Integrable turbulence generated from modulational instability of cnoidal waves

TL;DR

This paper numerically investigates the nonlinear development of modulational instability in cnoidal waves within the focusing 1D NLS equation, revealing how integrable turbulence evolves and depends on soliton overlap.

Contribution

It provides new insights into the characteristics and asymptotic behavior of integrable turbulence generated from cnoidal wave MI, including energy oscillations and amplitude distributions.

Findings

Turbulence approaches a stationary state with oscillatory energy behavior.

Amplitude distribution varies from Rayleigh to non-Rayleigh depending on soliton overlap.

Two-soliton collisions can cause amplitude increases up to twice the original value.

Abstract

We study numerically the nonlinear stage of modulational instability (MI) of cnoidal waves, in the framework of the focusing one-dimensional Nonlinear Schrodinger (NLS) equation. Cnoidal waves are the exact periodic solutions of the NLS equation and can be represented as a lattice of overlapping solitons. MI of these lattices lead to development of "integrable turbulence" [Zakharov V.E., Stud. Appl. Math. 122, 219-234 (2009)]. We study the major characteristics of the turbulence for dn-branch of cnoidal waves and demonstrate how these characteristics depend on the degree of "overlapping" between the solitons within the cnoidal wave. Integrable turbulence, that develops from MI of dn-branch of cnoidal waves, asymptotically approaches to it's stationary state in oscillatory way. During this process kinetic and potential energies oscillate around their asymptotic values. The amplitudes of these oscillations decay with time as t^{-a}, 1<a<1.5, the phases contain nonlinear phase shift decaying as t^{-1/2}, and the frequency of the oscillations is equal to the double maximal growth rate of the MI, s=2g_{max}. In the asymptotic stationary state the ratio of potential to kinetic energy is equal to -2. The asymptotic PDF of wave amplitudes is close to Rayleigh distribution for cnoidal waves with strong overlapping, and is significantly non-Rayleigh one for cnoidal waves with weak overlapping of solitons. In the latter case the dynamics of the system reduces to two-soliton collisions, which occur with exponentially small rate and provide up to two-fold increase in amplitude compared with the original cnoidal wave.