The short pulse hierarchy

We study a new hierarchy of equations containing the short pulse equation, which describes the evolution of very short pulses in nonlinear media, and the elastic beam equation, which describes nonlinear transverse oscillations of elastic beams under tension. We show that the hierarchy of equations is integrable. We obtain the two compatible Hamiltonian structures. We construct an infinite series of both local and nonlocal conserved charges. A Lax description is presented for both systems. For the elastic beam equations we also obtain a nonstandard Lax representation.

[1]  On recursion operators , 2001 .

[2]  S. Okubo,et al.  A systematic study of the Toda lattice , 1989 .

[3]  S. Lou,et al.  Symmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations , 1994 .

[4]  Christopher K. R. T. Jones,et al.  Ultra-short pulses in linear and nonlinear media , 2004, nlin/0408020.

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

[6]  J. Verosky,et al.  Negative powers of Olver recursion operators , 1991 .

[7]  M. Wadati,et al.  Nonlinear Transverse Oscillation of Elastic Beams under Tension , 1981 .

[8]  J. C. Brunelli,et al.  On an integrable hierarchy derived from the isentropic gas dynamics , 2004 .

[9]  S. Okubo,et al.  The integrability condition for dynamical systems , 1988 .

[10]  Paul Adrien Maurice Dirac,et al.  Lectures on Quantum Mechanics , 2001 .

[11]  Franco Magri,et al.  A Simple model of the integrable Hamiltonian equation , 1978 .

[12]  On the nonlocal equations and nonlocal charges associated with the Harry Dym hierarchy , 2002, nlin/0207041.

[13]  Sergei Sakovich,et al.  The Short Pulse Equation Is Integrable , 2005 .

[14]  Deformed Harry Dym and Hunter–Zheng equations , 2003, nlin/0307043.

[15]  PROPERTIES OF AN ALTERNATE LAX DESCRIPTION OF THE KdV HIERARCHY , 1995, hep-th/9501095.

[16]  C. E. Wayne,et al.  Propagation of ultra-short optical pulses in cubic nonlinear media , 2004 .

[17]  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 .

[18]  Mark Adler,et al.  On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-devries type equations , 1978 .

[19]  M. Shmakova,et al.  Hierarchy of lower Korteweg–de Vries equations and supersymmetry structure of Miura transformations , 1993 .

[20]  Kimiaki Konno,et al.  New Integrable Nonlinear Evolution Equations , 1979 .

[21]  Darryl D. Holm,et al.  A New Integrable Shallow Water Equation , 1994 .

[22]  J. C. Brunelli.  PSEUDO: applications of streams and lazy evaluation to integrable models , 2004, Comput. Phys. Commun..