Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg–de Vries equation

Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically

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

Development of level set methods for computing the semiclassical limit of Schrödinger equations with potentials

In this thesis, several level set methods are developed and analyzed for computing multi-valued solutions to the semiclassical limits of Schroedinger equations. Both formulation and numerical results are obtained for level set method. Superposition is also proved via let set method setting. Meanwhile, multi-valued solutions of the Euler-Poisson equations are also analyzed and computed using level set formulation via field space. Multi-scale computation and homogenization are studied for a class of Schroedinger equations. A Bloch band based level set method is developed with a series of numerical examples

High-frequency wave-propagation: error analysis for analytical and numerical approximations

In this thesis we investigate a specific type of semilinear hyperbolic systems with highly oscillatory initial data. This type of systems is numerically very challenging to treat since the solutions are highly oscillatory in space and time. The goal is to derive suitable analytical and numerical approximations. Based on the classical slowly varying envelope approximation (SVEA), an improved error estimate is proven for this analytical approximation. The envelope equation avoids oscillations in space, making this approximation attractive for numerical computations. Furthermore, more accurate analytical approximations are obtained by extending the ansatz of the SVEA. In addition to the analytical study of the SVEA two numerical time integrators are constructed and analyzed without any step-size restrictions. Numerical examples are provided to illustrate the theoretical results. Finally, a complementary approach is presented which address both problems, the oscillations in space and time, simultaneously

Light Beams in Liquid Crystals

This reprint collects recent articles published on "Light Beams in Liquid Crystals", both research and review contributions, with specific emphasis on liquid crystals in the nematic mesophase. The editors, Prof. Gaetano Assanto (NooEL, University of Rome "Roma Tre") and Prof. Noel F. Smyth (School of Mathematics, University of Edinburgh), are among the most active experts worldwide in nonlinear optics of nematic liquid crystals, particularly reorientational optical solitons ("nematicons") and other all-optical effects

Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation

Solitons and breathers are localized solutions of integrable systems that can be viewed as “particles” of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media, these “integrable” gases present a fundamental interest for nonlinear physics. We develop an analytical theory of breather and soliton gases by considering a special, thermodynamic-type limit of the wave-number–frequency relations for multiphase (finite-gap) solutions of the focusing nonlinear Schrödinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator, and it yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the “background” Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, noninteracting breathers (solitons) to a special limiting state, which we term a breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathers (solitons). For a nonhomogeneous breather gas, we derive a full set of kinetic equations describing the slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating the efficacy of the developed general theory with broad implications for nonlinear optics, superfluids, and oceanography. In particular, our work provides the theoretical underpinning for the recently observed remarkable connection of the soliton gas dynamics with the long-term evolution of spontaneous modulational instability