Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials

We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order $\alpha >1$. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyze the propagation of localized waves when $\alpha$ is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic KdV equation, and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with H\"older-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When $\alpha \rightarrow 1^+$, we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed, and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile

Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation

We study nonlinear waves in Newton's cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz's contact forces. We formally derive a spatially discrete modulation equation, for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations. The fully-nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that we call the discrete p-Schroedinger (DpS) equation. It consists of a spatial discretization of a generalized Schroedinger equation with p-Laplacian, with fractional p>2 depending on the exponent of Hertz's contact force. We show that the DpS equation admits explicit periodic travelling wave solutions, and numerically find a plethora of standing wave solutions given by the orbits of a discrete map, in particular spatially localized breather solutions. Using a modified Lyapunov-Schmidt technique, we prove the existence of exact periodic travelling waves in the chain of beads, close to the small amplitude modulated waves given by the DpS equation. Using numerical simulations, we show that the DpS equation captures several other important features of the dynamics in the weakly nonlinear regime, namely modulational instabilities, the existence of static and travelling breathers, and repulsive or attractive interactions of these localized structures

Nonlinear wave patterns in the complex KdV and nonlinear Schrodinger equations

This thesis is on the theory of nonlinear waves in physics. To begin with, we develop from first principles the theory of the complex Korteweg-de Vries (KdV) equation as an equation for the complex velocity of a weakly nonlinear wave in a shallow, ideal fluid. We show that this is completely consistent with the well-known theory of the real KdV equation as a special case, but has the advantage of directly giving complete information about the motion of all particles within the fluid. We show that the complex KdV equation also has conserved quantities which are completely consistent with the physical interpretation of the real KdV equation. When a periodic wave solution to the real KdV equation is expanded in the quasi-monochromatic approximation, it is known that the amplitude of the wave envelope is described by the nonlinear Schrodinger (NLS) equation. However, in the complex KdV equation, we show that the fundamental modes of the velocity are described by the split NLS equations, themselves a special case of the Ablowitz-Kaup-Newell-Segur system. This is a directly physical interpretation of the split NLS equations, which were primarily introduced as only a mathematical construct emerging from the Zakharov-Shabat equations. We also discuss an empirically obtained symmetry of the rational solutions to the KdV equations, which seems to have been unnoticed until now. Solutions which can be written in terms of Wronskian determinants are well-known; however, we show that these are actually part of a more general family of rational solutions. We show that a linear combination of the Wronskians of orders $n$ and $n+2$ generates a new, multi-peak rational solution to the KdV equation. We next move on to the integrable extensions of the NLS equation. These incorporate higher order nonlinear and dispersive terms in such a way that the system keeps the same conserved quantities, and is thus completely integrable. We obtain the general solution of the doubly-periodic solutions of the class I extension of the NLS equation, and discuss several special cases. These are the most general one-parameter first order solutions of the (class I) extended NLS equation. Building on this, we also discuss second order solutions to the extended NLS equation. We obtain the general 2-breather solutions, and discuss several special cases; among them, semirational breathers, the degenerate breather solution, the second-order rogue wave, and the rogue wave triplet solution. We also discuss the breather to soliton conversion, which is a solution which does not exist in the basic NLS equation where only the lowest order dispersive and nonlinear terms are present. Finally, we discuss a few possibilities for future research based on the work done in this thesis

Linear and nonlinear dynamics in stratified shear flows

Stably stratified shear flows, in which a less dense layer of fluid lies above and moves counter to a more dense layer below, are ubiquitous in geophysical fluid dynamics. These are often found to be unstable if the non-dimensional Richardson number Ri, quantifying the strength of stratification to shear, is sufficiently low. This is of particular importance in oceanography, where shear instabilities are conjectured to be important in the generation of turbulence in the deep ocean, an area of huge uncertainty in contemporary climate models. The Miles-Howard theorem tells us that for a steady, inviscid, parallel shear flow, if the local Richardson number is everywhere greater than one quarter, the flow is stable to infinitesimal perturbations. Though an important result, the strong restrictions in the applicability of this theorem mean care must be used when applying the criterion of Ri > 1/4 for stability. This thesis explores some of these limitations, beginning with an overview in chapter 1. Chapter 2 explores the infinitesimal restriction of the Miles-Howard theorem, by asking whether finite-amplitude perturbations could lead to significant nonlinear behaviour, in a so-called subcritical instability. It is found that while the classical Kelvin-Helmholtz instability does indeed exhibit subcriticality, nonlinear steady states are found only just above Ri = 1/4. Chapter 3 investigates in detail a hitherto unknown linear instability, which was discovered in chapter 2. Behaving similarly to the classic Holmboe instability, it exists for Ri > 1/4 when viscosity is introduced, and reveals new insights into the possible physical interpretations of stratified shear instability. Chapter 4 revisits the results of chapter 2 but considers two cases of the Prandtl number Pr, the ratio of diffusivity of the momentum to density. When Pr = 0.7, as is approximately the case for air, a simple supercritical instability is found. However, for Pr = 7, corresponding approximately to water, strong subcritical behaviour is observed, and it is demonstrated that finite-amplitude perturbations can trigger Kelvin-Helmholtz-like behaviour well above Ri = 1/4. Chapter 5 considers the time-varying, non-parallel flow of an oblique internal gravity wave incident on a shear layer. Using direct-adjoint looping, it is shown that the disturbances which maximise energy after a certain time, so-called linear optimal perturbations, can be convective-like rolls in the spanwise direction, rather than a shear instability, calling into question the relevance of the classical shear instabilities in oceanography. Chapter 6 concludes the thesis with a discussion of the implications of the results