PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

On the geometric structure of characteristic vector fields related with nonlinear equations of the Hamilton-Jacobi type

Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Cartan-Monge geometric approach to the characteristics method for Hamilton-Jacobi type equations and nonlinear partial differential equations of higher orders is analyzed. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of first order is analyzed, the tensor fields of special structure are constructed for defining characteristic vector fields naturally related with nonlinear partial differential equations of higher orders. The generalized characteristics method is developed in the framework of the symplectic theory within geometric Monge and Cartan pictures. The related characteristic vector fields are constructed making use of specially introduced tensor fields, carrying the symplectic structure. Based on their inherited geometric properties, the related functional-analytic Hopf-Lax type solutions to a wide class of boundary and Cauchy problems for nonlinear partial differential equations of Hamilton-Jacobi type are studied. For the non-canonical Hamilton-Jacobi equations there is stated a relationship between their solutions and a good specified functional-analytic fixed point problem, related with Hopf-Lax type solutions to specially constructed dual canonical Hamilton-Jacobi equations.
Rocznik
Strony
89--111
Opis fizyczny
Bibliogr. 31 poz., rys., wykr.
Twórcy
autor
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland, natpats@yahoo.com
Bibliografia
  • [1] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, NY, 1978.
  • [2] J.T Schwartz, Nonlinear function analysis, Gordon and Breach Sci. Publishers, New York, 1969.
  • [3] M.G. Crandall, H. Ishii, P.L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bulletin of AMS, (1992) N1, 1-67.
  • [4] J.E. Marsden, S. Shkoler, Multi-symplectic geometry, covariant Hamiltonians, and water waves, Math. Proceed. Cambr. Philos. Soc. 124 (1997), 1-23.
  • [5] T.J. Bridges, Multi-symplectic structures and wave propagation, Cambr. Philos. Soc. 121 (1997), 147-190.
  • [6] G.A. Bliss, Lectures on the calculus of variations, Chicago, Illinois, USA, 1946.
  • [7] T.J. Bridges, F.E. Laine-Pearson, Nonlinear counter - propagating waves, multi-symplectic geometry, and the instability of standing waves, SIAM J. Appl. Math. 64 (2004) 6, 2096-2120.
  • [8] O.I. Mokhov, Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations. Topics in Topology and Mathematical Physics, Ed. S.P. Novikov, Providence, RI, 1995, 121-151.
  • [9] M.J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia, 1981.
  • [10] M.J. Ablowitz, A.S. Fokas, Complex variables: introduction and applications, Cambridge University Press, 1997.
  • [11] S.N. Kruzhkov, Generalized solutions to nonlinear equations of first order with many independent variables, I, Mathem. USSR Sbornik 70 (112) (1966) 3, 395-415.
  • [12] S.N. Kruzhkov, Generalized solutions to nonlinear equations of first order with many independent variables, II, Mathem. USSR Sbornik 70 (114) (1967) 3, 109-134.
  • [13] V.P. Maslov, Asymptotical methods of solution of pseudo-differential equations, Nauka Publisher, Moscow, 1976.
  • [14] S. Benton, The Hamilton-Jacobi equation: a global approach, Academic Press. USA, 1977.
  • [15] P. Garabedian, Partial differential equations, Wiley Publ., 1964.
  • [16] L. Hormander, The analysis of linearpartial differential operators, Springer-Verlag, Berlin, 1985
  • [17] V.I. Arnold, Introduction into partial differential equations. Interfactor, Moscow, 2002
  • [18] N.K. Prykarpatska, M. Pytel-Kudela, On the structure of characteristics surfaces related with partial differential equations of first and higher orders, Part 1. Opuscula Mathematica 25/2 (2005), 299-306
  • [19] N.K. Prykarpatska, D.L. Blackmore, A.K. Prykarpatsky, M. Pytel-Kudela, On the inf-type extremality solutions to Hamilton-Jacobi equations, their regularity properties and some generalizations, Miskolc Mathematical Notes, 4 (2003), 157-180
  • [20] N.K. Prykarpatska, E. Wachnicki, An exact functional-analityc representation of solutions to a Hamilton-Jacobi equation of Riccati type, Matematychni Studii, 26/2 (2006), 134-140
  • [21] A.K. Prykarpatsky, I.V. Mykytiuk, Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects, Dordrecht-Boston-London: Kluwer Academic Publishers, 1998, 553 p.
  • [22] L.C. Evans, Partial Differential equations, AMS, USA, 1998, 632 p.
  • [23] R. Abracham, J. Marsden, Foundations of mechanics, Cummings, NY, USA, 1978, 806 p.
  • [24] Yu.A. Mitropolski, N.N. Bogoliubov (Jr.), A.K. Prykarpatsky, V.Hr. Samoilenko, Integrable dynamical systems: Differential-geometric and spectral aspects, Naukova Dumka, Kiev, 1987, 296 p (in Russian).
  • [25] F. John, Partial differential equations, Springer, Berlin, 1970, 457 p.
  • [26] V.I. Arnold, Lectures on partial differential equations, Fazis, Moscow, 1999, 175 p (in Russian).
  • [27] Z. Kamont, Równania różniczkowe cząstkowe pierwszego rzędu, Gdańsk University Publisher, Poland, 2003, 303 p.
  • [28] E. Cartan, Systems of differential forms, Herman, 1934, Paris, 260 p.
  • [29] N.K. Prykarpatska, On the structure of characteristic surfaces related with nonlinear partial differential equations of first and higher orders, Part 2, Nonlinear Oscillations, 4 (2005) 8, 137-145.
  • [30] B.R. Weinberg, Asymptotical methods in equations of mathematical physics, Moscow State University Publisher, 1982, 294 p.
  • [31] O.Ye. Hentosh, M.M. Prytula, A.K Prykarpatsky, Differential-geometric integrability fundamentals of nonlinear dynamical systems on functional menifolds, Lviv University Publisher, Lviv, Ukraine, 2005, 408 p.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGH4-0008-0009
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.