cahtНаучный вестник МГТУ ГАCivil Aviation High Technologies2079-06192542-0119Moscow State Technical University of Civil Aviation (MSTU CA)caht-1061Research ArticleСтатьиСИММЕТРИИ И ИНТЕГРИРУЕМОСТЬ ПО ЛАКСУ ОБОБЩЕННОГО УРАВНЕНИЯ ПРУДМАНА - ДЖОНСОНАSYMMETRIES AND LAX INTEGRABILITY OF THE GENERALIZED PROUDMAN-JOHNSON EQUATIONМорозовО. И.MorozovO. I.

доктор физ.-мат. наук, профессор кафедры прикладной математики,

г. Краков

Doctor in phys.-math. sciences, professor of Department of Applied Mathematics,

Krakow

oimorozov@gmail.comУниверситет Науки и ТехнологииПольшаAGH University of Science and TechnologyPoland2017030520172029499Copyright © Морозов О.И., 20172017Морозов О.И.Morozov O.I.This work is licensed under a Creative Commons Attribution 4.0 License.https://avia.mstuca.ru/jour/article/view/1061

Изучаются локальные симметрии обобщённого уравнения Прудмана-Джонсона. Симметрии дифференциального уравнения в частных производных могут использоваться для нахождения его инвариантных решений.В частности, если <р есть производящая функция симметрии для уравнения Н = 0, то ϕ-инвариантные решениясуть решения переопределённой совместной системы H = 0, ϕ = 0. Показано, что алгебра Ли локальных симмет-рий обобщённого уравнения Прудмана-Джонсона является бесконечной. Найдены некоторые случаи, когда редуцированное при помощи симметрий уравнение сводится к обыкновенным дифференциальным уравнениям, которые интегрируются в квадратурах, что позволяет построить соответствующие точные решения. Дифференциальные накрытия (или структуры продолжения Волквиста-Истабрука, или представления нулевой кривизны, или интегрируемые расширения и так далее) весьма важны в геометрии уравнений в частных производных. Теория дифференциальных накрытий есть естественный язык для работы с обратной задачей теории рассеивания в случае солитонных уравнений, преобразований Бэклунда, операторов рекурсии, нелокальных симметрий и нелокальных законов сохранения, преобразований Дарбу и деформаций нелинейных уравнений. В последнем разделе статьи показано, что при некоторых значениях параметра, входящего в уравнение Прудмана-Джонсона, оно обладает дифференциальным накрытием. Это свойство также называется интегрируемостью по Лаксу.

We study local symmetries of the generalized Proudman-Johnson equation. Symmetries of a partial differential equation may be used to find its invariant solutions. In particular, if <р is a characteristic of a symmetry for a PDE Н = О then the <р-invariant solution of the PDE is a solution to the compatible over-determined system Н = О, < р = О. We show that the Lie algebra of local symmetries for the generalized Proudman-Johnson equation is infinite-dimensional. Reductions of equation with respect to the local symmetries provide ordinary differential equations that describe invariant solutions. For a certain value of the parameter entering the equation we find some cases when the reduced ODE is integrable by quadratures and thus allows one to construct exact solutions. Differential coverings (or Wahlquist-Estabrook prolongation structures, or zero-curvature representations, or integrable extensions, etc.) are of great importance in geometry of PDEs. The theory of coverings is a natural framework for dealing with inverse scattering constructions for soliton equations, Bäcklund transformations, recursion operators, nonlocal symmetries and nonlocal conservation laws, Darboux transformations, and deformations of nonlinear PDEs. In the last section we show that in the case of a certain value of the parameter entering the equation it has a differential covering. This property is referred to as a Lax integrability.

обобщённое уравнение Прудмана-Джонсонадифференциальное накрытиелокальные симметрииинвариантные решенияпреобразование Бэклундаgeneralized Proudman-Johnson equationlocal symmetry of differential equationinvariant solutiondifferential coveringBäcklund transformationReferencesProudman I., Johnson K. Boundary-layer growth near a rear stagnation point. J. Fluid Mech, 1962, pp. 161-168Proudman I., Johnson K. Boundary-layer growth near a rear stagnation point. J. Fluid Mech, 1962, pp. 161–168.Okamoto H., Zhu, J. Some similarity solutions of the Navier-Stokes equations and related topics. Taiwanese J. Math, 2000, A, pp. 65-103Okamoto H., Zhu, J. Some similarity solutions of the Navier-Stokes equations and related topics. Taiwanese J. Math, 2000, A, pp. 65–103.Chen X., Okamoto H. Global existence of solutions to the Proudman-Johnson equation. Proc. Japan Acad. 2000. pp. 149-152Chen X., Okamoto H. Global existence of solutions to the Proudman-Johnson equation. Proc. Japan Acad. 2000. pp. 149–152.Chen X., Okamoto H. Global existence of solutions to the generalized Proudman-Johnson equation. Proc. Japan Acad., 2002. A, pp. 136-139Chen X., Okamoto H. Global existence of solutions to the generalized Proudman-Johnson equation. Proc. Japan Acad., 2002. A, pp. 136–139.Morozov O.I. Contact equaivalence of the generalized Hunter-Saxton and the Euler-Poisson equation. 2004. Preprint www.arXiv.org: math-ph/0406016/Morozov O.I. Contact equaivalence of the generalized Hunter-Saxton and the Euler-Poisson equation. 2004. Preprint www.arXiv.org: math-ph/0406016/Morozov O.I. Linearizability and integrability of the generalized Calogero-Hunter-Saxton equation. Scientific Bulletin of the Moscow State Technical University of Civil Aviation, 2006, issue 114, pp. 34-42. (in Russian)Morozov O.I. Linearizability and integrability of the generalized Calogero-Hunter-Saxton equation. Scientific Bulletin of the Moscow State Technical University of Civil Aviation, 2006, issue 114, pp. 34–42. (in Russian)Krasil’shchik I.S., Vinogradov A.M. Nonlocal symmetries and the theory of coverings. Acta Appl. Math. 1984, pp. 79-86Krasil’shchik I.S., Vinogradov A.M. Nonlocal symmetries and the theory of coverings. Acta Appl. Math. 1984, pp. 79–86.Krasil’shchik I.S., Lychagin V.V., Vinogradov A.M. Geometry of jet spaces and nonlinear partial differential equations. Gordon and Breach, New York. 1986Krasil’shchik I.S., Lychagin V.V., Vinogradov A.M. Geometry of jet spaces and nonlinear partial differential equations. Gordon and Breach, New York. 1986.Krasil’shchik I.S., Vinogradov A.M. Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math. 1989, pp. 161-209Krasil’shchik I.S., Vinogradov A.M. Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations. Acta Appl. Math. 1989, pp. 161–209.Beals R., Sattinger D.H., Szmigielski J. Inverse scattering solutions of the Hunter-Saxton equation. Applicable Analysis, 2001, issue 3-4, pp. 255-269Beals R., Sattinger D.H., Szmigielski J. Inverse scattering solutions of the Hunter–Saxton equation. Applicable Analysis, 2001, issue 3–4, pp. 255–269.Reyes E.G. The soliton content of the Camassa-Holm and Hunter-Saxton equations. Proceedings of Institute of Math., NAS of Ukraine, 2002, Part I, pp. 201-208Reyes E.G. The soliton content of the Camassa-Holm and Hunter-Saxton equations. Proceedings of Institute of Math., NAS of Ukraine, 2002, Part I, pp. 201–208.Baran H., Marvan M. Jets. A software for differential calculus on jet spaces and diffieties. Available online at http://jets.math.slu.czBaran H., Marvan M. Jets. A software for differential calculus on jet spaces and diffieties. Available online at http://jets.math.slu.czAbramowitz M., Stegun I.A., (Eds). Handbook of mathematical functions, 10th print, National Bureau of Standards, 1972Abramowitz M., Stegun I.A., (Eds). Handbook of mathematical functions, 10th print, National Bureau of Standards, 1972.Whittaker E.T., Watson G.N. A course of modern analysis. 4th Edition, Cambridge University Press, Cambridge, 1927, Reprinted 2002Whittaker E.T., Watson G.N. A course of modern analysis. 4th Edition, Cambridge University Press, Cambridge, 1927, Reprinted 2002.Kovacic J. An algorithm for solving second order linear homogeneus differential equations, J. Symbolic Comput, 1986, pp. 3-43Kovacic J. An algorithm for solving second order linear homogeneus differential equations, J. Symbolic Comput, 1986, pp. 3–43.Wahlquist H.D., Estabrook F.B. Prolongation structures of nonlinear evolution equations. J. Math. Phys., 1975, vol. 16, no. 1, pp. 1-7Wahlquist H.D., Estabrook F.B. Prolongation structures of nonlinear evolution equations. J. Math. Phys., 1975, vol. 16, no. 1, pp. 1–7.Zakharov V.E., Shabat A.B. Integration of nonlinear equations of mathematical physics by the method of inverse scattering, II, Funct. Anal. Appl, 1980, pp. 166-174Zakharov V.E., Shabat A.B. Integration of nonlinear equations of mathematical physics by the method of inverse scattering, II, Funct. Anal. Appl, 1980, pp. 166–174.Bryant R.L., Griffiths Ph.A. Characteristic cohomology of differential systems (II): conservation laws for a class of parabolic equations, Duke Math. J., 1995, pp. 531-676Bryant R.L., Griffiths Ph.A. Characteristic cohomology of differential systems (II): conservation laws for a class of parabolic equations, Duke Math. J., 1995, pp. 531–676.

The authors declare that there are no conflicts of interest present.