Bound-state soliton gas as a limit of adiabatically growing integrable turbulence

Numerical direct scattering transform for breathers

“…An accurate treatment of the data obtained in experiments or numerical simulations requires further development of periodic and quasi-periodic DST procedures, particularly for the norming constants of the A breathers. In addition, the existence of the variety of data analysis approaches, such as windowing or periodization methods and finite gap theory or theta-functions framework, raises the question of finding direct links between different wave field scattering data interpretations [21,55,[63][64][65].…”

“…Another important direction represents the rapidly growing field of integrable turbulence, where the DST algorithms can be used to study phase correlations of individual breathers as well as phase correlations in gases of breathers [55,[69][70][71][72][73][74][75]. Note that both IST and DST algorithms face rapid growth of numerical errors when increasing the number of discrete spectrum components, which can be coped with using high-precision arithmetic as recently suggested in [55,57,76]. We highlight that the complete characterization of the breather's parameters benefits the IST analysis of experimental data on coherent structures in optics, hydrodynamics and other nonlinear waveguides described by nearly integrable models [21,32,34,77,78].…”

“…Note that the parameters x0 and θ are equal to the observed in the physical space position and phase of a breather only for the one-breather solution. In the presence of other breathers or dispersive waves, the observed position and phase of a breather may differ considerably from the IST parameters x0 and θ; see the discussion of a similar situation in the case of interacting solitons in [55].…”

“…Namely, to mitigate the round-off errors of the transfer matrix computation, which come from operations with exponentially small and large numbers, we used the high-precision for the matrix multiplications in equation (A 8); see appendix A(b). In addition, to cope with the so-called anomalous errors of the DST procedure we computed the discrete eigenvalues with the 100-digits accuracy as well using the high-precision Newton method to compute zeros of the coefficient a tr ; see our recent works elaborating on these issues [55,57]. Although in our case, the computation time of a single DST run supplemented by the 100-digits precision constitutes a few seconds, the DST procedure can become highly resource-consuming when dealing with large wavefields containing hundreds of coherent structures [55].…”

“…In addition, to cope with the so-called anomalous errors of the DST procedure we computed the discrete eigenvalues with the 100-digits accuracy as well using the high-precision Newton method to compute zeros of the coefficient a tr ; see our recent works elaborating on these issues [55,57]. Although in our case, the computation time of a single DST run supplemented by the 100-digits precision constitutes a few seconds, the DST procedure can become highly resource-consuming when dealing with large wavefields containing hundreds of coherent structures [55]. Note that we chose the precision by making a few tests of the DST output data and taking 100 digits as a highup boundary estimate so that a more optimal choice of precision can significantly speed up the computations.…”