There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

/pdf/linear-and-nonlinear-waves-4ds9thgmph.pdf

Linear and nonlinear waves

Dark optical solitons: physics and applications

In recent years two nonlinear dispersive partial differential equations have attracted a lot of attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahoney and Korteweg-de Vries equations. In particular, they accomodate wave breaking phenomena.

/pdf/the-hydrodynamical-relevance-of-the-camassa-holm-and-mcluwibdlr.pdf

The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

An inverse scattering method is developed for the Camassa–Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard inverse scattering transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.

/pdf/inverse-scattering-transform-for-the-camassa-holm-equation-4qg9hgn7dx.pdf

Inverse Scattering Transform for the Camassa-Holm equation

Integrable Hamiltonian Hierarchies

The direct and the inverse scattering problems for the generalised Zakharov-Shabat spectral problem L related to a given semi-simple Lie algebra are solved. The fundamental analytic solutions and the minimal set of scattering data F are constructed. The mapping from the set of potentials to F is shown to be a generalised Fourier transform over the sets ( Psi ), ( Phi ) of the adjoint solutions of L. The expansions for the potential of L and its variations over ( Psi ), ( Phi ) are obtained and used to investigate the corresponding non-linear evolution equations (NLEE) on homogeneous spaces. These are shown to be spectral expansions for the generating operators Lambda +or- which play a fundamental role in the theory of the NLEE. The approach is explicitly gauge covariant, which allows one to investigate also the corresponding gauge-equivalent NLEE.

Generalised Fourier transforms for the soliton equations. Gauge-covariant formulation

Based on the completeness relation for the squared solutions of the Lax operator L, we show that a subset of nonlocal equations from the hierarchy of nonlocal nonlinear Schrodinger equations (NLS) is a completely integrable system. The spectral properties of the Lax operator indicate that there are two types of soliton solutions. The relevant action-angle variables are parametrized by the scattering data of the Lax operator. The notion of the symplectic basis, which directly maps the variations of the potential of L to the variations of the action-angle variables has been generalized to the nonlocal case. We also show that the inverse scattering method can be viewed as a generalized Fourier transform. Using the trace identities and the symplectic basis, we construct the hierarchy Hamiltonian structures for the nonlocal NLS equations.

Vladimir S. Gerdjikov

Papers

Inverse Scattering Transform for the Camassa-Holm equation

Integrable Hamiltonian Hierarchies

Generalised Fourier transforms for the soliton equations. Gauge-covariant formulation

Complete integrability of nonlocal nonlinear Schrödinger equation

The generating operator for the n × n linear system