Solitary wave billiards

In the present work we explore the concept of solitary wave billiards. That is, instead of a point particle, we examine a solitary wave in an enclosed region and examine its collision with the boundaries and the resulting trajectories in cases which for particle billiards are known to be integrable and for cases that are known to be chaotic. A principal conclusion is that solitary wave billiards are generically found to be chaotic even in cases where the classical particle billiards are integrable. However, the degree of resulting chaoticity depends on the particle speed and on the properties of the potential. Furthermore, the nature of the scattering of the deformable solitary wave particle is elucidated on the basis of a negative Goos-Hänchen effect which, in addition to a trajectory shift, also results in an effective shrinkage of the billiard domain.

Figure 1

Dependence of the norm and the width of the solitary wave with respect to its frequency for the relevant family of solutions.

Figure 2

Left: Form of the square potential barrier given by (12). Right: The trajectory of a classical point particle (right), in this potential.

Figure 3

Evolution of the center-of-mass of the finite-width solitary wave with $\alpha =4$ and $v_0=0.02$, for $\omega =0.5$ (left) and $\omega =0.95$ (right). The final time of both simulations is $t=2\times {10}^5$. Notice that, in this figure and the other ones showing square barriers, the axis limits correspond to the barrier boundaries. See also companion movies movie1.gif (corresponding to $\omega =0.5$) and movie2.gif (corresponding to $\omega =0.95$) in the Supplemental Material [40] depicting the evolution of the solitary wave dynamics, in which the whole domain is shown with the barrier limits being depicted as white lines.

Figure 4

Evolution of the center of mass of the solitary waves with $v_0=0.1$ (left panel) and $v_0=0.8$ (right panel), in the square barrier potential (12). In both cases, $\alpha =4$ and $\omega =0.5$. The final times of the simulations are $t=4\times {10}^4$ and $t=5\times {10}^3$, respectively. See also companion movies movie3.gif and movie4.gif in the Supplemental Material [40] depicting the evolution of the solitary wave dynamics.

Figure 5

Left panel: Evolution of the projected solitary wave density when interacting with a wall barrier of height $\alpha =4$ located at $y=0$; the initial velocity of the coherent structure is $v_0=0.1$ and the initial angle $\theta ={cos}^{-1}3/5$. Right panel: Path followed by the center of mass of the wave together with a scheme indicating some relevant parameters like the Goos-Hänchen shift $\mathrm{\Delta }$ or the domain shrinking $\sigma$.

Figure 6

Dependence of the domain shrinking caused by the Goos-Hänchen effect for $\theta ={cos}^{-1}(3/5)$ (left panels) and $\theta ={cos}^{-1}(4/5)$ (right panels) for fixed frequency $\omega =0.5$ (top panels) and fixed initial speed $v_0=0.1$ (bottom panels). Dirichlet boundary conditions are denoted by $\alpha \to \infty$.

Figure 7

Trajectory of a classical particle in a shrunk domain with a Goos-Hänchen shift corresponding to a solitary wave in a square barrier potential of height $\alpha =4$ with the domain shrinking corresponding to $v_0=0.02$ and $\omega =0.5$. The left panel shows the evolution until $t=2\times {10}^5$, whereas right panel captures the early stage up to $t=34880$ and compares it with the evolution of the the center of mass of the solitary wave (full blue line). One can observe that up to the second collision with the barrier, the trajectories overlap, while at later times, they slightly deviate from each other.

Figure 8

Left: Evolution of the center-of-mass of the finite-width solitary wave with $\alpha =4, v_0=0.1$ and $\omega =0.5$ launched from $(-15,15)$ with an initial angle $\theta ={45}^{\circ }$. See companion movie movie5.gif in the Supplemental Material [40], in which the launching point is marked by a white dot in order to observe that the solitary wave comes back to that point. Right: Trajectory of a classical point particle launched from the same site and initial angle.

Figure 9

Left panels: Form of the Bunimovich stadium and Sinai billiard potentials. Right panels: Evolution of the center of mass of the solitary wave with $v_0=0.02$ in a Bunimovich stadium (left panel) and a Sinai billiard (right panel). See also companion movies movie6.gif and movie7.gif in [40] depicting the evolution of the billiard dynamics. The final time of both simulations is $t=3.2\times {10}^5$.