Метод оберненої задачі та неспадні розв’язки нелінійних еволюційних рівнянь

Ключові слова: метод оберненої задачі розсіювання, нелінійні еволюційні рівняння, скінченнозонні фони, спектр канторівського типу, асимптотика за великим часом.

Егорова И.Е. Метод обратной задачи и неубывающие решения нелинейных эволюционных уравнений. - Рукопись.

Диссертация на соискание ученой степени доктора физико-математических наук по специальности 01.01.03 - математическая физика. Інститут математики НАН Украины, Киев, 2010.

Работа посвящена разработке метода обратной задачи и его использованию для интегрирования некоторых нелинейных уравнений типа КдФ с неубывающими данными типа ступеньки, либо почти периодическими. Построена теория рассеяния для одномерного оператора Шредингера и оператора Якоби с асимптотически конечнозонными коэффициентами, имеющими разные фоновые асимптотики на полуосях. Получены характеристические свойства данных рассеяния, позволившие решить прямую и обратную задачи в классе возмущений, имеющих заданную гладкость и число конечных моментов. Результаты использованы для решения методом обратной задачи рассеяния задач Коши для уравнений KдФ, мКдФ и иерархии Тоды с асимтотически конечнозонными начальными условиями типа ступеньки. Получены точные асимтотические формулы, описывающие поведение решения цепочки Тода с начальными условиями типа ступеньки при больших временах (распад решения в асимптотический ряд солитонов). Проинтегрированы уравнения КдФ и дефокусирующего НШ в классах почти периодических функций, имеющих спектр канторовского типа. Доказано, что эти решения являются равномерными почти периодическими функциями по времени. Методами теории рассеяния исследованы некоторые свойства дискретного спектра комплексных матриц Якоби.

Ключевые слова: метод обратной задачи рассеяния, нелинейные эволюционные уравнения, конечнозонные фоны, спектр канторовского типа, асимптотика при больших временах.

розсіювання шрьодінгер ланцюжок тод

Egorova I. Ye. Inverse problem method and non decreasing solutions of nonlinear evolutionary equations. - Manuscript.

The doctoral thesis (in physics and mathematics, specialization 01.01.03 - mathematical physics). Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, 2010.

The work is devoted to the extension of the inverse scattering method for the finite gap backgrounds and its application to the integration of certain KdV-like nonlinear equations with the step-like finite-gap initial conditions. The scattering theory for the finite-gap backgrounds is constructed. Precisely, the properties of the transformation operators that corresponds to the Jacobi and one-dimensional Schrцdinger operators with asymptotically finite-gap coefficients are investigated; the necessary and sufficient conditions of analytical nature for the scattering data of the Schrцdinger operators with step-like potentials with different finite-gap backgrounds on half axes and the finite second moments of perturbations are obtained; the same properties for the step-like finite-gap Jacobi operators with the finite first moments are studied; the Marchenko equations are derived for both operators, and the characteristic properties of their kernels in the case of the finite second or higher moments of perturbations, including also smooth perturbations, are investigated. In the inverse problem setting the uniqueness of the solution for the left and right Marchenko equations is studied. The uniqueness theorem, which asserts the coincidence of the operators coefficients, restored from the left and right scattering data, is proved. Moreover, for the Jacobi operators the uniqueness theorem is proved in the most general case of the finite first moment of perturbation. Thus, the thesis contains a thorough investigation of the direct/inverse scattering problem for the Schrцdinger and the Jacobi operators with step-like finite-gap coefficients in the prescribed classes of perturbations.

These results are applied for the integration of the associated nonlinear evolutionary equations, such as KdV, modified KdV and the Toda hierarchy equations. In particular, by the generalized inverse scattering transform the Cauchy problem for the KdV equation with asymptotically finite-gap step-like initial conditions is solved. For such initial conditions the corresponding perturbations are assumed to have a prescribed smoothness and decaying rate at infinity. The method of investigation, developed in the thesis, allows also to solve the Cauchy problem for the KdV equation in the Schwartz class of perturbations of finite-gap step-like backgrounds. This result is new even in the case of ordinary step-like initial data with two constant backgrounds. By the Miura transformation the Cauchy problem is also solved for the mKdV equation with asymptotically finite-gap step-like initial conditions with the prescribed (minimal for the existence of the classical solution) smoothness of the initial perturbations, and the number of their finite moments. The inverse scattering transform is used for the integration of the Toda hierarchy equations in a class of asymptotically finite-gap step-like sequences with the finite first moments of the perturbations.

The long-time asymptotical formulas, that describe the behaviour of the solutions of Toda lattices with step-like initial conditions, which are asymptotically constant on one half axis and asymptotically finite-gap on the other one, are obtained. If the continuous spectrum of the associated Jacobi operator has a band of the spectrum of multiplicity one away from the spectrum of the Laplacian, it is proved, that the solution near the wave front can be expanded into a certain asymptotical series of solitons. The precise formulas for those solitons are given.

There is another type of non decreasing initial conditions, for which the inverse problem method for the integration of corresponding nonlinear equations is applied, namely, the set of the reflectionless almost periodic potentials that have nowhere dense spectrum. For these classes of initial conditions the following problems are studied: the inverse spectral problem is solved for the Schrцdinger operator with the reflectionless potential, which has a Cantor type spectrum of positive measure. For such type of spectrum the constrains should be imposed on the lengths of gaps and the distances between them, in particular, the gaps of higher orders are superexponentially small. By using the generalized Jacobi inversion problem it is proved, that the reconstructed unique potential is the Bohr almost periodic function. This potential is considered further as an initial condition for the Cauchy problem for the KdV equation. It is proved, that the solution of the problem under discussion exists, is unique, and is a uniformly almost periodic function with respect to both the time and the space.

A similar inverse spectral problem is solved for the self-adjoint reflectionless Dirac operator with the spectrum of the Craig class. A typical representative of this class is a nowhere dense set of the positive measure. It is proved, that the reconstructed unique potential is the Bohr almost periodic complex-valued function. The associated Cauchy problem for the defocusing nonlinear Schrцdinger equation is studied. The solution exists, is unique and is a complex-valued uniformly almost periodic function with respect to each variable.

The KdV and defocusing NS equations are integrated also in a class of limit periodic functions, which are fast approximated by periodic functions of growing periods (smooth and complex counterparts of the Pastur--Tkachenko class). A typical representative of this class has the Cantor type spectrum. It is proved, that these limit periodic (by the space variable) solutions are the Bohr almost periodic functions with respect to the time variable.

The scattering problem method is applied for the investigation of properties of the discrete spectrum of complex Jacobi matrices (non self-adjoint Jacobi operators on the half-axis with the Dirichlet boundary conditions). In particular, the discrete analogs of the Pavlov theorems for compact complex perturbations of the discrete Laplacian, whose coefficients decay exponentially fast, are obtained. The localization bounds of the discrete spectrum for complex Jacobi matrices which are compact perturbations of 2-periodic real-valued matrix, are described.

Key words: inverse scattering transform, nonlinear evolutionary equations, finite-gap backgrounds, Cantor type spectrum, long time asymptotics.

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