In the continuous limit $\varrho \to V_0/\beta_0$, so that $E_{\text{crit}}$ and $T_{\text{crit}}$ recover the expressions in the thermodynamic limit $E_{\text{crit}}^*$ in Eq.(B.4) and $T_{\text{crit}}^*$ in Eq.(B.2).

Beyond the thermodynamic limit, we remove the assumption $\mu = \beta_0$ for $E \le E_{\text{crit}}$. Starting from $N = \sum_p T/(\beta_p - \mu)$ and remarking that $n_0^{\text{eq}} = T/(\beta_0 - \mu)$, we get:

$$ \frac{n_0^{\text{eq}}}{N}(\mu) = \frac{1}{-(\mu - \beta_0) \sum_p (\beta_p - \mu)^{-1}}, \qquad (\text{B.10}) $$

$$ \frac{E}{E_{\text{crit}}}(\mu) = \frac{\sum_p \frac{\beta_p}{\beta_p - \mu}}{(1 + (M-1)/\varrho) \sum_p \frac{\beta_0}{\beta_p - \mu}}, \qquad (\text{B.11}) $$

where (B.11) is obtained from $E = T \sum_p \beta_p/(\beta_p - \mu)$ with $T = N/\sum_p (\beta_p - \mu)^{-1}$. The parametric plot of Eqs.(B.10)-(B.11) with respect to $\mu$, provides the condensed fraction $n_0^{\text{eq}}/N$ vs $E/E_{\text{crit}}$.

# Appendix C Derivation of the kinetic equation with strong disorder [Eq.(37)].

## Appendix C.1 Primary asymptotics

The starting point is the NLS Eq.(34) written in the mode basis, i.e., Eq.(35). We consider the regime where linear propagation dominates over disorder, which in turn dominates over the nonlinearity. Accordingly, we introduce a small dimensionless parameter $\varepsilon$ and we consider the regime $\beta_p \to \beta_p$, $C \to \varepsilon C$, $\gamma \to \varepsilon^2\gamma$. For propagation distances of order $\varepsilon^{-2}$, the rescaled mode amplitudes $a_j^\varepsilon(z) = a_j(z/\varepsilon^2)$ satisfy

$$ \partial_z a_j^\varepsilon = -i \frac{\beta_j}{\varepsilon^2} a_j^\varepsilon + i \gamma \sum_{l,m,n=0}^{M-1} S_{jlmn} a_l^\varepsilon a_m^\varepsilon \bar{a}_n^\varepsilon - \frac{i}{\varepsilon} \sum_{l=0}^{M-1} C_{jl} \left(\frac{z}{\varepsilon^2}\right) a_l^\varepsilon, $$

where the bar stands for complex conjugation. We set $c_j^\varepsilon(z) = a_j^\varepsilon(z) \exp(i \frac{\beta_j}{\varepsilon^2} z)$. The amplitudes $c_j^\varepsilon(z)$ satisfy:

$$ \begin{aligned} \partial_z c_j^\varepsilon = & i\gamma \sum_{l,m,n=0}^{M-1} S_{jlmn} c_l^\varepsilon c_m^\varepsilon \bar{c}_n^\varepsilon \exp \left( i \frac{\beta_j - \beta_l - \beta_m + \beta_n}{\varepsilon^2} z \right) \\ & - \frac{i}{\varepsilon} \sum_{l=0}^{M-1} C_{jl} \left( \frac{z}{\varepsilon^2} \right) c_l^\varepsilon \exp \left( i \frac{\beta_j - \beta_l}{\varepsilon^2} z \right). \end{aligned} \qquad (\text{C.1}) $$

This is the usual diffusion approximation framework [145]. We get the following Proposition.

**Proposition:** The random process $(c_j^\varepsilon(z))_{j=0}^{M-1}$ converges in distribution in $C^0([0, \infty), \mathbb{C}^M)$, the space of continuous functions from $[0, \infty)$ to $\mathbb{C}^M$, to the Markov process $(c_j(z))_{j=0}^{M-1}$ with infinitesimal generator $\mathcal{L}$:

$$ \mathcal{L} = \mathcal{L}_1 + \mathcal{L}_2 + \mathcal{L}_3 + \mathcal{L}_4 + \mathcal{L}_5, \qquad (\text{C.2}) $$
with

$$ \mathcal{L}_1 = \frac{1}{2} \sum_{j,l=0,j \neq l}^{M-1} \Gamma_{jl}^{\text{OD}} (\mathfrak{c}_j \overline{\mathfrak{c}}_j \partial_{\epsilon_l} \partial_{\overline{\epsilon}_l} + \mathfrak{c}_l \overline{\mathfrak{c}}_l \partial_{\epsilon_j} \partial_{\overline{\epsilon}_j} - \mathfrak{c}_j \overline{\mathfrak{c}}_j \partial_{\epsilon_l} \partial_{\epsilon_l} - \overline{\mathfrak{c}_j \overline{\mathfrak{c}}_j} \partial_{\overline{\epsilon}_l} \partial_{\overline{\epsilon}_l}), $$

$$ \mathcal{L}_2 = \frac{1}{2} \sum_{j,l=0}^{M-1} \Gamma_{jl}^{\text{D}} (\mathfrak{c}_j \overline{\mathfrak{c}}_l \partial_{\epsilon_j} \partial_{\overline{\epsilon}_l} + \overline{\mathfrak{c}}_j \mathfrak{c}_l \partial_{\overline{\epsilon}_j} \partial_{\epsilon_l} - \mathfrak{c}_j \mathfrak{c}_l \partial_{\epsilon_j} \partial_{\epsilon_l} - \overline{\mathfrak{c}}_j \overline{\mathfrak{c}}_l \partial_{\overline{\epsilon}_j} \partial_{\overline{\epsilon}_l}), $$

$$ \mathcal{L}_3 = \frac{1}{2} \sum_{j=0}^{M-1} \Gamma_{jj}^{\text{OD}} (\mathfrak{c}_j \partial_{\epsilon_j} + \overline{\mathfrak{c}}_j \partial_{\overline{\epsilon}_j}) + i \hat{\Gamma}_{jj}^{\text{OD}} (\mathfrak{c}_j \partial_{\epsilon_j} - \overline{\mathfrak{c}}_j \partial_{\overline{\epsilon}_j}), $$

$$ \mathcal{L}_4 = -\frac{1}{2} \sum_{j=0}^{M-1} \Gamma_{jj}^{\text{D}} (\mathfrak{c}_j \partial_{\epsilon_j} + \overline{\mathfrak{c}}_j \partial_{\overline{\epsilon}_j}), $$

$$ \mathcal{L}_5 = i\gamma \sum_{l,m,n=0}^{M-1} \delta_{jlmn}^{K} S_{jlmn} (\mathfrak{c}_l \mathfrak{c}_m \overline{\mathfrak{c}}_n \partial_{\epsilon_j} - \overline{\mathfrak{c}_l \mathfrak{c}_m} \overline{\mathfrak{c}}_n \partial_{\overline{\epsilon}_j}), $$

where $\delta_{jlmn}^K = \mathbf{1}_{\beta_j - \beta_l - \beta_m + \beta_n = 0}$. In this definition we use the classical complex derivative: if $\zeta = \zeta_r + i\zeta_i$, then $\partial_{\zeta} = (1/2)(\partial_{\zeta_r} - i\partial_{\zeta_i})$ and $\partial_{\bar{\zeta}} = (1/2)(\partial_{\bar{\zeta}_r} + i\partial_{\bar{\zeta}_i})$, and the coefficients of the operator $\mathcal{L}_k$ ($k = 1, \dots, 5$) are defined for $j, l = 0, \dots, M-1$, as follows:

- For all $j \neq l$, $\Gamma_{jl}$ and $\hat{\Gamma}_{jl}^{\text{OD}}$ are given by

$$ \Gamma_{jl}^{\text{OD}} = 2 \int_{0}^{\infty} \mathcal{R}_{jl}(z) \cos ((\beta_l - \beta_j)z) dz, \qquad (\text{C.3}) $$

$$ \hat{\Gamma}_{jl}^{\text{OD}} = 2 \int_{0}^{\infty} \mathcal{R}_{jl}(z) \sin ((\beta_l - \beta_j)z) dz, \qquad (\text{C.4}) $$

with $\mathcal{R}_{jl}(z)$ defined by

$$ \mathcal{R}_{jl}(z) = \mathbb{E}[C_{jl}(0)C_{jl}(z)] = \iint u_j(\mathbf{r})u_j(\mathbf{r}')\mathbb{E}[\delta V(0, \mathbf{r})\delta V(z, \mathbf{r}')]u_l(\mathbf{r})u_l(\mathbf{r}')d\mathbf{r}d\mathbf{r}'. $$

- For all $j, l = 0, \dots, M-1$:

$$ \Gamma_{jl}^{\text{D}} = \int_{0}^{\infty} \mathbb{E}[C_{jj}(0)C_{ll}(z)] dz + \int_{0}^{\infty} \mathbb{E}[C_{ll}(0)C_{jj}(z)] dz. \qquad (\text{C.5}) $$

- For all $j = 0, \dots, M-1$:

$$ \Gamma_{jj}^{\text{OD}} = - \sum_{l=0, l \neq j}^{M-1} \Gamma_{jl}^{\text{OD}}, \qquad \hat{\Gamma}_{jj}^{\text{OD}} = - \sum_{l=0, l \neq j}^{M-1} \hat{\Gamma}_{jl}^{\text{OD}}. \qquad (\text{C.6}) $$

## Appendix C.2 Secondary asymptotics

We observe that $\Gamma^{\text{OD}}$ and $\hat{\Gamma}^{\text{OD}}$ depend on the power spectral density of the random index perturbation evaluated at the difference of distinct frequencies
$\beta_p - \beta_l$, while $\Gamma^D$ depends on the power spectral density of the index perturbation evaluated at zero-frequency. Therefore, when $L_{lin} = 1/\beta_0 \ll l_\beta$, then $\Gamma^D$ is larger than $\Gamma^{OD}, \hat{\Gamma}^{OD}$. We consider this regime by introducing a small dimensionless parameter $\eta$ with $\Gamma^D \to \Gamma^D, \Gamma^{OD} \to \eta^2\Gamma^{OD}, \hat{\Gamma}^{OD} \to \eta^2\hat{\Gamma}^{OD}, \gamma \to \eta\gamma$.

For propagation distances of order $\eta^{-2}$, we introduce the rescaled mode amplitudes $c_j^\eta(z) = c_j(z/\eta^2)$. By the Proposition presented above, it is a Markov process with infinitesimal generator $\mathcal{L}^\eta$:

$$ \mathcal{L}^\eta = \mathcal{L}_1 + \eta^{-2}\mathcal{L}_2 + \mathcal{L}_3 + \eta^{-2}\mathcal{L}_4 + \eta^{-1}\mathcal{L}_5, \quad (C.7) $$

where the operators $\mathcal{L}_k$ ($k = 1, \dots, 5$) are given above. By (C.7) the second-order moments satisfy for $j \neq j'$:

$$ \begin{aligned} \partial_z \mathbb{E}[\mathbf{c}_j^\eta \overline{\mathbf{c}_{j'}^\eta}] = & -\frac{1}{2\eta^2}(\Gamma_{jj}^D + \Gamma_{j'j'}^D - 2\Gamma_{jj'}^D)\mathbb{E}[\mathbf{c}_j^\eta \overline{\mathbf{c}_{j'}^\eta}] \\ & + \frac{1}{2}(\Gamma_{jj}^{\mathrm{OD}} + \Gamma_{j'j'}^{\mathrm{OD}})\mathbb{E}[\mathbf{c}_j^\eta \overline{\mathbf{c}_{j'}^\eta}] + \frac{i}{2}(\hat{\Gamma}_{jj}^{\mathrm{OD}} - \hat{\Gamma}_{j'j'}^{\mathrm{OD}})\mathbb{E}[\mathbf{c}_j^\eta \overline{\mathbf{c}_{j'}^\eta}] \\ & + i\frac{\gamma}{\eta}\sum_{l,m,n=0}^{M-1} \delta_{jlmn}^K S_{jlmn} \mathbb{E}[\overline{\mathbf{c}_j^\eta \mathbf{c}_l^\eta \mathbf{c}_m^\eta \mathbf{c}_n^\eta}] - i\frac{\gamma}{\eta}\sum_{l,m,n=0}^{M-1} \delta_{j'lmn}^K S_{j'lmn} \mathbb{E}[\overline{\mathbf{c}_j^\eta \mathbf{c}_l^\eta \mathbf{c}_m^\eta \mathbf{c}_n^\eta}], \end{aligned} $$

up to negligible terms in $\eta$. Note that $\Gamma_{jj}^D + \Gamma_{j'j'}^D - 2\Gamma_{jj'}^D = \int_{-\infty}^{\infty} \mathbb{E}[(C_{jj}(0) - C_{j'j'}(0))(C_{jj}(z) - C_{j'j'}(z))]dz$ is positive (it is the power spectral density evaluated at 0 frequency of the stationary process $C_{jj}(z) - C_{j'j'}(z)$ by Bochner's theorem). Therefore $\mathbb{E}[\mathbf{c}_j^\eta \overline{\mathbf{c}_{j'}^\eta}]$ is exponentially damped and

$$ \mathbb{E}[\mathbf{c}_j^\eta \overline{\mathbf{c}_{j'}^\eta}] = O(\eta). \quad (C.8) $$

If $j = j'$, then the mean square amplitudes $n_p^\eta(z) = \mathbb{E}[|\mathbf{c}_j^\eta(z)|^2]$ satisfy

$$ \partial_z n_p^\eta = \sum_{l=0, l \neq j}^{M-1} \Gamma_{jl}^{\mathrm{OD}} (n_l^\eta - n_p^\eta) - 2 \frac{\gamma}{\eta} \sum_{l,m,n=0}^{M-1} \delta_{jlmn}^K S_{jlmn} \mathrm{Im} \{ \mathbb{E}[\overline{\mathbf{c}_j^\eta \mathbf{c}_l^\eta \mathbf{c}_m^\eta \mathbf{c}_n^\eta}] \}. \quad (C.9) $$

By (C.7) the fourth-order moments satisfy

$$ \begin{aligned} \partial_z \mathbb{E}[\overline{\mathbf{c}_j^\eta \mathbf{c}_l^\eta \mathbf{c}_m^\eta \mathbf{c}_n^\eta}] = & -\frac{1}{2\eta^2} G_{jlmn}^D \mathbb{E}[\overline{\mathbf{c}_j^\eta \mathbf{c}_l^\eta \mathbf{c}_m^\eta \mathbf{c}_n^\eta}] + i \frac{\gamma}{\eta} Y_{jlmn}^\eta \\ & + \sum_{j', l', m', n'} M_{jlmn, j' l' m' n'} \mathbb{E}[\overline{\mathbf{c}_j^\eta \mathbf{c}_l^\eta \mathbf{c}_m^\eta \mathbf{c}_{n'}^\eta}], \end{aligned} \quad (C.10) $$

up to negligible terms in $\eta$. The coefficients $G_{jlmn}^D$ and the sixth-order moment
$Y_{jlmn}^{\eta}$ are given by

$$G_{jlmn}^{\mathrm{D}} = \Gamma_{ll}^{\mathrm{D}} + \Gamma_{mm}^{\mathrm{D}} + \Gamma_{nn}^{\mathrm{D}} + \Gamma_{jj}^{\mathrm{D}} + 2\Gamma_{lm}^{\mathrm{D}} - 2\Gamma_{ln}^{\mathrm{D}} - 2\Gamma_{lj}^{\mathrm{D}} - 2\Gamma_{mn}^{\mathrm{D}} - 2\Gamma_{mj}^{\mathrm{D}} + 2\Gamma_{nj}^{\mathrm{D}}, \qquad (C.11)$$

$$Y_{jlmn}^{\eta} = \sum_{l',m',n'=0}^{M-1} \delta_{ll'm'n'}^{K} S_{ll'm'n'} \mathbb{E}[\mathbf{c}_{l'}^{\eta} \mathbf{c}_{m'}^{\eta} \overline{\mathbf{c}_{n'}^{\eta} \mathbf{c}_{m}^{\eta}} \overline{\mathbf{c}_{j}^{\eta}}] + \delta_{ml'm'n'}^{K} S_{ml'm'n'} \mathbb{E}[\mathbf{c}_{l}^{\eta} \mathbf{c}_{m}^{\eta} \overline{\mathbf{c}_{n}^{\eta} \mathbf{c}_{m}^{\eta}} \overline{\mathbf{c}_{j}^{\eta}}] \\ - \delta_{nl'm'n'}^{K} S_{nl'm'n'} \mathbb{E}[\mathbf{c}_{l}^{\eta} \mathbf{c}_{m}^{\eta} \overline{\mathbf{c}_{n}^{\eta} \mathbf{c}_{l'}^{\eta}} \overline{\mathbf{c}_{j}^{\eta}}] - \delta_{jl'm'n'}^{K} S_{jl'm'n'} \mathbb{E}[\mathbf{c}_{l}^{\eta} \mathbf{c}_{m}^{\eta} \overline{\mathbf{c}_{n}^{\eta} \mathbf{c}_{l'}^{\eta}} \overline{\mathbf{c}_{m}^{\eta}} \overline{\mathbf{c}_{n'}^{\eta}}], \qquad (C.12)$$

up to negligible terms in $\eta$. The tensor $M_{jlmn,j'l'm'n'}$ involves the coefficients $\Gamma^{\mathrm{OD}}$ and $\hat{\Gamma}^{\mathrm{OD}}$. Note that we have $G_{jlmn}^{\mathrm{D}} = \int_{-\infty}^{\infty} \mathbb{E}[(C_{ll}(0) + C_{mm}(0) - C_{nn}(0) - C_{jj}(0))(C_{ll}(z) + C_{mm}(z) - C_{nn}(z) - C_{jj}(z))]dz \ge 0$. Therefore, we find from (C.10) that

$$\mathbb{E}[\overline{\mathbf{c}_j^{\eta} \mathbf{c}_l^{\eta} \mathbf{c}_m^{\eta} \mathbf{c}_n^{\eta}}] = \frac{2i\eta\gamma}{G_{jlmn}^{\mathrm{D}}} Y_{jlmn}^{\eta} + O(\eta^2).$$

By substituting into (C.9) and by using Isserlis formula for the sixth-order moments that appear in the expression (C.12) of $Y_{jlmn}^{\eta}$ we obtain the kinetic Eq.(37):

$$\partial_z n_p^{\eta} = \sum_{l=0, l \neq j}^{M-1} \Gamma_{jl}^{\mathrm{OD}} (n_l^{\eta} - n_p^{\eta}) + 8\gamma^2 \sum_{l,m,n=0}^{M-1} \frac{\delta_{jlmn}^{K} S_{jlmn}^{2}}{G_{jlmn}^{\mathrm{D}}} (n_l^{\eta} n_m^{\eta} n_p^{\eta} + n_l^{\eta} n_m^{\eta} n_n^{\eta} - n_p^{\eta} n_n^{\eta} n_m^{\eta} - n_p^{\eta} n_n^{\eta} n_l^{\eta}). \qquad (C.13)$$

The second term in (C.13) has a form analogous to the conventional collision term of the wave turbulence kinetic equation [40]. Exploiting the invariances properties of the tensors $S_{jlmn}$ and $G_{jlmn}^{\mathrm{D}}$, as well as the property $G_{jlmn}^{\mathrm{D}} \ge 0$, it can be shown that the collision term conserves the particle number $N = \sum_p n_p$, the energy $E = \sum_p \beta_p n_p$, and exhibits a H-theorem of entropy growth $\partial_z S^{\mathrm{neq}}(z) \ge 0$, where the nonequilibrium entropy reads $S^{\mathrm{neq}}(z) = \sum_p \log[n_p(z)]$ (note that, for simplicity we omitted to write the superscript $\eta$). The entropy growth saturates at thermal equilibrium. The distribution that maximizes the entropy, under the constraints that $N$ and $E$ are conserved, then corresponds to the RJ equilibrium $n_p^{\mathrm{RJ}} = T/(\beta_p - \mu)$.

## Appendix C.3 Degenerate modes

In this section we assume that the modes may be degenerate. The detailed derivation of the kinetic equation accounting for mode degeneracy is cumbersome and will be reported elsewhere. Here we report the main results.
There are $G$ distinct wavenumbers:

$$ \{\beta^{(g)}, g = 1, \dots, G\}, $$

and the mode indices can be partitioned into $G$ groups $\mathcal{G}^{(g)}, g = 1, \dots, G$:

$$ \mathcal{G}^{(g)} = \{p = 1, \dots, N, \beta_p = \beta^{(g)}\}. $$

We obtain the kinetic equation:

$$ \partial_z n^{(g)} = 8\gamma^2 \sum_{g_1,g_2,g_3=1}^{G} \delta^{(gg_1g_2g_3)} q^{(gg_1g_2g_3)} (n^{(g)} n^{(g_3)} n^{(g_2)} \\ + n^{(g)} n^{(g_3)} n^{(g_1)} - n^{(g_1)} n^{(g_2)} n^{(g)} - n^{(g_1)} n^{(g_2)} n^{(g_3)}), $$

where

$$ q^{(gg_1g_2g_3)} = \frac{1}{|\mathcal{G}^{(g)}|} \sum_{j \in \mathcal{G}^{(g)}, l \in \mathcal{G}^{(g_1)}, m \in \mathcal{G}^{(g_2)}, n \in \mathcal{G}^{(g_3)}} S_{jlmn} Q^{(gg_1g_2g_3)}_{jlmn} $$

where

$$ \mathbf{Q}^{(gg_1g_2g_3)} = (Q^{(gg_1g_2g_3)}_{jlmn})_{j \in \mathcal{G}^{(g)}, l \in \mathcal{G}^{(g_1)}, m \in \mathcal{G}^{(g_2)}, n \in \mathcal{G}^{(g_3)}} \\ = (\mathbf{M}^{(gg_1g_2g_3)})^{-1}((S_{jlmn})_{j \in \mathcal{G}^{(g)}, l \in \mathcal{G}^{(g_1)}, m \in \mathcal{G}^{(g_2)}, n \in \mathcal{G}^{(g_3)}}). $$

The tensor $\mathbf{M}^{(gg_1g_2g_3)}$ (seen as a $q \times q$ matrix with $q = |\mathcal{G}^{(g)}||\mathcal{G}^{(g_1)}||\mathcal{G}^{(g_2)}||\mathcal{G}^{(g_3)}|$) is given by:

$$ \sum_{j' \in \mathcal{G}^{(g)}, l' \in \mathcal{G}^{(g_1)}, m' \in \mathcal{G}^{(g_2)}, n' \in \mathcal{G}^{(g_3)}} M^{(gg_1g_2g_3)}_{jlmn, j'l'm'n'} n_{j'l'm'n'} \\ = \sum_{l' \in \mathcal{G}^{(g_1)}, m' \in \mathcal{G}^{(g_2)}} 2\gamma_{ll'mm'} n_{jl'm'n} + \sum_{n' \in \mathcal{G}^{(g_3)}, j' \in \mathcal{G}^{(g)}} 2\gamma_{nn'jj'} n_{j'lmn'} \\ - \sum_{l' \in \mathcal{G}^{(g_1)}, n' \in \mathcal{G}^{(g_3)}} 2\gamma_{ll'nn'} n_{jl'mn'} - \sum_{l' \in \mathcal{G}^{(g_1)}, j' \in \mathcal{G}^{(g)}} 2\gamma_{ll'jj'} n_{j'l'mn} \\ - \sum_{m' \in \mathcal{G}^{(g_2)}, n' \in \mathcal{G}^{(g_3)}} 2\gamma_{mm'nn'} n_{jlm'n'} - \sum_{m' \in \mathcal{G}^{(g_2)}, j' \in \mathcal{G}^{(g)}} 2\gamma_{mm'jj'} n_{j'lm'n} \\ + \sum_{l', l'' \in \mathcal{G}^{(g_1)}} \gamma_{l'll'll''} n_{jl'mn} + \sum_{m', m'' \in \mathcal{G}^{(g_2)}} \gamma_{m''m'mm''} n_{jlm'n} \\ + \sum_{n', n'' \in \mathcal{G}^{(g_3)}} \gamma_{n''n'nn''} n_{jlmn'} + \sum_{j', j'' \in \mathcal{G}^{(g)}} \gamma_{j''j'jj''} n_{j'lmn}. $$

where

$$ \gamma_{pqp'q'} = 2 \int_{0}^{\infty} \mathbb{E}[C_{pq}(z)C_{p'q'}(0)] e^{i(\beta_p - \beta_{q'})z} dz. $$
Appendix C.4 Model of disorder implemented in the simulations

We have considered in the numerical simulations an elliptical parabolic potential $V(\mathbf{x}) = q_x x^2 + q_y y^2$, with

$$u_{p_x, p_y}(x, y) = \sqrt{\kappa_x \kappa_y} (\pi p_x! p_y! 2^{p_x+p_y})^{-1/2} H_{p_x}(\kappa_x x) H_{p_y}(\kappa_y y) \exp\left[-\frac{(\kappa_x^2 x^2 + \kappa_y^2 y^2)}{2}\right],$$

the normalized Hermite-Gaussian functions with corresponding eigenvalues $\beta_p = \beta_{p_x, p_y} = \beta_{0x}(p_x + 1/2) + \beta_{0y}(p_y + 1/2)$, with $\kappa_x = (q_x/\alpha)^{1/4}$, $\kappa_y = (q_y/\alpha)^{1/4}$, $\beta_{0x} = 2\sqrt{\alpha q_x}$, $\beta_{0y} = 2\sqrt{\alpha q_y}$, and the radii of the fundamental mode $r_{0x} = 1/\kappa_x = \sqrt{2\alpha/\beta_{0x}}$, $r_{0y} = 1/\kappa_y = \sqrt{2\alpha/\beta_{0y}}$.

We have considered the following form of model of disorder: $\delta V(\mathbf{x}, z) = \mu(z) \cos(\kappa_x b_x x) \cos(\kappa_y b_y y)$, with $\mathbb{E}[\mu(0)\mu(z)] = \sigma_\beta^2 f(z)$, $f(z) = \exp(-|z|/l_\beta)$. The advantage of this model is that the matrices $C$, $\Gamma^D$, $\Gamma^{OD}$ can be computed in analytical form. We have

$$C_{nk}(z) = \mu(z) C_{n_x k_x}^0 C_{n_y k_y}^0 = \mu(z) \int u_{n_x}(x) \cos(\kappa_x b_x x) u_{k_x}(x) dx \int u_{n_y}(y) \cos(\kappa_y b_y y) u_{k_y}(y) dy.$$

Then we have for $j_x, j_y, l_x, l_y \ge 0$: $C_{j,j+2l} = \mu(z) C_{j_x,j_x+2l_x}^0 C_{j_y,j_y+2l_y}^0$ where we denote for $s = x$ or $s = y$:

$$C_{j_s, j_s+2l_s}^0 = (-1)^{l_s} b_s^{2l_s} \exp(-b_s^2/4) L_{j_s}^{2l_s} (b_s^2/2) \frac{\sqrt{j_s!/(j_s+2l_s)!}}{2^{l_s}}$$

and $C_{j_s, j_s+2l_s+1}^0 = 0$, where $L_j^l$ is the generalized Laguerre polynomial [194, formula 7.388.7]. In particular $C_{j_s j_s}^0 = \exp(-b_s^2/4) L_{j_s} (b_s^2/2)$, where $L_j$ is the Laguerre polynomial. For $j_x, j_y, l_x, l_y \ge 0$, we have $\Gamma_{jl}^D = 2\sigma_\beta^2 l_\beta C_{j_x j_x}^0 C_{j_y j_y}^0 C_{l_x l_x}^0 C_{l_y l_y}^0$. For $n_x, k_x, n_y, k_y \ge 0$ we obtain:

$$\Gamma_{n,k}^{\mathrm{OD}} = \frac{2\sigma_{\beta}^{2} l_{\beta} \mathcal{R}_{n_x, k_x}^{0} \mathcal{R}_{n_y, k_y}^{0}}{1 + l_{\beta}^{2} [\beta_{0x}(n_x - k_x) + \beta_{0y}(n_y - k_y)]^{2}},$$

where $\mathcal{R}_{j_s, j_s+2l_s+1}^0 = 0$ and

$$\mathcal{R}_{j_s, j_s+2l_s}^0 = b_s^{4l_s} \exp(-b_s^2/2) L_{j_s}^{2l_s} (b_s^2/2)^2 (j_s!/(j_s+2l_s)!) 2^{-2l_s}.$$

In order to avoid high values of $\Gamma_{n,k}^{OD}$, we have considered an irrational ratio $\beta_{0x}/\beta_{0y} = \sqrt{2}$, so that $\beta_{0x}(n_x - k_x) + \beta_{0y}(n_y - k_y) \ne 0$. Parameters are $(b_x = 0.4, b_y = 0.5)$ in Figs. 8-9, and $(b_x = 0.4, b_y = 0.3)$ in Fig. 10. In all cases we considered $M=46$ modes. Note that the value of $L_{nl} = 1/(\gamma N/A_{eff}^0)$ in the simulations is computed by considering that all the power $N$ is in the fundamental mode of effective area $A_{eff}^0 = 1/\int |u_0|^4(\mathbf{r})d\mathbf{r}$.

To implement the disorder in the simulations of the NLS Eq.(35), we considered an *exact discretization* of the Ornstein-Uhlenbeck process. The propagation axis is divided in intervals with deterministic lengths $\Delta z$, with $\Delta z < l_\beta$.
The random function $\mu(z)$ is stepwise constant over each elementary interval $z \in [k\Delta z, (k+1)\Delta z)$, where $\mu_0 \sim \mathcal{N}(0, \sigma_\beta^2/2)$ denotes the Gaussian distribution,

$$ \mu_k = \sqrt{1 - 2\Delta z/l_\beta \mu_{k-1} + \sqrt{2\Delta z/l_\beta} \mathcal{N}(0, \sigma_\beta^2/2)}, $$

with $\mathcal{N}(0, \sigma_\beta^2/2)$ all independent and identically distributed.

## Acknowledgments

This work was supported by: the EU - Next Generation EU under the Italian National Recovery and Resilience Plan (NRRP), Mission 4, CUP B53C22004050001, partnership on "Telecommunications of the Future" (PE00000001 - program "RESTART"), the European Innovation Council Pathfinder Open MULTISCOPE (101185664), the Italian Ministerial grant PRIN2022 "SAFE" (2022ESAC3K), Sapienza University of Rome Seed PNR (SP12218480C7D1E9), the Italian Ministry of Health under the Ricerca Finalizzata 2021 program (Project Code RF-2021-12373094), the Russian Science Foundation (21-72-30024-II), the Centre national de la recherche scientifique (CNRS), Conseil régional de Bourgogne Franche-Comté, iXCore Research Fondation, Agence Nationale de la Recherche (ANR-23-CE30-0021, ANR-19-CE46-0007, ANR-15-IDEX-0003, ANR-21-ESRE-0040). Some of the calculations were carried out using HPC resources from DNUM CCUB (Centre de Calcul, Université de Bourgogne).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## References

[1] G. P. "Agrawal, "Fiber-optic communication systems", "John Wiley & Sons", "2012".

[2] C. Lin, V. Nguyen, W. French, Wideband near-ir continuum (0.7–2.1 µm) generated in low-loss optical fibres, Electronics Letters 14 (25) (1978) 822–823.

[3] I. Cristiani, C. Lacava, G. Rademacher, B. J. Puttnam, R. S. Luís, C. Antonelli, A. Mecozzi, M. Shtaif, D. Cozzolino, D. Bacco, et al., Roadmap on multimode photonics, Journal of Optics 24 (8) (2022) 083001.
[4] P. Horak, F. Poletti, Multimode Nonlinear Fibre Optics: Theory and Applications, Recent Progress in Optical Fiber Research (Jan. 2012).
doi:10.5772/27489.
URL https://www.intechopen.com/books/recent-progress-in-optical-fiber-research/multimode-nonlinear-fibre-optics-theory-and-applications

[5] L. G. Wright, D. N. Christodoulides, F. W. Wise, Controllable spatiotemporal nonlinear effects in multimode fibres, Nat. Photonics (2015) 1–5.

[6] L. G. Wright, S. Wabnitz, D. N. Christodoulides, F. W. Wise, Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves, Phys. Rev. Lett. 115 (2015) 223902.

[7] A. Hasegawa, Self-confinement of multimode optical pulse in a glass fiber, Opt. Lett. 5 (1980) 416–417.

[8] S. Longhi, Modulation instability and space-time dynamics in nonlinear parabolic-index optical fibers, Opt. Lett. 28 (2003) 2363–2365.

[9] W. H. Renninger, F. W. Wise, Optical solitons in graded-index multimode fibres, Nat. Commun. 4 (2012) 1719.

[10] K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, S. Wabnitz, Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves, Phys. Rev. Lett. 116 (2016) 183901. doi:10.1103/PhysRevLett.116.183901.
URL https://link.aps.org/doi/10.1103/PhysRevLett.116.183901

[11] P. Aschieri, J. Garnier, C. Michel, V. Doya, A. Picozzi, Condensation and thermalization of classsical optical waves in a waveguide, Physical Review A 83 (3) (2011) 033838.

[12] K. Baudin, A. Fusaro, K. Krupa, J. Garnier, S. Rica, G. Millot, A. Picozzi, Classical rayleigh-jeans condensation of light waves: Observation and thermodynamic characterization, Phys. Rev. Lett. 125 (2020) 244101. doi:10.1103/PhysRevLett.125.244101.
URL https://link.aps.org/doi/10.1103/PhysRevLett.125.244101

[13] F. O. Wu, A. U. Hassan, D. N. Christodoulides, Thermodynamic theory of highly multimoded nonlinear optical systems, Nature Photonics 13 (2019) 776–782. doi:10.1038/s41566-019-0501-8.
URL http://dx.doi.org/10.1038/s41566-019-0501-8

[14] M. Parto, F. O. Wu, P. S. Jung, K. Makris, D. N. Christodoulides, Thermodynamic conditions governing the optical temperature and chemical potential in nonlinear highly multimoded photonic systems, Opt. Lett. 44 (16) (2019) 3936–3939. doi:10.1364/OL.44.003936.
URL https://opg.optica.org/ol/abstract.cfm?URI=ol-44-16-3936
[15] K. G. Makris, F. O. Wu, P. S. Jung, D. N. Christodoulides, Statistical mechanics of weakly nonlinear optical multimode gases, Optics letters 45 (7) (2020) 1651-1654.

[16] L. G. Wright, F. O. Wu, D. N. Christodoulides, F. W. Wise, Physics of highly multimode nonlinear optical systems, Nature Physics 18 (9) (2022) 1018-1030.

[17] N. K. Efremidis, D. N. Christodoulides, Fundamental entropic processes in the theory of optical thermodynamics, Phys. Rev. A 103 (2021) 043517.
doi:10.1103/PhysRevA.103.043517.
URL https://link.aps.org/doi/10.1103/PhysRevA.103.043517

[18] M. A. Selim, G. G. Pyrialakos, F. O. Wu, Z. Musslimani, K. G. Makris, M. Khajavikhan, D. Christodoulides, Thermalization of the ablowitz-ladik lattice in the presence of non-integrable perturbations, Opt. Lett. 48 (8) (2023) 2206-2209.
doi:10.1364/OL.489165.
URL https://opg.optica.org/ol/abstract.cfm?URI=ol-48-8-2206

[19] G. G. Pyrialakos, H. Ren, P. S. Jung, M. Khajavikhan, D. N. Christodoulides, Thermalization dynamics of nonlinear non-hermitian optical lattices, Phys. Rev. Lett. 128 (2022) 213901.
doi:10.1103/PhysRevLett.128.213901.
URL https://link.aps.org/doi/10.1103/PhysRevLett.128.213901

[20] F. O. Wu, P. S. Jung, M. Parto, M. Khajavikhan, D. N. Christodoulides, Thermodynamic optical pressures in tight-binding nonlinear multimode photonic systems, Communications Physics 3 (2020) 216.

[21] N. Efremidis, D. Christodoulides, Thermodynamic optical pressures in tight-binding nonlinear multimode photonic systems, Communications Physics 5 (2022) 286.

[22] H. Ren, G. G. Pyrialakos, F. O. Wu, P. S. Jung, N. K. Efremidis, M. Khajavikhan, D. N. Christodoulides, Nature of optical thermodynamic pressure exerted in highly multimoded nonlinear systems, Phys. Rev. Lett. 131 (2023) 193802.
doi:10.1103/PhysRevLett.131.193802.
URL https://link.aps.org/doi/10.1103/PhysRevLett.131.193802

[23] N. K. Efremidis, D. N. Christodoulides, Statistical mechanics and pressure of composite multimoded weakly nonlinear optical systems, Opt. Lett. 49 (10) (2024) 2777-2780.
doi:10.1364/OL.511787.
URL https://opg.optica.org/ol/abstract.cfm?URI=ol-49-10-2777

[24] H. Ren, G. G. Pyrialakos, F. O. Wu, N. K. Efremidis, M. Khajavikhan, D. N. Christodoulides, Dalton's law of partial optical thermodynamic pressures in highly multimoded nonlinear photonic systems, Opt. Lett. 49 (7) (2024) 1802-1805.
doi:10.1364/OL.517715.
URL https://opg.optica.org/ol/abstract.cfm?URI=ol-49-7-1802
[25] J. Garnier, A. Fusaro, K. Baudin, C. Michel, K. Krupa, G. Millot, A. Picozzi, Wave condensation with weak disorder versus beam self-cleaning in multimode fibers, Phys. Rev. A 100 (2019) 053835. doi: 10.1103/PhysRevA.100.053835.
URL https://link.aps.org/doi/10.1103/PhysRevA.100.053835

[26] E. Podivilov, F. Mangini, O. Sidelnikov, M. Ferraro, M. Gervaziev, D. Kharenko, M. Zitelli, M. Fedoruk, S. Babin, S. Wabnitz, Thermalization of orbital angular momentum beams in multimode optical fibers, Physical Review Letters 128 (24) (2022) 243901.

[27] F. O. Wu, Q. Zhong, H. Ren, P. S. Jung, K. G. Makris, D. N. Christodoulides, Thermalization of light's orbital angular momentum in nonlinear multimode waveguide systems, Physical Review Letters 128 (12) (2022) 123901.

[28] K. Rasmussen, T. Cretegny, P. G. Kevrekidis, N. Gronbech-Jensen, Statistical mechanics of a discrete nonlinear system, Phys. Rev. Lett. 84 (2000) 3740.

[29] R. Jordan, C. Josserand, Self-organization in nonlinear wave turbulence, Phys. Rev. E 61 (2000) 1527.

[30] R. Jordan, B. Turkington, C. L. Zirbel, A mean-field statistical theory for the nonlinear Schrödinger equation, Physica D 137 (2000) 353.

[31] R. Rumpf, A. C. Newell, Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems, Phys. Rev. Lett. 87 (2001) 054102.

[32] R. Rumpf, A. C. Newell, Localization and coherence in nonintegrable systems, Physica D 184 (2003) 162.

[33] B. Rumpf, Simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities, Phys. Rev. E 69 (2004) 016618.

[34] L. Leuzzi, C. Conti, V. Folli, L. Angelani, G. Ruocco, Phase diagram and complexity of mode-locked lasers: From order to disorder, Phys. Rev. Lett. 102 (2009) 083901.

[35] C. Conti, L. Leuzzi, Complexity of waves in nonlinear disordered media, Phys. Rev. B 83 (2011) 134204.

[36] N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella, G. Gigli, L. L., C. Conti, Experimental evidence of replica symmetry breaking in random lasers, Nature Commun. 6 (2015) 6058.

[37] D. Pierangeli, A. Tavani, F. Di Mei, A. Agranat, C. Conti, E. DelRe, Observation of replica symmetry breaking in disordered nonlinear wave propagation, Nature Commun. 8 (2017) 1501.
[38] K. Baudin, J. Garnier, A. Fusaro, N. Berti, C. Michel, K. Krupa, G. Millot, A. Picozzi, Observation of light thermalization to negative-temperature rayleigh-jeans equilibrium states in multimode optical fibers, Phys. Rev. Lett. 130 (2023) 063801. doi:10.1103/PhysRevLett.130.063801.
URL https://link.aps.org/doi/10.1103/PhysRevLett.130.063801

[39] A. L. M. Muniz, F. O. Wu, P. S. Jung, M. Khajavikhan, D. N. Christodoulides, U. Peschel, Observation of photon-photon thermodynamic processes under negative optical temperature conditions, Science 379 (6636) (2023) 1019-1023. arXiv:https://www.science.org/doi/pdf/10.1126/science.ade6523, doi:10.1126/science.ade6523.
URL https://www.science.org/doi/abs/10.1126/science.ade6523

[40] V. E. Zakharov, V. S. L'vov, G. Falkovich, Kolmogorov Spectra of Turbulence I, Springer; Berlin, 1992.

[41] S. Nazarenko, Wave Turbulence, Lecture Notes in Physics 825, Springer, 2011.

[42] A. C. Newell, R. Rumpf, Wave turbulence, Annual Review of Fluid Mechanics 43 (2011) 59-78.

[43] S. Galtier, Physics of Wave Turbulence, Cambridge University Press, 2022.

[44] A. Eloy, O. Boughdad, M. Albert, P.-E. Larré, F. Mortessagne, M. Bellec, C. Michel, Experimental observation of turbulent coherent structures in a superfluid of light, Europhysics Letters 134 (2021) 26001.

[45] M. Baker-Rasooli, W. Liu, T. Aladjidi, A. Bramati, Q. Glorieux, Turbulent dynamics in a two-dimensional paraxial fluid of light, Phys. Rev. A 108 (2023) 063512. doi:10.1103/PhysRevA.108.063512.
URL https://link.aps.org/doi/10.1103/PhysRevA.108.063512

[46] K. Hammani, B. Kibler, C. Finot, A. Picozzi, Emergence of rogue waves from optical turbulence, Phys. Lett. A 374 (2010) 3585.

[47] B. Kibler, K. Hammani, C. Finot, A. Picozzi, Rogue waves, rational solitons and wave turbulence theory, Phys. Lett. A 375 (2011) 3149.

[48] M. Onorato, S. Residori, U. Bortolozzo, A. Montina, F. Arecchi, Rogue waves and their generating mechanisms in different physical contexts, Physics Reports (2013).

[49] J. Dudley, G. Genty, A. Mussot, A. Chabchoub, F. Dias, Rogue waves and analogies in optics and oceanography, Nature Reviews Physics 1 (2019) 675.

[50] B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, A. Picozzi, Experimental signature of optical wave thermalization through super-continuum generation in photonic crystal fiber, Opt. Express 17 (2009) 7392.
[51] B. Barviau, B. Kibler, A. Picozzi, Wave turbulence description of super-continuum generation: influence of self-steepening and higher-order dispersion, Opt. Express 79 (2009) 063840.

[52] B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, A. Picozzi, Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber, Phys. Rev. E 84 (2011) 066605.

[53] B. Barviau, J. Garnier, G. Xu, B. Kibler, G. Millot, A. Picozzi, Truncated thermalization of incoherent optical waves through supercontinuum generation in photonic crystal fibers, Physical Review A 87 (3) (2013) 035803.

[54] F. Meng, C. Lapre, C. Billet, T. Sylvestre, J.-M. Merolla, C. Finot, S. K. Turitsyn, G. Genty, J. Dudley, Intracavity incoherent supercontinuum dynamics and rogue waves in a broadband dissipative soliton laser, Nat Commun 12 (2021) 5567.

[55] T. Congy, G. A. El, G. Roberti, A. Tovbis, S. Randoux, P. Suret, Statistics of extreme events in integrable turbulence, Phys. Rev. Lett. 132 (2024) 207201. doi:10.1103/PhysRevLett.132.207201.
URL https://link.aps.org/doi/10.1103/PhysRevLett.132.207201

[56] P. Suret, S. Randoux, A. Gelash, D. Agafontsev, B. Doyon, G. El, Soliton gas: Theory, numerics, and experiments, Phys. Rev. E 109 (2024) 061001. doi:10.1103/PhysRevE.109.061001.
URL https://link.aps.org/doi/10.1103/PhysRevE.109.061001

[57] C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, S. Rica, Condensation of classical nonlinear waves, Phys. Rev. Lett. 95 (2005) 263901.

[58] C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, J. Fleischer, Observation of the kinetic condensation of classical waves, Nature Physics 8 (2012) 470.

[59] U. Bortolozzo, J. Laurie, S. Nazarenko, S. Residori, Optical wave turbulence and the condensation of light, J. Opt. Soc. Am. B 26 (2009) 2280.

[60] J. Laurie, U. Bortolozzo, S. Nazarenko, S. Residori, One-dimensional optical wave turbulence: Experiment and theory, Phys. Rep. 514 (2012) 121.

[61] A. Chiocchetta, P. Larré, I. Carusotto, Thermalization and bose-einstein condensation of quantum light in bulk nonlinear media, Reviews of Modern Physics 115 (2016) 24002.

[62] S. Pitois, S. Lagrange, H. R. Jauslin, A. Picozzi, Velocity locking of incoherent nonlinear wave packets, Phys. Rev. Lett. 97 (2006) 033902.

[63] P. Suret, S. Randoux, H. Jauslin, A. Picozzi, Anomalous thermalization of nonlinear wave systems, Phys. Rev. Lett. 104 (2010) 054101.
[64] N. Šantić, A. Fusaro, S. Salem, J. Garnier, A. Picozzi, R. Kaiser, Nonequilibrium precondensation of classical waves in two dimensions propagating through atomic vapors, Phys. Rev. Lett. 120 (2018) 055301.
doi:10.1103/PhysRevLett.120.055301.
URL https://link.aps.org/doi/10.1103/PhysRevLett.120.055301

[65] A. Kurnosov, L. J. Fernández-Alcázar, A. Ramos, B. Shapiro, T. Kottos, Optical kinetic theory of nonlinear multimode photonic networks, Phys. Rev. Lett. 132 (2024) 193802.
doi:10.1103/PhysRevLett.132.193802.
URL https://link.aps.org/doi/10.1103/PhysRevLett.132.193802

[66] M. Lian, Y. Geng, Y.-J. Chen, Y. Chen, J.-T. Lü, Coupled thermal and power transport of optical waveguide arrays: Photonic wiedemann-franz law and rectification effect, Phys. Rev. Lett. 133 (2024) 116303.
doi:10.1103/PhysRevLett.133.116303.
URL https://link.aps.org/doi/10.1103/PhysRevLett.133.116303

[67] C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, A. Picozzi, Thermalization and condensation in an incoherently pumped passive optical cavity, Phys. Rev. A 84 (2011) 033848.

[68] E. Turitsyna, S. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. Babin, E. Podivilov, D. Churkin, G. Falkovich, S. Turitsyn, The laminar-turbulent transition in a fibre laser, Nature Photonics 7 (10) (2013) 783–786.

[69] E. Turitsyna, G. Falkovich, V. Mezentsev, S. Turitsyn, Optical turbulence and spectral condensate in long-fiber lasers, Phys. Rev. A 80 (2009) 031804.

[70] E. Turitsyna, G. Falkovich, A. El-Taher, X. Shu, P. Harper, S. Turitsyn, Optical turbulence and spectral condensate in long fibre lasers, Proc. R. Soc. A 468 (2012) 2145.

[71] D. Churkin, I. Kolokolov, E. Podivilov, I. D. Vatnik, M. A. Nikulin, S. S. Vergeles, I. S. Terekhov, V. V. Lebedev, G. Falkovich, S. A. Babin, S. K. Turitsyn, Wave kinetics of random fibre lasers, Nature Communications 6 (2015) 6214.

[72] A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux, G. Millot, D. Christodoulides, Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics, Physics Reports 542 (1) (2014) 1–132, optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics.
doi:https://doi.org/10.1016/j.physrep.2014.03.002.
URL https://www.sciencedirect.com/science/article/pii/S0370157314001203

[73] S. K. Turitsyn, S. A. Babin, E. G. Turitsyna, G. E. Falkovich, E. Podivilov, D. Churkin, Optical wave turbulence, Eds. V. Shira and S. Nazarenko, Wave Turbulence, World Scientific Series on Nonlinear Science Series A: Volume 83, 2013.
[74] V. N. Tsytovich, Nonlinear effects in plasma, Plenum; New-York, 1970.

[75] K. Hasselmann, On the non-linear energy transfer in a gravity-wave spectrum. part 1. general theory, J. Fluid Mech. 12 (1962) 481–500.

[76] K. Hasselmann, On the non-linear energy transfer in a gravity-wave spectrum. part 2. conservation theorems; wave-particle analogy; irreversibility, J. Fluid Mech. 15 (1963) 273–281.

[77] S. Dyachenko, A. C. Newell, A. Pushkarev, V. E. Zakharov, Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation, Physica D 57 (1992) 96.

[78] V. Zakharov, F. Dias, A. Pushkarev, One-dimensional wave turbulence, Phys. Rep. 398 (2004) 1.

[79] D. J. Benney, P. G. Saffman, Nonlinear interactions of random waves in dispersive medium, Proc. R. Soc. London Ser. A 289 (1966) 301.

[80] A. Newell, The closure problem in a system of random gravity waves, Rev. of Geophys. 6 (1968) 1.

[81] D. J. Benney, A. Newell, Random wave closure, Stud. Appl. Math. 48 (1969) 29.

[82] A. C. Newell, S. Nazarenko, L. Biven, Wave turbulence and intermittency, Physica D 152-153 (2001) 520–550.

[83] K. Huang, Statistical Mechanics, Wiley, 1963.

[84] K. Krupa, C. Louot, V. Couderc, M. Fabert, R. Guenard, B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A. Bendahmane, R. Dupiol, G. Millot, S. Wabnitz, Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber, Opt. Lett. 41 (24) (2016) 5785–5788. doi:10.1364/OL.41.005785.

[85] K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, V. Couderc, Spatial beam self-cleaning in multimode fibres, Nat. Photonics 11 (2017) 234–241.

[86] Z. Liu, L. G. Wright, D. N. Christodoulides, F. W. Wise, Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber, Opt. Lett. 41 (16) (2016) 3675–3678. doi:10.1364/OL.41.003675.

[87] L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, F. W. Wise, Self-organized instability in graded-index multimode fibres, Nat. Photonics 10 (2016) 771–776.

[88] M. Ferraro, F. Mangini, M. Zitelli, S. Wabnitz, On spatial beam self-cleaning from the perspective of optical wave thermalization in multimode graded-index fibers, Advances in Physics: X 8 (1) (2023). doi:10.1080/23746149.2023.2228018.
[89] U. Teğin, B. Rahmani, E. Kakkava, D. Psaltis, C. Moser, Single-mode output by controlling the spatiotemporal nonlinearities in mode-locked femtosecond multimode fiber lasers, Advanced Photonics 2 (5) (2020) 056005–056005.

[90] N. O. Moussa, T. Mansuryan, C.-H. Hage, M. Fabert, K. Krupa, A. Tonello, M. Ferraro, L. Leggio, M. Zitelli, F. Mangini, et al., Spatiotemporal beam self-cleaning for high-resolution nonlinear fluorescence imaging with multimode fiber, Scientific Reports 11 (1) (2021) 18240.

[91] S. Wehbi, T. Mansuryan, K. Krupa, M. Fabert, A. Tonello, M. Zitelli, M. Ferraro, F. Mangini, Y. Sun, S. Vergnole, et al., Continuous spatial self-cleaning in multimode fiber for self-referenced multiplex cars imaging, Optics Express 30 (10) (2022) 16104–16114.

[92] T. Hansson, A. Tonello, T. Mansuryan, F. Mangini, M. Zitelli, M. Ferraro, A. Niang, R. Crescenzi, S. Wabnitz, V. Couderc, Nonlinear beam self-imaging and self-focusing dynamics in a GRIN multimode optical fiber: theory and experiments, Optics Express 28 (16) (2020) 24005. arXiv: 2005.07280, doi:10.1364/oe.398531.

[93] Y. Leventoux, G. Granger, K. Krupa, A. Tonello, G. Millot, M. Ferraro, F. Mangini, M. Zitelli, S. Wabnitz, S. Février, V. Couderc, 3d time-domain beam mapping for studying nonlinear dynamics in multimode optical fibers, Optics Letters 46 (1) (2020) 66–69.

[94] O. S. Sidelnikov, E. V. Podivilov, M. P. Fedoruk, S. Wabnitz, Random mode coupling assists kerr beam self-cleaning in a graded-index multimode optical fiber, Optical Fiber Technology 53 (2019) 101994. doi:https://doi.org/10.1016/j.yofte.2019.101994. URL https://www.sciencedirect.com/science/article/pii/S1068520019301580

[95] H. Degueldre, J. Metzger, T. Geisel, F. R., Random focusing of tsunami waves, Nature Physics 259 (2016) 12.

[96] P. Caramazza, O. Moran, R. Murray-Smith, D. Faccio, Transmission of natural scene images through a multimode fibre, Nature Communications 2029 (2019) 10.

[97] N. Cherroret, T. Karpiuk, B. Grémaud, C. Miniatura, Thermalization of matter waves in speckle potentials, Phys. Rev. A 92 (2015) 063614. doi:10.1103/PhysRevA.92.063614. URL https://link.aps.org/doi/10.1103/PhysRevA.92.063614

[98] N. Cherroret, T. Scoquart, D. Delande, Coherent multiple scattering of out-of-equilibrium interacting bose gases, Annals of Physics 435 (2021) 168543, special Issue on Localisation 2020. doi:https://doi.org/10.1016/j.aop.2021.168543.
URL https://www.sciencedirect.com/science/article/pii/S0003491621001494

[99] Z. Wang, W. Fu, Y. Zhang, H. Zhao, Wave-turbulence origin of the instability of anderson localization against many-body interactions, Phys. Rev. Lett. 124 (2020) 186401. doi:10.1103/PhysRevLett.124.186401.
URL https://link.aps.org/doi/10.1103/PhysRevLett.124.186401

[100] S. Nazarenko, A. Soffer, M.-B. Tran, On the wave turbulence theory for the nonlinear schrödinger equation with random potentials, Entropy 21 (9) (2019). doi:10.3390/e21090823.
URL https://www.mdpi.com/1099-4300/21/9/823

[101] A. Ramos, C. Shi, L. Fernández-Alcázar, D. Christodoulides, T. Kottos, Theory of localization-hindered thermalization in nonlinear multimode photonics, Communications Physics 6 (2023) 189.

[102] G. G. Pyrialakos, F. O. Wu, P. S. Jung, H. Ren, K. G. Makris, Z. H. Musslimani, M. Khajavikhan, T. Kottos, D. Christodoulides, Slow-down of thermalization and the emergence of prethermal dynamics in disordered optical lattices, Phys. Rev. Res. 6 (2024) 013072. doi:10.1103/PhysRevResearch.6.013072.
URL https://link.aps.org/doi/10.1103/PhysRevResearch.6.013072

[103] A. Fusaro, J. Garnier, K. Krupa, G. Millot, A. Picozzi, Dramatic acceleration of wave condensation mediated by disorder in multimode fibers, Phys. Rev. Lett. 122 (2019) 123902. doi:10.1103/PhysRevLett.122.123902.
URL https://link.aps.org/doi/10.1103/PhysRevLett.122.123902

[104] N. Berti, K. Baudin, A. Fusaro, G. Millot, A. Picozzi, J. Garnier, Interplay of thermalization and strong disorder: Wave turbulence theory, numerical simulations, and experiments in multimode optical fibers, Phys. Rev. Lett. 129 (2022) 063901. doi:10.1103/PhysRevLett.129.063901.
URL https://link.aps.org/doi/10.1103/PhysRevLett.129.063901

[105] G. Pariente, V. Gallet, A. Borot, O. Gobert, F. Quéré, Space-time characterization of ultra-intense femtosecond laser beams, Nat. Photonics 10 (8) (2016) 547-553. doi:10.1038/nphoton.2016.140.

[106] M. D. Gervaziev, I. Zhdanov, D. S. Kharenko, V. A. Gonta, V. M. Volosi, E. V. Podivilov, S. A. Babin, S. Wabnitz, Mode decomposition of multimode optical fiber beams by phase-only spatial light modulator, Laser Phys. Lett. 18 (1) (2021) 015101. doi:10.1088/1612-202X/abcf27.
URL https://doi.org/10.1088/1612-202X/abcf27https://iopscience.iop.org/article/10.1088/1612-202X/abcf27
[107] H. Pourbeyram, P. Sidorenko, F. O. Wu, N. Bender, L. Wright, D. N. Christodoulides, F. Wise, Direct observations of thermalization to a rayleigh-jeans distribution in multimode optical fibres, Nature Physics 18 (6) (2022) 685–690.

[108] F. Mangini, M. Gervaziev, M. Ferraro, D. S. Kharenko, M. Zitelli, Y. Sun, V. Couderc, E. V. Podivilov, S. A. Babin, S. Wabnitz, Statistical mechanics of beam self-cleaning in multimode optical fibers, Opt. Express 30 (7) (2022) 10850–10865. doi:10.1364/OE.449187. URL https://opg.optica.org/oe/abstract.cfm?URI=oe-30-7-10850

[109] F. Mangini, M. Ferraro, W. A. Gemechu, Y. Sun, M. Gervaziev, D. Kharenko, S. Babin, V. Couderc, S. Wabnitz, On the maximization of entropy in the process of thermalization of highly multimode nonlinear beams, Optics Letters 49 (12) (2024) 3340–3343.

[110] M. Ferraro, F. Mangini, F. Wu, M. Zitelli, D. Christodoulides, S. Wabnitz, Calorimetry of photon gases in nonlinear multimode optical fibers, Physical Review X 14 (2) (2024) 021020.

[111] F. Mangini, M. Ferraro, Y. Sun, M. Gervaziev, P. Parra-Rivas, D. S. Kharenko, V. Couderc, S. Wabnitz, Modal phase-locking in multimode nonlinear optical fibers, Optics Letters 48 (14) (2023) 3677–3680.

[112] E. V. Podivilov, F. Mangini, O. S. Sidelnikov, M. Ferraro, M. Gervaziev, D. S. Kharenko, M. Zitelli, M. P. Fedoruk, S. A. Babin, S. Wabnitz, Thermalization of Orbital Angular Momentum Beams in Multimode Optical Fibers, Phys. Rev. Lett. 128 (24) (2022) 243901. doi:10.1103/PhysRevLett.128.243901. URL https://link.aps.org/doi/10.1103/PhysRevLett.128.243901

[113] J. C. Maxwell, Ii. on the dynamical theory of gases, Proceedings of the Royal Society of London 15 (1867) 167–171. doi:10.1098/rspl.1866.0039.

[114] A. Mecozzi, C. Antonelli, M. Shtaif, Coupled manakov equations in multimode fibers with strongly coupled groups of modes, Opt. Express 20 (21) (2012) 23436–23441. doi:10.1364/OE.20.023436. URL http://www.opticsexpress.org/abstract.cfm?URI=oe-20-21-23436

[115] A. Mecozzi, C. Antonelli, M. Shtaif, Nonlinear propagation in multimode fibers in the strong coupling regime, Opt. Express 20 (11) (2012) 11673–11678. doi:10.1364/OE.20.011673. URL https://opg.optica.org/oe/abstract.cfm?URI=oe-20-11-11673
[116] S. Mumtaz, R.-J. Essiambre, G. P. Agrawal, Nonlinear Propagation in Multimode and Multicore Fibers: Generalization of the Manakov Equations, Journal of Lightwave Technology 31 (3) (2013) 398-406. doi: 10.1109/JLT.2012.2231401.

[117] Y. Xiao, R.-J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, G. P. Agrawal, Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers, Opt. Express 22 (26) (2014) 32039-32059. doi:10.1364/OE.22.032039.
URL http://www.opticsexpress.org/abstract.cfm?URI=oe-22-26-32039

[118] K. Ho, J. Kahn, Linear propagation effects in mode-division multiplexing systems, J. Lightwave Tech. 32 (2014) 4.

[119] I. Kaminow, T. Li, A. Willner, Optical Fiber Telecommunications, Systems and Networks, Sixth Ed., Elsevier, 2013.

[120] G. P. Agrawal, Physics and Engineering of Graded-Index Media, Cambridge University Press, Cambridge, 2023. doi:10.1017/9781009282086.
URL https://www.cambridge.org/core/books/physics-and-engineering-of-gradedindex-media/328D1D61DEB81F78CC7AAD69658A4904

[121] S. A. Babin, A. G. Kuznetsov, O. S. Sidelnikov, A. A. Wolf, I. N. Nemov, S. I. Kablukov, E. V. Podivilov, M. P. Fedoruk, S. Wabnitz, Spatio-spectral beam control in multimode diode-pumped Raman fibre lasers via intracavity filtering and Kerr cleaning, Sci. Rep. 11 (1) (2021) 21994. doi:10.1038/s41598-021-01491-0.
URL https://doi.org/10.1038/s41598-021-01491-0https://www.nature.com/articles/s41598-021-01491-0

[122] D. S. Kharenko, M. D. Gervaziev, A. G. Kuznetsov, E. V. Podivilov, S. Wabnitz, S. A. Babin, Mode-resolved analysis of pump and Stokes beams in LD-pumped GRIN fiber Raman lasers, Opt. Lett. 47 (5) (2022) 1222-1225. doi:10.1364/OL.449119.
URL https://www.osapublishing.org/ol/abstract.cfm?doi=10.1364/OL.449119https://opg.optica.org/abstract.cfm?URI=ol-47-5-1222

[123] A. Biasi, O. Evnin, B. A. Malomed, Fermi-pasta-ulam phenomena and persistent breathers in the harmonic trap, Phys. Rev. E 104 (2021) 034210. doi:10.1103/PhysRevE.104.034210.
URL https://link.aps.org/doi/10.1103/PhysRevE.104.034210

[124] A. Biasi, O. Evnin, B. A. Malomed, Obstruction to ergodicity in nonlinear schrödinger equations with resonant potentials, Phys. Rev. E 108 (2023) 034204. doi:10.1103/PhysRevE.108.034204.
URL https://link.aps.org/doi/10.1103/PhysRevE.108.034204
[125] V. E. Zakharov, A. O. Korotkevich, A. N. Pushkarev, A. I. Dyachenko, Mesoscopic wave turbulence, JETP Letters 82 (8) (2005) 487-491. doi: 10.1134/1.2150867.

[126] E. Kartashova, S. Nazarenko, O. Rudenko, Resonant interactions of non-linear water waves in a finite basin, Phys. Rev. E 78 (2008) 016304. doi:10.1103/PhysRevE.78.016304.
URL https://link.aps.org/doi/10.1103/PhysRevE.78.016304

[127] E. Kartashova, Discrete wave turbulence, Europhysics Letters 87 (4) (2009) 44001. doi:10.1209/0295-5075/87/44001.
URL https://dx.doi.org/10.1209/0295-5075/87/44001

[128] V. S. L'vov, S. Nazarenko, Discrete and mesoscopic regimes of finite-size wave turbulence, Phys. Rev. E 82 (2010) 056322. doi:10.1103/PhysRevE.82.056322.
URL https://link.aps.org/doi/10.1103/PhysRevE.82.056322

[129] R. Hassaini, N. Mordant, Confinement effects on gravity-capillary wave turbulence, Phys. Rev. Fluids 3 (2018) 094805. doi:10.1103/PhysRevFluids.3.094805.
URL https://link.aps.org/doi/10.1103/PhysRevFluids.3.094805

[130] S. Chibbaro, G. Dematteis, C. Josserand, L. Rondoni, Wave-turbulence theory of four-wave nonlinear interactions, Phys. Rev. E 96 (2017) 021101. doi:10.1103/PhysRevE.96.021101.
URL https://link.aps.org/doi/10.1103/PhysRevE.96.021101

[131] S. Chibbaro, G. Dematteis, L. Rondoni, 4-wave dynamics in kinetic wave turbulence, Physica D: Nonlinear Phenomena 362 (2018) 24-59. doi:https://doi.org/10.1016/j.physd.2017.09.001.
URL https://www.sciencedirect.com/science/article/pii/S0167278917301847

[132] S. Kuksin, A. Maiocchi, The effective equation method, Eds. E. Tobisch, New Approaches to Nonlinear Waves, Springer, New York, 2015.

[133] G. During, A. Picozzi, S. Rica, Breakdown of weak-turbulence and non-linear wave condensation, Physica D 238 (2009) 1524.

[134] Y. Lvov, S. Nazarenko, R. West, Wave turbulence in bose-einstein condensates, Physica D: Nonlinear Phenomena 184 (1) (2003) 333-351, complexity and Nonlinearity in Physical Systems - A Special Issue to Honor Alan Newell. doi:https://doi.org/10.1016/S0167-2789(03)00239-2.
URL https://www.sciencedirect.com/science/article/pii/S0167278903002392

[135] K. Baudin, A. Fusaro, J. Garnier, N. Berti, K. Krupa, I. Carusotto, S. Rica, G. Millot, A. Picozzi, Energy and wave-action flows underlying rayleigh-jeans thermalization of optical waves propagating in a multimode fiber(a),
Europhysics Letters 134 (1) (2021) 14001. doi:10.1209/0295-5075/134/14001.
URL https://dx.doi.org/10.1209/0295-5075/134/14001

[136] E. V. Podivilov, D. S. Kharenko, V. A. Gonta, K. Krupa, O. S. Sidelnikov, S. Turitsyn, M. P. Fedoruk, S. A. Babin, S. Wabnitz, Hydrodynamic 2D Turbulence and Spatial Beam Condensation in Multimode Optical Fibers, Phys. Rev. Lett. 122 (10) (2019) 103902. arXiv:1811.10239, doi:10.1103/PhysRevLett.122.103902.
URL http://arxiv.org/abs/1811.10239https://link.aps.org/doi/10.1103/PhysRevLett.122.103902

[137] V. Zakharov, Weak turbulence in media with a decay spectrum, Journal of Applied Mechanics and Technical Physics 6 (1965) 22.

[138] V. Zakharov, On the weak turbulence spectrum in plasma without magnetic field, Sov. Phys. JETP 24 (1967) 455.

[139] A. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers, Doklady Akademiia Nauk SSSR 30 (1941) 301.

[140] Y. Zhu, B. Semisalov, G. Krstulovic, S. Nazarenko, Self-similar evolution of wave turbulence in gross-pitaevskii system, Phys. Rev. E 108 (2023) 064207. doi:10.1103/PhysRevE.108.064207.
URL https://link.aps.org/doi/10.1103/PhysRevE.108.064207

[141] K. Baudin, J. Garnier, A. Fusaro, C. Michel, K. Krupa, G. Millot, A. Picozzi, Rayleigh-jeans thermalization vs beam cleaning in multimode optical fibers, Optics Communications 545 (2023) 129716.

[142] W. Xiong, C. Hsu, Y. e. a. Bromberg, Complete polarization control in multimode fibers with polarization and mode coupling, Light Sci Appl 7 (2018) 54.

[143] M. B. Shemirani, W. Mao, R. A. Panicker, J. M. Kahn, Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling, Journal of Lightwave Technology 27 (10) (2009) 1248–1261. doi:10.1109/JLT.2008.2005066.

[144] X. Shen, J. M. Kahn, M. A. Horowitz, Compensation for multimode fiber dispersion by adaptive optics, Opt. Lett. 30 (22) (2005) 2985–2987. doi: 10.1364/OL.30.002985.
URL https://opg.optica.org/ol/abstract.cfm?URI=ol-30-22-2985

[145] J.-P. Fouque, J. Garnier, G. Papanicolaou, K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, 2007.
[146] P. Haldar, S. Mu, B. Georgeot, J. Gong, C. Miniatura, G. Lemarié, Rayleigh-jeans prethermalization and wave condensation in a nonlinear disordered floquet system, EPL 144 (2024) 63001.

[147] K. Krupa, G. G. Castañeda, A. Tonello, A. Niang, D. S. Kharenko, M. Fabert, V. Couderc, G. Millot, U. Minoni, D. Modotto, S. Wabnitz, Nonlinear polarization dynamics of Kerr beam self-cleaning in a graded-index multimode optical fiber, Opt. Lett. 44 (1) (2019) 171–174.
doi:10.1364/OL.44.000171.

[148] W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, H. Cao, Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling, Phys. Rev. Lett. 117 (2016) 053901. doi: 10.1103/PhysRevLett.117.053901.
URL https://link.aps.org/doi/10.1103/PhysRevLett.117.053901

[149] M. Zitelli, V. Couderc, M. Ferraro, F. Mangini, P. Parra-Rivas, Y. Sun, S. Wabnitz, Spatiotemporal mode decomposition of ultrashort pulses in linear and nonlinear graded-index multimode fibers, Photon. Res. 11 (5) (2023) 750–756. doi:10.1364/PRJ.484271.
URL https://opg.optica.org/prj/abstract.cfm?URI=prj-11-5-750

[150] R. Olshansky, Mode coupling effects in graded-index optical fibers, Applied Optics 14 (1975) 935.

[151] D. Marcuse, Theory of dielectric optical waveguides, Academic Press, New York, 1974.

[152] J. Garnier, Intensity fluctuations in random waveguides, Commun. Math. Sci. 18 (2020) 947.

[153] G. P. Agrawal, Nonlinear Fiber Optics, sixth Edition, Elsevier, 2019.
doi:10.1016/B978-0-12-397023-7.00001-2.
URL https://linkinghub.elsevier.com/retrieve/pii/B9780123970237000012

[154] M. Ferraro, F. Mangini, M. Zitelli, A. Tonello, A. De Luca, V. Couderc, S. Wabnitz, Femtosecond nonlinear losses in multimode optical fibers, Photonics Research 9 (12) (2021) 2443–2453.

[155] M. Gervaziev, I. Zhdanov, D. Kharenko, V. Gonta, V. Volosi, E. Podivilov, S. Babin, S. Wabnitz, Mode decomposition of multimode optical fiber beams by phase-only spatial light modulator, Laser Physics Letters 18 (1) (2020) 015101.

[156] F. Mangini, M. Ferraro, A. Tonello, V. Couderc, S. Wabnitz, High-temperature wave thermalization spoils beam self-cleaning in nonlinear multimode grating fibers, Optics Letters 48 (18) (2023) 4741–4744.
[157] M. Ferraro, F. Mangini, M. Zitelli, R. Jauberteau, Y. Sun, P. Parra-Rivas, K. Krupa, A. Tonello, V. Couderc, S. Wabnitz, Spatial beam cleaning in multimode grating fibers: Polarization effects, IEEE Photonics Journal (2023).

[158] M. Ferraro, F. Mangini, Y. Leventoux, A. Tonello, M. Zitelli, T. Mansuryan, Y. Sun, S. Fevrier, K. Krupa, D. Kharenko, et al., Multimode optical fiber beam-by-beam cleanup, Journal of Lightwave Technology 41 (10) (2023) 3164–3174.

[159] G. Steinmeyer, Generalized thermodynamics of optical multimode systems and the breach of the second law of thermodynamics, in: 2023 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC), IEEE, 2023, pp. 1–1.

[160] L. Zanaglia, J. Garnier, S. Rica, R. Kaiser, S. Wabnitz, C. Michel, V. Doya, A. Picozzi, Bridging rayleigh-jeans and bose-einstein condensation of a guided fluid of light with positive and negative temperatures, Phys. Rev. A 110 (2024) 063530. doi:10.1103/PhysRevA.110.063530.
URL https://link.aps.org/doi/10.1103/PhysRevA.110.063530

[161] M. J. Davis, S. A. Morgan, K. Burnett, Simulations of bose fields at finite temperature, Phys. Rev. Lett. 87 (2001) 160402. doi:10.1103/PhysRevLett.87.160402.
URL https://link.aps.org/doi/10.1103/PhysRevLett.87.160402

[162] P. B. Blakie, M. J. Davis, Projected gross-pitaevskii equation for harmonically confined bose gases at finite temperature, Phys. Rev. A 72 (2005) 063608. doi:10.1103/PhysRevA.72.063608.
URL https://link.aps.org/doi/10.1103/PhysRevA.72.063608

[163] L. Onsager, Statistical hydrodynamics, Il Nuovo Cimento 6 (1949) 279.

[164] N. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperatures, Physical Review 103 (1956) 20.

[165] M. Baldovin, S. Iubini, R. Livi, A. Vulpiani, Statistical mechanics of systems with negative temperature, Physics Reports 923 (2021) 1–50, statistical mechanics of systems with negative temperature. doi:https://doi.org/10.1016/j.physrep.2021.03.007.
URL https://www.sciencedirect.com/science/article/pii/S0370157321001204

[166] M. Onorato, Y. Lvov, G. Dematteis, S. Chibbaro, Wave turbulence and thermalization in one-dimensional chains, Physics Reports 1040 (2023) 1–36, wave Turbulence and thermalization in one-dimensional chains. doi:https://doi.org/10.1016/j.physrep.2023.09.006.
URL https://www.sciencedirect.com/science/article/pii/S0370157323003046
[167] S. Braun, J. P. Ronzheimer, M. Schreiber, S. S. Hodgman, T. Rom, I. Bloch, U. Schneider, Negative absolute temperature for motional degrees of freedom, Science 339 (6115) (2013) 52–55. arXiv:https://www.science.org/doi/pdf/10.1126/science.1227831, doi:10.1126/science.1227831. URL https://www.science.org/doi/abs/10.1126/science.1227831

[168] G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. A. Baker, T. A. Bell, H. Rubinsztein-Dunlop, M. J. Davis, T. W. Neely, Giant vortex clusters in a two-dimensional quantum fluid, Science 364 (6447) (2019) 1264–1267. arXiv:https://www.science.org/doi/pdf/10.1126/science.aat5718, doi:10.1126/science.aat5718. URL https://www.science.org/doi/abs/10.1126/science.aat5718

[169] S. P. Johnstone, A. J. Groszek, P. T. Starkey, C. J. Billington, T. P. Simula, K. Helmerson, Evolution of large-scale flow from turbulence in a two-dimensional superfluid, Science 364 (6447) (2019) 1267–1271. arXiv:https://www.science.org/doi/pdf/10.1126/science.aat5793, doi:10.1126/science.aat5793. URL https://www.science.org/doi/abs/10.1126/science.aat5793

[170] J. Fienup, Phase retrieval algorithms: a comparison, Applied Optics 21 (1982) 2758.

[171] Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, M. Segev, Phase retrieval with application to optical imaging: A contemporary overview, IEEE Signal Processing Magazine 32 (3) (2015) 87–109. doi: 10.1109/MSP.2014.2352673.

[172] M. Fabert, M. Săpântan, K. Krupa, A. Tonello, Y. Leventoux, S. Février, T. Mansuryan, A. Niang, B. Wetzel, G. Millot, et al., Coherent combining of self-cleaned multimode beams, Scientific reports 10 (1) (2020) 20481.

[173] M. Guasoni, J. Garnier, B. Rumpf, D. Sugny, J. Fatome, F. Amrani, G. Millot, A. Picozzi, Incoherent fermi-pasta-ulam recurrences and unconstrained thermalization mediated by strong phase correlations, Phys. Rev. X 7 (2017) 011025. doi:10.1103/PhysRevX.7.011025. URL https://link.aps.org/doi/10.1103/PhysRevX.7.011025

[174] G. G. Pyrialakos, M. S. Kirsch, R. M. Altenkirch, J. Beck, H. Ren, M. A. Selim, P. S. Jung, M. Khajavikhan, A. Szameit, M. Heinrich, et al., Optical condensation of light via joule-thomson expansion, in: CLEO: Fundamental Science, Optica Publishing Group, 2024, pp. FTu3R-3.

[175] N. Bogoliubov, On the theory of superfluidity, J. Phys. (USSR) 11 (1947) 23.

[176] Q. Fontaine, T. Bienaimé, S. Pigeon, E. Giacobino, A. Bramati, Q. Glorieux, Observation of the bogoliubov dispersion in a fluid of light, Phys. Rev. Lett. 121 (2018) 183604. doi:10.1103/PhysRevLett.121.183604. URL https://link.aps.org/doi/10.1103/PhysRevLett.121.183604
[177] T. Frisch, Y. Pomeau, S. Rica, Transition to dissipation in a model of super-flow, Phys. Rev. Lett. 69 (1992) 1644–1647. doi:10.1103/PhysRevLett.69.1644
URL https://link.aps.org/doi/10.1103/PhysRevLett.69.1644

[178] I. Carusotto, Superfluid light in bulk nonlinear media, Proceedings of the Royal Society A 470 (2014) 2169.

[179] C. Michel, O. Boughdad, M. Albert, P.-E. Larré, M. Bellec, Superfluid motion and drag-force cancellation in a fluid of light, Nature Communications 9 (2018) 2108.

[180] M. C. Braidotti, R. Prizia, C. Maitland, F. Marino, A. Prain, I. Starshynov, N. Westerberg, E. M. Wright, D. Faccio, Measurement of penrose superradiance in a photon superfluid, Phys. Rev. Lett. 128 (2022) 013901. doi:10.1103/PhysRevLett.128.013901
URL https://link.aps.org/doi/10.1103/PhysRevLett.128.013901

[181] G. Situ, J. Fleischer, Dynamics of the berezinskii-kosterlitz-thouless transition in a photon fluid, Nature Photonics 14 (2020) 517.

[182] B. Rumpf, A. C. Newell, V. E. Zakharov, Turbulent transfer of energy by radiating pulses, Phys. Rev. Lett. 103 (2009) 074502.

[183] K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, S. Wabnitz, Spatiotemporal light-beam compression from nonlinear mode coupling, Phys. Rev. A 97 (2018) 043836. doi:10.1103/PhysRevA.97.043836.

[184] M. Labaz, P. Sidorenko, Spatial-spectral complexity in kerr beam self-cleaning, Optics Letters 49 (11) (2024) 2902–2905.

[185] G. Genty, J. Li, J. Ruelle, J. M. Dudley, Real-time full-field spatial measurement of multimode fibers in the nonlinear regime, in: Real-time Measurements, Rogue Phenomena, and Single-Shot Applications X, SPIE, 2025, p. PC1334803.

[186] L. G. Wright, S. Wabnitz, D. N. Christodoulides, F. W. Wise, Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves, Phys. Rev. Lett. 115 (2015) 223902. doi:10.1103/PhysRevLett.115.223902.

[187] O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, S. K. Turitsyn, Coexistence of collapse and stable spatiotemporal solitons in multimode fibers, Physical Review A 97 (1 2018). doi:10.1103/PhysRevA.97.013841.

[188] C. Mas Arabí, A. Kudlinski, A. Mussot, M. Conforti, Geometric parametric instability in periodically modulated graded-index multimode fibers, Phys. Rev. A 97 (2018) 023803. doi:10.1103/PhysRevA.97.023803.
URL https://link.aps.org/doi/10.1103/PhysRevA.97.023803
[189] M. A. Eftekhar, L. G. Wright, M. S. Mills, M. Kolesik, R. A. Correa, F. W. Wise, D. N. Christodoulides, Versatile supercontinuum generation in parabolic multimode optical fibers, Opt. Express 25 (8) (2017) 9078–9087.
doi:10.1364/OE.25.009078.

[190] K. Krupa, C. Louot, V. Couderc, M. Fabert, R. Guenard, B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A. Bendahmane, R. Dupiol, G. Millot, S. Wabnitz, Spatiotemporal characterization of supercontinuum extend-ing from the visible to the mid-infrared in a multimode graded-index opti-cal fiber, Opt. Lett. 41 (24) (2016) 5785–5788. doi:10.1364/OL.41.005785. URL http://ol.osa.org/abstract.cfm?URI=ol-41-24-5785

[191] G. L. Galmiche, Z. S. Eznaveh, M. A. Eftekhar, J. A. Lopez, L. G. Wright, F. Wise, D. Christodoulides, R. A. Correa, Visible supercontinuum gener-ation in a graded index multimode fiber pumped at 1064 nm, Opt. Lett. 41 (11) (2016) 2553–2556. doi:10.1364/OL.41.002553.

[192] G. Xu, J. Garnier, B. Rumpf, A. Fusaro, P. Suret, S. Randoux, A. Kudlinski, G. Millot, A. Picozzi, Origins of spectral broadening of incoherent waves: Catastrophic process of coherence degradation, Phys. Rev. A 96 (2017) 023817. doi:10.1103/PhysRevA.96.023817.
URL https://link.aps.org/doi/10.1103/PhysRevA.96.023817

[193] K. V.V., L. Vazquez, Nonlinear Random Waves, Singapore, 1994.

[194] I. Gradstein, I. Ryzhik, Tables of Integrals, Sums, Series, and Products, Academic Press, New York, 1980.