Differential-algebraic approach to constructing representations of commuting differentiations in functional spaces and its application to nonlinear integrable dynamical systems

Abstract There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only one conserved quantity is analyzed in detail, the corresponding Lax type representations of differentiations are constructed for an infinite hierarchy of nonlinear dynamical systems of the Burgers and Korteweg–de Vries type. A related infinite bi-Hamiltonian hierarchy of Lax type dynamical systems is constructed.

[1]  Peter D. Lax,et al.  Almost Periodic Solutions of the KdV Equation , 1976 .

[2]  I. Gel'fand,et al.  Integrable nonlinear equations and the Liouville theorem , 1979 .

[3]  A. R. Forsyth.  Theory of Differential Equations , 1961 .

[4]  Maciej Błaszak,et al.  Multi-Hamiltonian Theory of Dynamical Systems , 1998 .

[5]  C. Godbillon.  Géométrie différentielle et mécanique analytique , 1969 .

[6]  A. K Prikarpatskiĭ,et al.  Algebraic integrability of nonlinear dynamical systems on manifolds : classical and quantum aspects , 1998 .

[7]  A. Jamiołkowski.  Book reviewApplications of Lie groups to differential equations : Peter J. Olver (School of Mathematics, University of Minnesota, Minneapolis, U.S.A): Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, XXVI+497pp. , 1989 .

[8]  T. Wolf,et al.  Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I , 2004, nlin/0412003.

[9]  Y. Prykarpatsky.  Finite dimensional local and nonlocal reductions of one type of hydrodynamic systems , 2002 .

[10]  I. Gel'fand,et al.  Poisson brackets and the kernel of the variational derivative in the formal calculus of variations , 1976 .

[11]  I. Gel'fand,et al.  A Lie algebra structure in a formal variational calculation , 1976 .

[12]  I. Gel'fand,et al.  The calculus of jets and nonlinear Hamiltonian systems , 1978 .

[13]  A. Prykarpatsky,et al.  Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg–de Vries hydrodynamical equations , 2010, 1005.2660.

[14]  G. M.,et al.  Theory of Differential Equations , 1902, Nature.

[15]  G. Wilson.  On the quasi-hamiltonian formalism of the KdV equation , 1988 .

[16]  I. Gel'fand,et al.  The resolvent and Hamiltonian systems , 1977 .

[17]  I︠u︡. G. Borisovich.  Introduction to Differential and Algebraic Topology , 1995 .

[18]  D. Blackmore,et al.  The Lax integrability of a two-component hierarchy of the Burgers type dynamical systems within asymptotic and differential-algebraic approaches , 2013, 1309.5267.

[19]  A. Mednykh,et al.  On the volume of a spherical octahedron with symmetries , 2009 .

[20]  H. Tasso,et al.  HAMILTONIAN FORMULATION OF ODD BURGERS HIERARCHY , 1996 .

[21]  Denis Blackmore,et al.  Nonlinear Dynamical Systems of Mathematical Physics: Spectral and Symplectic Integrability Analysis , 2011 .

[22]  A. Prykarpatsky,et al.  Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds , 1998 .

[23]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .