Table of contents

To investigate the Helmholtz resonator a model is developed based on the theory of selfadjoint extensions of symmetric operators in a space with indefinite metric. In the case of a small opening compared to the wavelength, approximations of any predetermined precision are obtained for the Green functions of the Dirichlet and Neumann problems for the Helmholtz resonator. The problem of resonances is considered in the framework of the Lax-Phillips approach. Formulae to determine the resonances with any required precision are obtained and substantiated.

Explicit analytic expressions are found for the spectrum and solutions of the discrete, inhomogeneous wave equation

with boundary conditions , where , , and . As a corollary a solution is given of the classical problem of finding an explicit analytic expression describing the vibrations of a string all the mass of which is concentrated at a finite number of equidistant points, which was the object of detailed study by Euler, D'Alembert, D. Bernoulli, Lagrange, Sturm, Routh, and others, who gave a solution of it in the particular case where the masses of all points are the same. The general solution of the problem turns out to be connected with a generalized quaternion algebra and properties of certain of its ideals, and this connection is used in an essential way in the proofs of the theorems.

The author works out methods of studying the problems indicated in the title, which, in contrast to previous methods, do not require the explicit solution of the nonlinear system of integral and differential equations that arises. A number of new results are obtained, including the first correct solution of Pu's well-known problem in its original formulation.

For systems of ordinary differential equations, the concept of particular integral is extended, enabling one to find several approaches to constructing the first integral and the last Jacobi multiplier. Methods are developed for distinguishing a center and focus, and for finding limit cycles of systems having particular algebraic integrals, taking their weight into account, and of systems having conditional particular integrals.

The Beltrami-Laplace operator on a homogeneous pseudo-Riemannian symmetric space of rank 1 is ultrahyperbolic in general. The present article gives a description of the eigenfunctions of this operator in a space of generalized functions with a restriction on their growth at infinity.

It is proved that mixing transformations and flows of rank 1 have mixing of any multiplicity and a minimal self-joining of any order.

Multisoliton solutions of the nonlinear Schrödinger equation are considered which satisfy the condition of finite density:

It is proved that all these solutions satisfy the inequalities

(, ) , which implies solvability of the Cauchy problem for the nonlinear Schrödinger equation with an initial function belonging to the closure of the set of nonreflecting potentials.

A rigid rank-2 transformation with trivial weak essential centralizer is constructed. A family of transformations with analogous properties but of infinite rank is also constructed.

The inverse problems of finding the free term and the coefficient of u(x,t) in a parabolic equation is considered. The Fredholm property of the linear inverse problem of finding the right-hand side of a special form is proved, along with global existence, uniqueness, and stability theorems for its solutions. A uniqueness theorem is established for the nonlinear inverse problem of determining the coefficient under restrictions in the form of inequalities containing no smallness conditions. The proof is carried out by a method of a priori estimates, with the use of the maximum principle for parabolic and elliptic equations. A connection between the uniqueness of the solution of the inverse problem and the completeness of a certain system of functions is established.

The basic results of the theory of A. T. Fomenko on the topological properties of integrable Hamiltonian systems with two degrees of freedom are used to obtain the topological classification of geodesic flows on the torus T² with a Bott integral that is quadratic in the impulses, to state a criterion for a system to be a Bott system in terms of the function of the metric on T², to explicitly calculate the Fomenko invariant W (an untagged molecule) and the Fomenko-Zieschang invariant W* (a tagged molecule), and to completely describe the place occupied by the systems under consideration in the molecular table of complexity.

This paper is devoted to a description of -regions, i.e., domains in the molecular table of Fomenko that are filled with integrable systems with constant energy surfaces that occur most frequently in physics. Namely, the -regions for , , , , and are computed explicitly. The -regions for an arbitrary three-dimensional constant energy submanifold are determined up to a finite number of points. These results make it possible to predict the topological properties of integrable Hamiltonian systems as yet not discovered in physics. The concepts of the order of torsion of integrable Hamiltonian systems and of a minimal system are also introduced, and the connection between these concepts and the concepts of complexity of systems and complexity of three-manifolds due to Matveev is indicated.

Under the assumptions that , , and a study is made of the problem of minimizing the functional in the class of absolutely continuous functions with and . A direct method is presented for investigating the regularity of solutions and their dependence on the parameters of the problem. An example is given of a problem in which is analytic, , , and all the sequences minimizing the functional in the class of admissible smooth functions converge to a nonsmooth function that is not a generalized solution of the Euler equation. An analogous example is given for the two-dimensional problem in the disk.