Solitons in a box-shaped wavefield with noise: perturbation theory and statistics

The propagation of nonlinear waves is well-described by a number of integrable models leading to the concept of the scattering data also known as the nonlinear Fourier spectrum. Here we investigate the fundamental problem of the nonlinear wavefield scattering data corrections in response to a perturbation of initial condition using inverse scattering transform theory. We present a complete theoretical linear perturbation framework to evaluate first-order corrections of the full set of the scattering data within the integrable one-dimensional focusing nonlinear Schrodinger (NLSE) equation, see our recent preprint [1]. The general scattering data portrait reveals nonlinear coherent structures - solitons - playing the key role in the wavefield evolution. Applying the developed theory to a classic box-shaped wavefield we solve the derived equations analytically for a single Fourier mode acting as a perturbation to the initial condition, thus, leading to the sensitivity closed-form expressions for basic soliton characteristics, i.e. the amplitude, velocity, phase and its position. With the appropriate statistical averaging we model the soliton noise-induced effects resulting in compact relations for standard deviations of soliton parameters. Relying on a concept of a virtual soliton eigenvalue we derive the probability of a soliton emergence or the opposite due to noise and illustrate these theoretical predictions with direct numerical simulations of the NLSE evolution. Note that the evolution of the box field within the NLSE model represents a classical so-called dam-break problem. A wide box-shaped field is unstable to long wave perturbations constituting the phenomena of modulation instability. In conclussion we discuss possible applications of the developed theory to these fundamental problems of physics of nonlinear waves.

The work was supported by Russian Science Foundation grant No. 20-71-00022.

[1] R. Mullyadzhanov and A. Gelash. Solitons in a box-shaped wavefield with noise: perturbation theory and statistics. arXiv preprint arXiv:2008.08874, 2020.