Universal regimes of strong turbulence in the multi-component Gross-Pitaevskii model

Abstract

The Gross-Pitaevskii (GP) model, also known as the nonlinear Schrödinger equation, is arguably the most universal model in classical and quantum physics, describing spectrally narrow or long-wavelength distributions of interacting waves or particles. Modern applications -- from oceanic and atmospheric flows to photonics and cold atoms -- predominantly involve states that are far from equilibrium, culminating in the regime of fully developed turbulence. To date, a consistent theoretical description of such states has only existed for weakly interacting quasiparticles. Here we present a theory of strong turbulence in the two-dimensional $N$-component Gross-Pitaevskii model for both repulsive and attractive interactions, corresponding to the defocusing and focusing cases, respectively. In the focusing case, we show that attraction is enhanced by multi-wave effects, leading to a critical-balance state independent of the pumping level. In the defocusing case, repulsion is suppressed by collective effects, giving rise to another type of universality in strong turbulence -- independence from the bare coupling constant. The theory is confirmed by analytical results in the many-component limit and by direct numerical simulations of the single-component GP model.

Figures (7)

• Figure 1: A schematic depiction of the transition from weak turbulence at large $k$ to strong turbulence at small $k$. The left panel shows the case with a fixed negative $\lambda$, with the dashed line corresponding to larger flux. The right panel shows the positive $\lambda$ case with a fixed flux, with the dashed line corresponding to larger coupling. The central panel shows two cases, with equal flux $Q$ and couplings $\lambda$ and $-\lambda$, respectively.
• Figure 2: Direct numerical simulation of the focusing Gross-Pitaevskii model with one field, $N=1$, $\lambda=-1$, with $k_{\min}=1/4$ and no low-$k$ damping, $f_k=0$, for different pumping rates. Left panel: Spectra for $\beta=1000$ and pumping rates $Q$ shown in the inset; black lines are approximations $n_k =T /[\mu +k^2]$ fitted up to pumping peak; inset shows the plateau values. The $n_k/k_{min}^2$ here should be compared with $n_k/(2\pi)^2$ in the theory section: the large $N$ theory prediction (\ref{['minus']}) gives $n_k/k_{min}^2 = 1/2\pi\approx .16$, larger by about a factor of $4$ than the value observed here for $N=1$. Right panel: Parameters $T(Q),\mu(Q)$ for $\beta=1000$ (solid lines) and $\beta=400$ (dashed lines).
• Figure 3: DNS of GP for $N=1$, $k_{\min}=1/8$, and the pumping rate is $Q=10^6$. Left panel: Lower (black) curve is for the focusing case with no low-$k$ sink, $f_k=0$; upper (purple) curve is for the defocusing case with the sink $f_k = -3000 k^{-1}$ for $k<k_p$. Middle and right panels: defocusing case for $f_k = -\gamma k^{-1}$ (middle) and $f_k = -\gamma k^{-2}$ (right) at $k<k_p$. The colors denote different values of $\gamma$ shown in the legend. Solid, dashed, and dotted lines are for $\lambda=1, 2, 4$, respectively. The solid straight line is the formula \ref{['plus']} without any fitting parameter.
• Figure 4: Left: Spectra obtained by DNS of the defocusing GP with $N=1$, $\lambda=1$, $Q=10^6$, $k_{\min} = 1/8$. Damping rate is $f_k = -\gamma k^{-1}$ at $k<k_p$ with two different values of $\gamma$. The values of action $\cal N$ are given in parentheses. Only the curve for high $\gamma$ agrees with the theoretical formula obtained for the $N\gg1$-component GP in \ref{['plus']}. Right: The probability density functions of $|\Psi|$ for the two spectra in the left panel.
• Figure 5: For the multi-component GP model the process of two wave scattering is dominated by bubble diagrams in the limit of a large number of components. These can be summed, allowing one to study the theory at strong nonlinearity.