Yves J M Tourigny

Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.

Explicit invariant measures for products of random matrices

We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL(2, C). The matrices in the product are such that one entry is gamma-distributed along a ray in the complex plane. When the ray is the positive real axis, the products are those associated with a continued fraction studied by Letac & Seshadri [Z. Wahr. Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the continued fraction is a generalised inverse Gaussian. We extend this result by finding the distribution for an arbitrary ray in the complex right-half plane, and thus compute the corresponding Lyapunov exponent explicitly. When the ray lies on the imaginary axis, the matrices in the infinite product coincide with the transfer matrices associated with a one-dimensional discrete Schrodinger operator with a random, gamma-distributed potential. Hence, the explicit knowledge of the Lyapunov exponent may be used to estimate the (exponential) rate of localisation of the eigenstates.

The numerical study of blowup with application to a nonlinear Schrodinger equation

Abstract We discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite-time unbounded growth. We first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate. As we shall demonstrate for a radial nonlinear Schrodinger equation, this approach is inadequate since different discretizations exhibit different growth rates. The spurious behaviour of discretizations in the neighbourhood of the singularity is discussed. A reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods.

Products of Random Matrices and Generalised Quantum Point Scatterers

To every product of 2×2 matrices, there corresponds a one-dimensional Schrödinger equation whose potential consists of generalised point scatterers. Products of random matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in SL(2,ℝ). We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.

The Lyapunov Exponent of Products of Random 2×2 Matrices Close to the Identity

We study products of arbitrary random real 2×2 matrices that are close to the identity matrix. Using the Iwasawa decomposition of SL(2,ℝ), we identify a continuum regime where the mean values and the covariances of the three Iwasawa parameters are simultaneously small. In this regime, the Lyapunov exponent of the product is shown to assume a scaling form. In the general case, the corresponding scaling function is expressed in terms of Gauss’ hypergeometric function. A number of particular cases are also considered, where the scaling function of the Lyapunov exponent involves other special functions (Airy, Bessel, Whittaker, elliptic). The general solution thus obtained allows us, among other things, to recover in a unified framework many results known previously from exactly solvable models of one-dimensional disordered systems.

Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension

AbstractWe consider the Schrödinger equation with a random potential of the form where w is a Lévy noise. We focus on the problem of computing the so-called complex Lyapunov exponent where N is the integrated density of states of the system, and γ is the Lyapunov exponent. In the case where the Lévy process is non-decreasing, we show that the calculation of Ω reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Lévy process. We review the known solvable cases—where Ω can be expressed in terms of special functions—and discover a new one.

The dynamics of Padé approximation

The analogy between the expansion of a real number as a continued fraction and the summation of a formal power series by means of Pade approximants is studied. The elementary dynamics of the map analogous to the well-known Gauss map of the remainder for the expression of a real number as a continued fraction is thereby developed.

A case study of methods of series summation: Kelvin-Helmholtz instability of finite amplitude

We compute the singularities of the solution of the Birkhoff-Rott equation that governs the evolution of a planar periodic vortex sheet. Our approach uses the Taylor series obtained by Meiron et al. [J. Fluid Mech. 114 (1982) 283] for a flat sheet subject initially to a sinusoidal disturbance of amplitude a. The series is then summed by using various generalisations of the Pade method. We find approximate values for the location and type of the principal singularity as a ranges from zero to infinity. Finally, the results are used as a basis to guide the choice of methods of summing series arising from problems in fluid mechanics.

The summation of series in several variables and its applications in fluid dynamics

Abstract We describe a method for the summation of series in powers of several variables and apply it to some problems of fluid dynamics. The summation method generates a sequence of algebraic approximants. The degree of the defining algebraic equations can either be held fixed, or else increased with the number of series coefficients used. Applications to Jeffery–Hamel flows and to porous-channel flows are discussed. In particular, we show how the method may be used to compute structurally unstable pitchfork bifurcations.

The Optimisation of the Mesh in First-Order Systems Least-Squares Methods

We describe an algorithm for optimising the mesh in the least-squares finite element discretisation of first-order systems of partial differential equations. The key feature of the method is that the optimisation process is based entirely on the solution of local PDE problems. We apply the algorithm to the Stokes equations for the flow of a viscous incompressible fluid, and to a convection diffusion equation where convection dominates.