Table of contents

In this survey the method of trajectory dynamical systems and trajectory attractors is described, and is applied in the study of the limiting asymptotic behaviour of solutions of non-linear evolution equations. This method is especially useful in the study of dissipative equations of mathematical physics for which the corresponding Cauchy initial-value problem has a global (weak) solution with respect to the time but the uniqueness of this solution either has not been established or does not hold. An important example of such an equation is the 3D Navier-Stokes system in a bounded domain. In such a situation one cannot use directly the classical scheme of construction of a dynamical system in the phase space of initial conditions of the Cauchy problem of a given equation and find a global attractor of this dynamical system. Nevertheless, for such equations it is possible to construct a trajectory dynamical system and investigate a trajectory attractor of the corresponding translation semigroup. This universal method is applied for various types of equations arising in mathematical physics: for general dissipative reaction-diffusion systems, for the 3D Navier-Stokes system, for dissipative wave equations, for non-linear elliptic equations in cylindrical domains, and for other equations and systems. Special attention is given to using the method of trajectory attractors in approximation and perturbation problems arising in complicated models of mathematical physics.

Bibliography: 96 titles.

A general theory unifying and generalizing the field theories for multiple integrals due to Weyl and Carathéodory is developed. Generalizations of the Legendre, Weyl, and Carathéodory transforms are introduced, as well as the associated connection and curvature on the fibre bundle underlying the multiple integrals.

Bibliography: 33 titles.

The Schur function expansion of Sato-Segal-Wilson KP -functions is reviewed. The case of -functions related to algebraic curves of arbitrary genus is studied in detail. Explicit expressions for the Plücker coordinate coefficients appearing in the expansion are obtained in terms of directional derivatives of the Riemann -function or Klein -function along the KP flow directions. By using the fundamental bi-differential it is shown how the coefficients can be expressed as polynomials in terms of Klein's higher-genus generalizations of Weierstrass' - and -functions. The cases of genus-two hyperelliptic and genus-three trigonal curves are detailed as illustrations of the approach developed here.

Bibliography: 53 titles.