solution of Eq. (1) and dotted blue curve corresponds to the potential well divided by $V_0$. Inset in (a) shows a zoom-in of the scattering near the potential region. Parameters: $u_0=2$, $V_0=22$, $\alpha =\sqrt{V_0}/5$, $x_0=-10$, $v_i=0.1$, $g=1$.

Figure 13

Two solitons scattered by the potential from both of its sides. (a) Two out-of-phase solitons $\mathrm{\Delta }\varphi =\pi$. (b) Two in-phase solitons $\mathrm{\Delta }\varphi =0$ with a symmetric soliton ejection. Ejection speed is $v_e=0.43$. (c) Single-soliton ejection with ejection speed $v_e=3.40$. (d) Asymmetric soliton ejection for $\mathrm{\Delta }\varphi =0.9\pi$. Ejection speed is $v_e=3.9$. Parameters: $u_0=\sqrt{2V_0},V_0=16,\alpha =\sqrt{2V_0},x_0=-3,v_i=0.1,g=1$.

Figure 14

Symmetric two-soliton ejection resulting from two out-of-phase input solitons, $\mathrm{\Delta }\varphi =\pi$. Ejection speed is $v_e=1.12$. Parameters: $u_0=2,V_0=22,\alpha =\sqrt{V_0}/6,x_0=-10,v_i=0.1,g=1$.