CONSTRUCTION OF CONSERVATION LAWS FOR NONLINEAR EQUATIONS CONNECTED WITH THE DIRAC SCATTERING OPERATOR

Introduction: the problem of integrability of nonlinear partial differential equations, even for the second order, is not always an obvious fact, since inding a general solution is possible only in rare cases. The proof of integrability can be justiied in many ways: by obtaining a large number of particular solutions, reducing to some exactly solvable reduction, and also constructing an infinite number of first integrals. Materials and methods of research: methods of the theory of solitons for equations possessing a Lax pair were used. To construct the conservation laws, we use the equation of isospectral deformation with a self-adjoint differential Dirac operator of the first and second kind. Under the condition that the functions occurring in the coeficients have a rapidly decreasing nature, the solution is represented in the form of a series expounded in negative powers of the spectral parameter. Results of the study: Eigenvalue equations with Dirac operators of the irst and second kind are investigated. A countable number of first integrals is found. Examples of nonlinear partial differential equations obtained with the help of an operator equation of zero curvature are given for which the scattering operator coincides with the Dirac operator. It is proved that such equations have a countable number of irst integrals. Discussion and conclusion: the author concluded that among the conservation laws obtained there is a Hamiltonian, all integrals of the motion are in involution with respect to the Poisson brackets. This suggests that nonlinear partial differential equations are Hamiltonian, which leads to their complete integrability.

оператор Дирака, уравнение изоспектральной деформации, нелинейные уравнения в частных производных, интегралы движения, уравнение Гамильтона, скобки Пуассона, первые интегралы, законы сохранения, Dirac operator, equation of isospectral deformation, nonlinear partial differential equations, integrals of motion, Hamilton's equation, Poisson brackets, irst integrals, conservation laws

1. Гельфанд И.М., Дикий Л.А. Резольвенты и гамильтоновы системы // Функц. Анализ и его прилож. 11, № 2, 1977. С. 11-27.

2. Гриневич П.Г. Преобразование рассеяния для двумерного оператора Шредингера с убывающим на бесконечности потенциалом // Успехи мат. Наук. Т. 55, вып. 6 (336). 2000.

3. Лакс П.Д. Интегралы нелинейных эволюционных уравнений и уединенные волны. // Математика, 13: 5. М.: Мир, 1969. С. 128150.

4. Ньюэлл А. Солитоны в математике и физике / пер. с англ. И.Р. Габитова и др.; под ред. А.В. Михайлова. М.: Мир, 1989.323 с.

5. Лэм Дж. Введение в теорию солитонов / под ред. В.Е. Захарова. М.: Мир, 1983. 294 с.

6. Редькина Т.В.Нелинейные уравнения, интегрируемые методами солитонной математики. Пары Лакса с самосопряженным оператором Дирака и с несамосопряженными операторами рассеяния. / LAP LAMBERT Academic Publishing. AV Akademikerverlag GmbH & Co. KG. Deutschland. 2013. 153 P.

7. Новикова О.В. Исследование нелинейного комплексного дифференциального уравнения в частных производных, обладающего парой Лакса: дис.. кандидата физ.-мат. наук. Воронеж, 2015.