Asymptotic soliton-like and asymptotic peakon-like solutions of the modified Camassa-Holm equation with variable coefficients and singular perturbation

The paper deals with the construction of the asymptotic soliton-like and the asymptotic peakon-like solutions to the modified Camassa-Holm equation with variable coefficicents and a singular perturbation. This equation is a generalization of the well known modified Camassa-Holm equation (\ref{CHE_cons_mod}) which is integrable system and in addition to the soliton solutions the equation has the peakon solutions. The novelty of the ideas of this paper lies in the development of a technique for constructing asymptotic peakon-like solutions. In the paper a general scheme of finding asymptotic approximation of any order is presented and accuracy of the asymptotic approximation is found. The obtained results are illustrated by examples both the soliton-like and the peakon-like solutions. For the examples the equations for the phase function as well as the main and the first terms of the soliton-like and peakon-like solutions are found. Moreover, for different values of a small parameter the graphs that demonstrate kind of the solutions are presented. The considered examples demonstrate that for an adequate description of the wave process it is enough obtain the main and the first terms of correspond asymptotic solutions. The results also confirm that the proposed technique can be used for constructing asymptotic wave-like solutions of other equations.Comment: 33 pages and 12 figure

Analysis and control of rogue waves in fibre lasers and in hydrodynamics: integrable turbulence framework

Understanding mechanisms underlying the formation of extreme events is the problem of primary importance in various domains of study including hydrodynamics, optics, forecasting natural disasters etc. In these domains, extreme events are known as RogueWaves (RWs). RWs are localised coherent structures of unusually large amplitude spontaneously emerging in nonlinear random wave fields, and as such, can have damaging effect on the environment (e.g. offshore engineering structures) or on the medium they propagate through (e.g. optical fibres). Within this PhD project several problems related to the emergence, control and manipulation of RWs in fibre optics and in hydrodynamics have been investigated. The particular emphasis is on the study of RWs emerging in the propagation of the so-called partially coherent waves described by the focusing nonlinear Schr¨odinger equation (fNLSE), the universal model for the propagation of modulationally unstable quasi-monochromatic wavepackets in a broad range of physical media. fNLSE belongs to the class of the completely integrable equations possessing deep mathematical structure and amenable to analytical methods such as Inverse Scattering Transform and Finite-Gap Integration. We use recent mathematical discoveries related to the semi classical, or small-dispersion, limit of fNLSE to investigate analytically, numerically and experimentally the formation of RWs within the framework of integrable turbulence—the emerging theory of random waves in integrable systems. The study of the RW formation in this project has also prompted the investigation of a closely related problem concerned with dynamics of soliton and breather gases as special types of integrable turbulence. The project’s findings fall in five categories: (i) the analytical description of the emergence of the so-called “heavy tails” in the probability distribution for the field intensity at the early stage of the development of integrable turbulence; (ii) the development and experimental realisation in a water tank of nonlinear spectral engineering, the IST-based method of control and manipulation of RWs; (iii) the development of the spectral theory of bidirectional soliton gases; (iv) numerical synthesis of breather gases and the verification of the recently developed spectral kinetic theory for such gases; (v) the investigation of the RWformation in the compression of broad optical pulses in the highly nonlinear propagation regimes, when the higher order effects such as self steepening, third order dispersion and Raman scattering need to be taken into account