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We study numerically the integrable turbulence in the framework of the focusing one-dimensional nonlinear Schrodinger equation using a new method -- the growing of turbulence. We add to the equation a weak controlled pumping term and start adiabatic evolution of turbulence from statistically homogeneous Gaussian noise. After reaching a certain level of average intensity, we switch off the pumping and realize that the grown up turbulence is statistically stationary. We measure its Fourier spectrum, the probability density function (PDF) of intensity and the autocorrelation of intensity. Additionally, we show that, being adiabatic, our method produces stationary states of the integrable turbulence for the intermediate moments of pumping as well. Presently, we consider only the turbulence of relatively small level of nonlinearity; however, even this moderate turbulence is characterized by enhanced generation of rogue waves.
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We examine integrable turbulence (IT) in the framework of the defocusing cubic one-dimensional nonlinear Schr{o}dinger equation. This is done theoretically and experimentally, by realizing an optical fiber experiment in which the defocusing Kerr nonlinearity strongly dominates linear dispersive effects. Using a dispersive-hydrodynamic approach, we show that the development of IT can be divided into two distinct stages, the initial, pre-breaking stage being described by a system of interacting random Riemann waves. We explain the low-tailed statistics of the wave intensity in IT and show that the Riemann invariants of the asymptotic nonlinear geometric optics system represent the observable quantities that provide new insight into statistical features of the initial stage of the IT development by exhibiting stationary probability density functions.
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 its 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.
We study numerically the integrable turbulence developing from strongly nonlinear partially coherent waves, in the framework of the focusing one-dimensional nonlinear Schrodinger equation. We find that shortly after the beginning of motion the turbulence enters a state characterized by a very slow evolution of statistics (the quasi-stationary state - QSS), and we concentrate on the detailed examination of the basic statistical functions in this state depending on the shape and the width of the initial spectrum. In particular, we show that the probability density function (PDF) of wavefield intensity is nearly independent of the initial spectrum and is very well approximated by a certain Bessel function representing an integral of the product of two exponential distributions. The PDF corresponds to the value of the second-order moment of intensity equal to 4, indicating enhanced generation of rogue waves. All waves of large amplitude that we have studied are very well approximated - both in space and in time - by the rational breather solutions of either the first (the Peregrine breather), or the second orders.
The quantum integrability of a class of massive perturbations of the parafermionic conformal field theories associated to compact Lie groups is established by showing that they have quantum conserved densities of scale dimension 2 and 3. These theories are integrable for any value of a continuous vector coupling constant, and they generalize the perturbation of the minimal parafermionic models by their first thermal operator. The classical equations-of-motion of these perturbed theories are the non-abelian affine Toda equations which admit (charged) soliton solutions whose semi-classical quantization is expected to permit the identification of the exact S-matrix of the theory.
Travelling waves arise in several areas of science, hence modification of travelling wave properties is of great interest. While many studies have demonstrated how to control the form or shape of a solitary travelling wave by employing soliton or dispersion management, far less is known about controlling the motion of a travelling wave while keeping its form unchanged. We present a technique for control of travelling wave motion using time-varying coefficients, which we refer to as wave management. The technique allows one to alter the trajectory of a travelling wave, slowing, stopping, or reversing the direction of the wave, all while ensuring that the wave form is unchanged, and we illustrate this through multiple examples. Our results suggest that wave management is a promising tool for applications where one needs to modify the motion of a wave while preserving its form, and we highlight several potential applications.
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