B. Tirozzi

Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations

The result of this paper is that any fast-decaying function can be represented as an integral over the canonical Maslov operator, on a special Lagrangian manifold, acting on a specific function. This representation enables one to construct effective explicit formulas for asymptotic solutions of a vast class of linear hyperbolic systems with variable coefficients.

Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity: I

where x is the spatial coordinate, t is time, D = D(x) is the basin depth (which we assume to be a smooth function of x), and g is the acceleration due to gravity. Assume that the point x = 0 corresponds to the shoreline. More precisely, D(x) > 0 for x > 0, D(0) = 0, D(x) = γx + O(x) for small x, where γ = 0, and Eqs. (0.1) are regarded in the domain x > 0 only. Let us pose the following Cauchy problem with initial data localized in a neighborhood of some point x = a, a > 0, for Eq. (0.1):

Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations

We study the behavior of the wave part of asymptotic solutions to the Cauchy problem for linearized shallow water equations with initial perturbations localized near the origin. The global representation for these solutions based on the generalized Maslov canonical operator was given earlier. The asymptotic solutions are also localized in the neighborhood of certain curves (fronts). The simplification of general formulas and the behavior of asymptotic solutions in a neighborhood of the regular part of fronts was also given earlier. Here the behavior of asymptotic solutions in a neighborhood of the focal point of the fronts is discussed in detail and the proof of formulas announced earlier for the wave equation is given. This paper can be regarded as a continuation of the paper in Russiian Journal of Mathematical Physics 15 (2), 192–221 (2008).

P-ADIC DYNAMIC SYSTEMS

Dynamic systems in non-Archimedean number fields (i.e., fields with non-Archimedean valuations) are studied. Results are obtained for the fields of p-adic numbers and complex p-adic numbers. Simple p-adic dynamic systems have a very rich structure—attractors, Siegel disks, cycles, and a new structure called a “fuzzy cycle”. The prime number p plays the role of a parameter of the p-adic dynamic system. Changing p radically changes the behavior of the system: attractors may become the centers of Siegel disks, and vice versa, and cycles of different lengths may appear or disappear.

Asymptotics of localized solutions of the one-dimensional wave equation with variable velocity. I. The Cauchy problem

We present a systematic study of the construction of localized asymptotic solutions of the one-dimensional wave equation with variable velocity. In part I, we discuss the solution of the Cauchy problem with localized initial data and zero right-hand side in detail. Our aim is to give a description of various representations of the solution, their geometric interpretation, computer visualization, and illustration of various general approaches (such as the WKB and Whitham methods) concerning asymptotic expansions. We discuss ideas that can be used in more complicated cases (and will be considered in subsequent parts of this paper) such as inhomogeneous wave equations, the linear surge problem, the small dispersion case, etc. and can eventually be generalized to the 2-(and n-) dimensional cases.

Representations of rapidly decaying functions by the Maslov canonical operator

The function V (x/μ) rapidly decreases as μ → +0 in the exterior of a small neighborhood of the point x = 0. In order to study asymptotic solutions of the Cauchy problem with initial condition of the form (2) for partial differential equations, the expression on the right-hand side of Eq. (2) can be rewritten as the Maslov canonical operator [1] K Λδ on the LagrangianmanifoldΛδ = {p = α, x = 0, α ∈ R n} (a plane), acting on the function V (α) defined on Λδ:

Rigorous results for the free energy in the Hopfield model

We prove that the free energy of the Hopfield model with a finite number of patterns can be represented in terms of an asymptotic series expansion in inverse powers of the neurons number. The series is Borel summable for large temperatures. We also establish mathematically some other interesting properties, partly used before in a seminal paper by Amit, Gutfreund and Sompolinsky.

P-ADIC DYNAMICAL SYSTEMS AND NEURAL NETWORKS

A p-adic model which describes a large class of neural networks is presented. In this model the states of neurons are described by digits in the canonical expansion of a p-adic number. Thus each p-adic number represents a configuration of firing and non firing neurons. The process of recognition of patterns is investigated in the P-adic framework. We study heteroassociative and autoassociative nets. P-adic dynamical systems are used to describe a feedback process for autoassociative nets.

Asymptotics of localized solutions of the one-dimensional wave equation with variable velocity. II. Taking into account a source on the right-hand side and a weak dispersion

In the present (second) part of the paper, we study the asymptotic behavior of the solution of the Cauchy problem for a nonhomogeneous wave equation and also consider (instead of the wave equation) an equation with added fourth derivatives containing a small parameter, i.e., include the effects of weak dispersion.

Asymptotic solutions of 2D wave equations with variable velocity and localized right-hand side

In the paper, we consider the Cauchy problem for the inhomogeneous wave equation with variable velocity and with a perturbation in the form of a right-hand side localized in space (near the origin) and in time. In particular, this problem is connected with the question about the creation of tsunami and Rayleigh waves. Using abstract operator theory and in particular Maslovs noncommutative analysis, we show that the solution is separated into two parts: the transient one, which is localized in a neighborhood of the origin and decreases in time and the propagating one, which propagates in space like the wave created by the momentary “equivalent source.” We present several examples covering a wide range of perturbation resulting in rather explicit formulas expressing the solutions it terms of the error function of complex argument.