Singular tori as attractors of four-wave-interaction systems

We study the spatiotemporal dynamics of the Hamiltonian four-wave interaction in its counterpropagating configuration. The numerical simulations reveal that, under rather general conditions, the four-wave system exhibits a relaxation process toward a stationary state. Considering the Hamiltonian system associated to the stationary state, we provide a global geometrical view of all the stationary solutions of the system. The analysis reveals that the stationary state converges exponentially toward a pinched torus of the Hamiltonian system in the limit of an infinite nonlinear medium. The singular torus thus plays the role of an attractor for the spatiotemporal wave system. The topological properties of the singular torus confer a robust character to the stationary solution when the boundary conditions or the length of the nonlinear medium are modified. Furthermore, an adiabatic approach of the boundary conditions reveals that singular tori also play a major role for the description of the spatiotemporal dynamics of the wave system.

Figure 1

Phase space $\mathcal{M}={\mathbb{S}}^2\times {\mathbb{S}}^2$.

Figure 2

(Color online) Projection in the space $\left({\Pi }_1,{\Pi }_2,{\Pi }_3=0\right)$ of the reduced phase space (solid line in blue) and of the surface $H=h$ (dashed line in red) for (a) identical powers $\left(J_0=S_0=1\right)$ and for (b) different powers $\left(S_0=J_0/2=1/2\right)$.

Figure 3

(Color online) Bifurcation diagram with the critical fibers (single and doubly pinched tori, circle, point) associated to critical points for (a) identical powers and for (b) different powers.

Figure 4

(Color online) (a) Values of the function $G\left(K,{\alpha }_0\right)$ for the boundary conditions $I_0^s=0.5$ and $I_L^p=-0.71$ and for a medium length of $L=1$. (b) Sign of the function $G\left(K,{\alpha }_0\right)$. The blue (dark gray) and red (light gray) regions correspond respectively to the negative and positive values of $G$. The green dots are associated to couples $\left(K,{\alpha }_0\right)$ computed from spatiotemporal numerical simulations for ${\varphi }_L^p$ varying from 0 to $2\pi$ by step of 0.1 $\left(S_0=J_0=1\right)$.

Figure 5

(Color online) (a) Function $G\left(K,{\alpha }_0\right)$ for the boundary conditions $I_0^s=0.5$ and $I_L^p=-0.71$ and for a medium length of $L=4$. (b) Zoom of the small part in violet of (a). (c) Sign of the function $G\left(K,{\alpha }_0\right)$ for the values of (b). The blue (dark gray) and red (light gray) regions correspond respectively to the negative and positive values $\left(S_0=J_0=1\right)$.

Figure 6

(Color online) Plot of the function $f$ in blue (dark gray) as a function of $\tau$ that has been used in the numerical simulations.

Figure 7

(Color online) Signal and pump stationary states in the adiabatic approach. The lengths are $L=20$ (a) and $L=50$ (b). The boundary conditions are $I_0^s=0.5$, ${\varphi }_0^s=\pi /4$, $I_L^p=-0.71$ and ${\varphi }_L^p=\pi$. The blue (dark gray) and red (light gray) curves represent respectively the signal and the pump.

Figure 8

(Color online) (a) Evolution of the positions of singular tori obtained by the theory in red (light gray) and corresponding evolution of the constants of motion obtained by solving numerically the PDE system [Eq. (2)] in blue (dark gray). The line (1) corresponds to the evolution of the double pinched torus while lines (2) and (3) are associated to critical fibers belonging to the boundary of the bifurcation diagram. The black arrow indicates the direction of the evolution. (b) Zoom of the green panel in (a): the space-time dynamics of the PDE system [Eq. (2)] follows adiabatically the singular pinched torus. The boundary conditions are $I_0^s=0.5$, ${\varphi }_0^s=\pi /4$, $I_L^p=-0.71$, and ${\varphi }_L^p=\pi$. The length of the medium is $L=20$.

Figure 9

(Color online) Evolution of the distance in the phase-space diagram $\rho =\sqrt{H^2+K^2}$ between the singular torus and the regular torus on which lies the stationary state for a given length $L$: the stationary state converges exponentially toward the singular torus.

Figure 10

(Color online) Functions $f_1$ in blue (dark gray) and $f_2$ in red (light gray) as a function of $\tau$ which are used in the study of the nonsimultaneous injection of the waves. The pump corresponding to $f_1$ is injected before the signal wave corresponding to $f_2$.

Figure 11

(Color online) (a) Evolution of the positions of singular tori in red (light gray) and of the point of coordinates $\left(K_{sp},H_{sp}\right)$ in blue (dark gray). The lines (1) et (2) correspond to the evolution of the pinched tori. The black arrow indicates the time evolution. The time interval between the two dashed lines corresponds to the injection of the signal wave. (b) Stationary state associated to the region in the pink square (upper square) on (a). (c) Stationary state associated to the region in the green square (lower square) on (a). The (final) boundary conditions are $I_0^s=0.5$, ${\varphi }_0^s=\pi /4$, $I_L^p=-0.71$ et ${\varphi }_L^p=\pi$. The length of the nonlinear medium is $L=5$.