МАТЁҚУБОВ МУҲАММАД МАХСУДОВИЧ МОСЛАНГАН МАНБАЛИ НОЧИЗИҚЛИ УМУМИЙ КОРТЕВЕГ-ДЕ ФРИЗ ТЕНГЛАМАСИНИ ДАВРИЙ ФУНКЦИЯЛАР СИНФИДА ИНТЕГРАЛЛАШ

Аннотация

OF DISSERTATION OF THE DOCTOR OF PHILOSOFHY (PhD) ON PHYSICAL AND MATEMATICAL SCIENCES URGENCH – 2018 40 The theme of dissertation of doctor of philosophy (PhD) on physical and mathematical sciences was registered at the Supreme Attestation Commission at the Cabinet of Ministers of the Republic of Uzbekistan under number B2017.3.FM110. Dissertation has been prepared at Urgench State University. The abstract of the dissertation is posted in three languages (uzbek, russian, english(resume)) on the website (www.ik-mat.urdu.uz) and the «Ziyonet» information and educational portal (www.ziyonet.uz). Scientific supervisor: Khasanov Aknazar Bekdurdievich Doctor of Physical and Mathematical Sciences, Professor Official opponents: Takhirov Jozil Ostanovich Doctor of Physical and Mathematical Sciences, Professor Hoitmetov Umid Azadovich Candidat of Physical and Mathematical Sciences Leading organization: National University of Uzbekistan Defense will take place «____» _____________2018 at _____ at the meeting of Scientific Council number PhD.28.12.2017.FM.55.01 at Urgench State University. (Address: 14 Kh. Olimjan str., Urgench city, 220100, Uzbekistan, Ph.: (99862)224-66-11, fax: (99862)224-67-00, e-mail: ik-mat.urdu@umail.uz Dissertation is possible to review in Information-resource centre at Urgench state University (is registered №______) (Address: 14 Kh. Olimjan str., Urgench city, 220100, Uzbekistan, Ph.: (99862)224-66-11, fax: (99862)224-67-00), e-mail: ursubox@gmail.com). Abstract of dissertation sent out on «____» _______________2018 year B.I.Abdullayev Chairman of scientific council on award of scientific degrees, D.F.M.S. A.A.Atamurotov Scientific secretary of scientific council on award of scientific degrees, C.F.M.S. A.A.Yakhshimratov Chairman of scientific Seminar under Scientific Council on award of scientific degrees, D.F.M.S. 41 INTRODUCTION (abstract of the PhD thesis) The aim of the research work is finding the solutions of nonlinear general Korteweg-de Vries equations with self-consistent source in the class of periodic functions via inverse spectral problems for the differential operators with periodic coefficients. The object of the research work consists of nonlinear general Korteweg-de Vries equations, loaded nonlinear general Korteweg-de Vries equations. Scientific novelty of the research work consists on the following: the method of evolutional nonlinear equations, which are integrable via Sturm-Liouville operator with periodic coefficients is given; the complete integrability of the loaded Korteweg-de Vries equation in the class of periodic functions is proved; the complete integrability of the higher order Korteweg-de Vries equation in the class of periodic functions is proved; the complete integrability of the higher order loaded Korteweg-de Vries equation in the class of periodic functions is proved; the complete integrability of loaded Korteweg-de Vries equation with self consistent source in the class of periodic functions is proved; the complete integrability of the higher order Korteweg-de Vries equation in the class of periodic functions, in which the free term does not depend on the spacial variable, is proven; the complete integrability of the higher order Korteweg-de Vries equation with self-consistent source in the class of periodic functions is proved; Implementation of the research results. The results obtained during the dissrtation research are practiced in the following areas: The results of integrating the loaded Korteweg-de Vries equations and integrating the higher order Korteweg-de Vries equations with a self-consistent source in the class of periodic functions are used in the project "Qualitative properties and bifurcations of differential equations and dynamical systems" of the Slovak Grant Agency VEGA (Confirmation from Bratislava, February 2018). The methods of integrating the higher Korteweg-de Vries equations with self-consistent resources in the class of periodic functions made it possible to estimate the phase changes of dynamical systems associated with evolutionary elliptic equations with self-consistent sources. The results of integrating the general nonlinear Korteweg-de Vries equation with a self-consistent source were used in the works carried out in the framework of the research project F-4-61 of the Urgench State University on the theme "Integration of nonlinear evolution equations with a self-consistent source by the inverse scattering method" 2012-2016 years. (Reference No. 89-03-1064 Ministry of Higher and Secondary Specialized Education of the Republic of Uzbekistan, March 3, 2017). The methods of integrating the higher Korteweg-de Vries equations with a self-consistent source in the class of periodic functions are used to find the solution of various evolutionary equations with the source. 42 The structure and volume of the thesis. The dissertation work consists: introduction, three chapters, conclusion and bibliography. The volume of the dissertatio