Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Abstract

We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive non-linear Schrödinger equation with a Gaussian random pulse as initial value function. Using an extension of the Thouless formula to non-Hermitian random operators, we calculate the corresponding average density of states. We also calculate the distribution of a set of scattering data of the Zakharov-Shabat operator that determine the asymptotics of the eigenfunctions. We analyze two cases, one with circularly symmetric complex Gaussian pulses and the other with real Gaussian pulses. We discuss the implications in the context of information transmission through non-linear optical fibers.

Appearance of bound states in random potentials with applications to soliton theory

“…[15] was that no solitons can possibly be created in the case of white Gaussian potential; a result that seems to be at odds with the finite DOS obtained in Ref. [16]. Therefore we shall later discuss the origin of such a discrepancy.…”

“…The nonexistence of BSs seems to be at odds with the results of Ref. [16] where nonvanishing DOS was obtained for a Stratonovich WGN in the limit L → ∞. Therefore we shall now spend some time and elucidate the nature of such a discrepancy.…”

“…[15]) and the nonvanishing DOS presented in Ref. [16]. The nonexistence of BSs obtained in the present paper holds, of course, only under the idealized conditions of Stratonovich white noise [i.e., the limit of infinitely narrow, symmetrically correlated potential Q(x)].…”

“…[15] while in Ref. [16] the density of such states was calculated in the self-averaging limit L → ∞. One of the results of Ref.…”

“…The average trace of the Green function was calculated in Ref. [16] via the Lyapunov exponent of system (14), but this is not strictly necessary. One can average the Green function directly and arrive at the same result (in the limit L → ∞)…”