h \nu_0$, where $K_T$ is the temperature-dependent phonon occupancy factor, and $h\nu_0$ is the average photon energy [@shevchenko2018tcom]. A general closed-form solution of the stochastic does not exist. In some special cases, e.g., for noisefree and lossless fibers, special solutions like, e.g., solitons, exist. Furthermore, we consider the in normalized form in the focusing regime, i.e., $\beta_2 < 0$, under the assumption of ideal distributed Raman amplification, i.e., $\alpha = 0$, $$\begin{aligned}
    \jmath\frac{\partial q(t,z)}{\partial z}  + \frac{\partial^2 q(t,z)}{\partial t^2} + 2 q(t,z) |q(t,z)|^2  =  0,
    \label{eqn:nlse_losless_noisefree}\end{aligned}$$ where $t=\tau / \sqrt{|\beta_2| L/2}$, $z=\ell / L$, $q = u\sqrt{\gamma L}/\sqrt{2}$, and $L$ is the length of the fiber. In this case, the is an integrable partial differential equation for which a pair of operators, called Lax pair, can be found. The eigenvalues of such an operator remain invariant during noiseless propagation and the Lax pair can be used to solve the partial differential equation. Solutions of [()]{} can be uniquely represented in terms of its eigenvalues via the so-called . For a given position $z$, the of a signal $q(t)$ (we drop the position $z$ for simplicity of presentation) with support on the time interval $t\in [t_1,t_2]$, is calculated by solving the 