Collisions and fusion of one- and two-dimensional solitons driven by potential troughs in the cubic-quintic nonlinear Schrödinger equations

TL;DR

The paper investigates collisions of 1D and 2D Gaussian and flat-top solitons in the cubic-quintic nonlinear Schrödinger equation with two intersecting potential troughs. It employs the modified squared-operator method to find stationary solitons and conducts extensive simulations to classify collision outcomes under tilted troughs, identifying quasi-elastic versus inelastic mergers and spontaneous symmetry breaking in FT-FT interactions. Width and power scale as $W \sim \mathrm{const}\cdot V_i^{-1/2}$ with $P \sim W$, and collisions between Gaussian-based solitons are largely quasi-elastic while FT-FT mergers create a single trapped soliton, breaking symmetry. These findings have implications for all-optical data processing and suggest extensions to Bose-Einstein condensates described by a quadratic-cubic GPE, including investigations of phase differences such as $\pi$ between colliding solitons.

Abstract

We study the formation and collision of one- and two-dimensional (1D and 2D) Gaussian-shaped and flat-top (FT) solitons in the framework of the nonlinear Schrödinger equation with the cubic-quintic nonlinearity and two intersecting potential troughs. We find that Gaussian-Gaussian and Gaussian-FT collisions between the solitons, steered by the troughs, are quasi-elastic, while the collisions between FT solitons may be either quasi-elastic or inelastic, in the form of merger into a single FT soliton, thus spontaneously breaking the symmetry between the steering troughs. The Gaussian-FT collisions, being overall quasi-elastic, generate weak radiation fields.

Figures (7)

• Figure 1: (a) The profiles of 1D solitons, pinned to the potential given by Eq. (\ref{['VE1D']}) with the single potential trough, with different values of $V_{i}$ and $x_{i}=0$. The blue, red, and green lines plot the results for $V_{i}=0.02,0.05,0.25$, respectively. (b) The half-peak width and power (the left and right vertical axes, respectively) of the 1D soliton vs. $V_{i}$. The black dashed line is the best fit of the $W\left(V_{i}\right)$ dependence to relation (\ref{['fit']}), with $\mathrm{const}=1.45$. Throughout this work, we use values $A=10$ and $b=9,$ which make it possible to represent generic results.
• Figure 2: Collisions of 1D solitons with $\Omega _{1}=\Omega _{2}=0.04$ in Eq. (\ref{['Omega']}): (a) the collision between two Gaussians solitons steered by the troughs with $V_{1,2}=0.2$; (b) the FT-Gaussian collision, with $V_{1}=0.2$, $V_{2}=0.02$; (c) the FT-FT collision with $V_{1,2}=0.02$. Panels (d)--(f) are similar to (a)--(c), but for the collisions of 1D solitons with $\Omega _{1}=0.04$, $\Omega _{2}=0$. In all panels, $\tilde{x}_{1}=10$, $\tilde{x}_{2}=-10$, and $|x|\leq 25$.
• Figure 3: (a) The contours of 2D solitons, supported by the single-trough cylindrical potential (\ref{['VE2D']}) with different values of $V_{i}$ and $x_{i}=0$: (a) a Gaussian soliton at $V_{i}=0.25$; (b) an FT soliton at $V_{i}=0.05$; (c) an FT soliton at $V_{i}=0.02$. (d) The half-peak width and total power (the blue and red curves, respectively) of the 2D solitons vs. the trough's inverse width $V_{i}$. The black dashed line is the best fit of the $W\left( V_{i}\right)$ dependence to relation (\ref{['fit']}), with $\mathrm{const}=1.35$.
• Figure 4: (a1) Collisions of 2D Gaussian solitons with ($V_{1}=0.2, V_{2}=0.2$) and ($\Omega_{1}=0.04, \Omega_{2}=0.04$). Contours of the collision in panel (a1) at different values of the propagation distance $z$: (a2) at $z=0$, (a3) at $z=200$, (a4) $z=400$, (a5) $z=800$. Panels (b1)--(b5) display the results similar to those in panels (a1)--(a5), but for ($\Omega_{1}=0.04,\Omega_{2}=0$). In all panels, $\tilde{x}_{1}=10$, $\tilde{x}_{2}=-10$, $\left\vert x,y\right\vert \leq 25.$
• Figure 5: (a1) The 2D collision of a flat-top soliton and a Gaussian soliton with $V_{1}=0.2$, $V_{2}=0.02$ at $\Omega _{1}=0.04$, $\Omega _{2}=0.04$. The contours of the collision in panel (a1) at different values of the propagation distance $z$: (a2) $z=0$; (a3) $z=150$; (a4) $z=400$; (a5) $z=700$. The results displayed in panels (b1)--(b5) are similar to those in panels (a1)--(a5), but for $\Omega _{1}=0.04$, $\Omega _{2}=0$. In all panels, $\tilde{x}_{1}=10$, $\tilde{x}_{2}=-10$, $x,y\in [-25,+25]$.