Dynamics of vortex-antivortex pairs and rarefaction pulses in liquid light

We present a numerical study of the cubic-quintic nonlinear Schrödinger equation in two transverse dimensions, relevant for the propagation of light in certain exotic media. A well-known feature of the model is the existence of flat-top bright solitons of fixed intensity, whose dynamics resembles the physics of a liquid. They support traveling wave solutions, consisting of rarefaction pulses and vortex-antivortex pairs. In this work, we demonstrate how the vortex-antivortex pairs can be generated in bright soliton collisions displaying destructive interference followed by a snake instability. We then discuss the collisional dynamics of the dark excitations for different initial conditions. We describe a number of distinct phenomena including vortex exchange modes, quasielastic flyby scattering, solitonlike crossing, fully inelastic collisions, and rarefaction pulse merging.

Figure 1

Radial profiles of the three solitons used to define the initial conditions in the examples below. Their powers $P=2\pi {\int }_0^{\infty }rf{\left(r\right)}^{2}dr$ are ${P|}_{\beta =0.15}=86.0, {P|}_{\beta =0.1802}=2055.5, {P|}_{\beta =0.1856}=30620$.

Figure 2

Three numerically computed traveling waves with $U=0.11$ [vortex-antivortex pair, (a)], $U=0.35$ [rarefaction pulse, (b)], and $U=0.71$ [faint rarefaction pulse, (c)]. The dark excitations are moving rightwards. The color scale displays the density ${\left|\psi \right|}^2$ and the arrows represent the current density $\mathbf{j}$.

Figure 3

The encounter of two bright solitons giving rise to a traveling vortex-antivortex pair. Initial conditions have ${\mathbf{x}}_{\mathbf{2}}=(-180,0), {\mathbf{v}}_{\mathbf{2}}=(0.2,0), {\varphi }_2=5$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller one has ${\beta }_2=0.1802$. The color code for ${\left|\psi \right|}^2$ is as in Fig. 2 and the range of the axes is $x\in [-270,270], y\in [-190,190]$. An animation is provided in the Supplemental Material [48].

Figure 4

Phase structure of the wave function. The plot corresponds to $z=2000$, Fig. 3. The region of the vortex-antivortex has been enlarged. (a) depicts ${\left|\psi \right|}^2$ with the same color scale of Fig. 2 and the arrows are a quiver diagram for $\mathbf{j}$ showing flows similar to Fig. 2. (b) corresponds to the interference pattern with a plane wave: ${|\psi (x,y,z=2000)+7exp(-100iy)|}^2$. The forklike structures prove the existence of a vortex-antivortex pair with charges $\pm 1$.

Figure 5

Head-on encounter of two vortex-antivortex pairs resulting in an exchange mode. Initial conditions are defined by Eq. (4) with $-{\mathbf{x}}_{\mathbf{2}}={\mathbf{x}}_{\mathbf{3}}=(180,0), {\mathbf{v}}_{\mathbf{2}}=-{\mathbf{v}}_{\mathbf{3}}=(0.2,0), {\varphi }_2=5.3, {\varphi }_3=5$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller ones have ${\beta }_2={\beta }_3=0.1802$. Color code and axes are defined as in Fig. 3. An animation is provided in the Supplemental Material [48].

Figure 6

Encounter at a right angle of two vortex-antivortex pairs resulting in an exchange mode. Initial conditions are defined by Eq. (4) with ${\mathbf{x}}_{\mathbf{2}}=(0,180), {\mathbf{x}}_{\mathbf{3}}=(180,0), {\mathbf{v}}_{\mathbf{2}}=(0,-0.2), {\mathbf{v}}_{\mathbf{3}}=(-0.2,0), {\varphi }_2={\varphi }_3=5$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller ones have ${\beta }_2={\beta }_3=0.1802$. The color code is defined as in Fig. 2. The range of the axes is $x,y\in [-270,270]$. An animation is provided in the Supplemental Material [48].

Figure 7

Encounter of two vortex-antivortex pairs, shifted with respect to each other along the direction transverse to propagation, resulting in pseudoelastic scattering. Initial conditions are defined by Eq. (4) with ${\mathbf{x}}_{\mathbf{2}}=(-180,-10), {\mathbf{x}}_{\mathbf{3}}=(180,10), {\mathbf{v}}_{\mathbf{2}}=(0.2,0), {\mathbf{v}}_{\mathbf{3}}=(-0.2,0), {\varphi }_2={\varphi }_3=5$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller ones have ${\beta }_2={\beta }_3=0.1802$. Color code and axes range are defined as in Fig. 3. An animation is provided in the Supplemental Material [48].

Figure 8

Flyby encounter of a rarefaction pulse with a vortex-antivortex pair. The pulse trajectory is modified because of the flows generated by the vortex-antivortex phase structure. The horizontal dashed line marks $y=0$, the path that the rarefaction pulse would follow in the absence of other excitations. Initial conditions are defined by Eq. (4) with ${\mathbf{x}}_{\mathbf{2}}=(-180,27), {\mathbf{x}}_{\mathbf{3}}=(240,0), {\mathbf{v}}_{\mathbf{2}}=(0.2,0), {\mathbf{v}}_{\mathbf{3}}=(-0.5,0), {\varphi }_2=4.9, {\varphi }_3=4$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller ones have ${\beta }_2=0.1802$ and ${\beta }_3=0.15$. Color code and axes range are defined as in Fig. 3. An animation is provided in the Supplemental Material [48].

Figure 9

Inelastic collision of a rarefaction pulse and a vortex-antivortex pair. The rarefaction pulse touches the antivortex, blends with it and, eventually, also gets connected to the dark region around the vortex. The resulting structure can be considered as a highly excited vortex-antivortex pair which continues to propagate within the liquid of light. Initial conditions are defined by Eq. (4) with ${\mathbf{x}}_{\mathbf{2}}=(-180,18), {\mathbf{x}}_{\mathbf{3}}=(240,0), {\mathbf{v}}_{\mathbf{2}}=(0.2,0), {\mathbf{v}}_{\mathbf{3}}=(-0.5,0), {\varphi }_2=4.9, {\varphi }_3=5$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller ones have ${\beta }_2=0.1802$ and ${\beta }_3=0.15$. Color code and axes range are defined as in Fig. 3. An animation is provided in the Supplemental Material [48].

Figure 10

Symmetric head-on encounter of two rarefaction pulses, which cross each other, losing a fraction of their energy in the process. Initial conditions are defined by Eq. (4) with $-{\mathbf{x}}_{\mathbf{2}}={\mathbf{x}}_{\mathbf{3}}=(180,0), {\mathbf{v}}_{\mathbf{2}}=-{\mathbf{v}}_{\mathbf{3}}=(0.5,0), {\varphi }_2={\varphi }_3=6$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller ones have ${\beta }_2={\beta }_3=0.15$. Color code and axes range are defined as in Fig. 3. An animation is provided in the Supplemental Material [48].

Figure 11

Symmetric head-on encounter of two rarefaction pulses resulting in a fully inelastic collision. The pulses annihilate each other and yield all their energy to a circular sound wave. Initial conditions are defined by Eq. (4) with $-{\mathbf{x}}_{\mathbf{2}}={\mathbf{x}}_{\mathbf{3}}=(180,0), {\mathbf{v}}_{\mathbf{2}}=-{\mathbf{v}}_{\mathbf{3}}=(0.5,0), {\varphi }_2={\varphi }_3=4.8$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller ones have ${\beta }_2={\beta }_3=0.15$. Color code and axes range are defined as in Fig. 3. An animation is provided in the Supplemental Material [48].

Figure 12

Two rarefaction pulses collide perpendicularly and merge. Initial conditions are defined by Eq. (4) with ${\mathbf{x}}_{\mathbf{2}}=(0,180),{\mathbf{x}}_{\mathbf{3}}=(180,0), {\mathbf{v}}_{\mathbf{2}}=(0,-0.5), {\mathbf{v}}_{\mathbf{3}}=(-0.5,0), {\varphi }_2={\varphi }_3=5$. The large soliton is the one with ${\beta }_1=0.1856$ and the smaller ones have ${\beta }_2={\beta }_3=0.15$. The color code is defined as in Fig. 2. The range of the axes is $x,y\in [-270,270]$. An animation is provided in the Supplemental Material [48].