Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media

Dispersive shock waves dominate wave-breaking phenomena in Hamiltonian systems. In the absence of loss, these highly irregular and disordered waves are potentially reversible. However, no experimental evidence has been given about the possibility of inverting the dynamics of a dispersive shock wave and turn it into a regular wavefront. Nevertheless, the opposite scenario, i.e., a smooth wave generating turbulent dynamics, is well studied and observed in experiments. Here we introduce a theoretical formulation for the dynamics in a highly nonlocal and defocusing medium described by the nonlinear Schroedinger equation. Our theory unveils a mechanism that enhances the degree of irreversibility. This mechanism explains why a dispersive shock cannot be reversed in evolution even for an arbitrarily small amount of loss. Our theory is based on the concept of nonlinear Gamow vectors, i.e., power-dependent generalizations of the counterintuitive and hereto-elusive exponentially decaying states in Hamiltonian systems. We theoretically show that nonlinear Gamow vectors play a fundamental role in nonlinear Schroedinger models: They may be used as a generalized basis for describing the dynamics of the shock waves and affect the degree of irreversibility of wave-breaking phenomena. Gamow vectors allow analytical calculation of the amount of breaking of time reversal with a quantitative agreement with numerical solutions. We also show that a nonlocal, nonlinear optical medium may act as a simulator for the experimental investigation of quantum irreversible models, as the reversed harmonic oscillator.

Figure 1

(a) GV $|{\mathfrak{f}}_n^{-}{\left(x\right)|}^2$ for increasing even order $n$; (b) ${\partial }_{x}arg\left[{\mathfrak{f}}_n^{-}\left(x\right)\right]$ for increasing even order; and (c) weight of the GV expansion of a Gaussian wave packet.

Figure 2

(a) Numerical solution of Eq. (3) with $P={10}^4$ and ${\sigma }^2=10$. (b) Projection on GV for increasing $n$ for $\alpha =0.3$ and $\gamma =8$; continuous lines are from Eq. (3), dots are from Eq. (20). (c) As in panel (b) for $\gamma =24$.

Figure 3

(a) Forward propagation of ${\psi }_F\left(x\right)$ for $\alpha =0$; (b) backward propagation corresponding to panel (a); (c) as in panel (a) for $\alpha =0.3$; (d) backward propagation corresponding to panel (b); (e) degree of reversibility vs $P$ for various $\alpha$; and (f) fractional degree of intrinsic irreversibility; the dashed line is the estimate after Eq. (34). Parameters: $L=2$, ${\sigma }^2=10$.

Figure 4

(a) Forward propagation of the ${\mathfrak{f}}_n^{-}$ truncated GV ($n=0$, $x_G=40$, $P=4$, $\alpha =0$, $\sigma =10$); (b) as in panel (a) for $n=2$; (c) backward propagation of ${\mathfrak{f}}_n^{+}$ truncated GV ($n=0$, $x_G=40$, $P=4$, $\alpha =0$, $\sigma =10$); and (d) as in panel (c) for $n=2$.