Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 096, 11 pages      arXiv:0706.0314      http://dx.doi.org/10.3842/SIGMA.2007.096

Lagrangian Approach to Dispersionless KdV Hierarchy

Amitava Choudhuri a, B. Talukdar a and U. Das b
a) Department of Physics, Visva-Bharati University, Santiniketan 731235, India
b) Abhedananda Mahavidyalaya, Sainthia 731234, India

Received June 05, 2007, in final form September 16, 2007; Published online September 30, 2007

Abstract
We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.

Key words: hierarchy of dispersionless KdV equations; Lagrangian approach; bi-Hamiltonian structure; variational symmetry.

pdf (224 kb)   ps (148 kb)   tex (14 kb)

References

  1. Zakharov V.E., Benney equations and quasiclassical approximation in the method of the inverse problem, Funct. Anal. Appl. 14 (1980), 89-98.
  2. Olver P.J., Nutku Y., Hamiltonian structures for systems of hyperbolic conservation laws, J. Math. Phys. 29 (1988), 1610-1619.
    Brunelli J.C., Dispersionless limit of integrable models, Braz. J. Phys. 30 (2000), 455-468, nlin.SI/0207042.
  3. Arik M., Neyzi F., Nutku Y., Olver P.J., Verosky J.M., Multi-Hamiltonian structure of the Born-Infeld equation, J. Math. Phys. 30 (1989), 1338-1344.
    Das A., Huang W.J., The Hamiltonian structures associated with a generalized Lax operator, J. Math. Phys. 33 (1992), 2487-2497.
    Brunelli J.C., Das A., Properties of nonlocal charges in the supersymmetric two boson hierarchy, Phys. Lett. B 354 (1995), 307-314, hep-th/9504030.
    Brunelli J.C., Das A., Supersymmetric two-boson equation, its reductions and the nonstandard supersymmetric KP hierarchy, Intern. J. Modern Phys. A 10 (1995), 4563-4599, hep-th/9505093.
    Brunelli J.C., Hamiltonian structures for the generalized dispersionless KdV hierarchy, Rev. Math. Phys. 8 (1996), 1041-1054, solv-int/9601001.
    Brunelli J.C., Das A., A Lax description for polytropic gas dynamics, Phys. Lett. A 235 (1997), 597-602, solv-int/9706005.
    Brunelli J.C., Das A., The sTB-B hierarchy, Phys. Lett. B 409 (1997), 229-238, hep-th/9704126.
    Brunelli J.C., Das A., A Lax representation for Born-Infeld equation, Phys. Lett. B 426 (1998), 57-63, hep-th/9712081.
  4. Calogero F., Degasperis A., Spectral transform and soliton, North-Holland Publising Company, New York, 1982.
  5. Olver P.J., Application of Lie groups to differential equation, Springer-Verlag, New York, 1993.
  6. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
  7. Frankel T., The geometry of physics, Cambridge University Press, UK, 1997.
  8. Ali Sk.G., Talukdar B., Das U., Inverse problem of variational calculus for nonlinear evolution equations, Acta Phys. Polon. B 38 (2007), 1993-2002, nlin.SI/0603037.
  9. Kaup D.J., Malomed B.A., The variational principle for nonlinear waves in dissipative systems, Phys. D 87 (1995), 155-159.
  10. Barcelos-Neto J., Constandache A., Das A., Dispersionless fermionic KdV, Phys. Lett. A 268 (2000), 342-351, solv-int/9910001.
  11. Zakharov V.E., Faddeev L.D., Korteweg-de Vries equation: a completely integrable Hamiltonian systems, Funct. Anal. Appl. 5 (1971), 18-27.
  12. Gardner C.S., Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system, J. Math. Phys. 12 (1971), 1548-1551.
  13. Hill E.L., Hamilton's principle and the conservation theorems of mathematical physics, Rev. Modern Phys. 23 (1951), 253-260.
  14. Gelfand I.M., Fomin S.V., Calculus of variations, Dover Publ., 2000.
  15. Chen H.H., Lee Y.C., Lin J.E., On a new hierarchy of symmetries for the Kadomtsev-Petviashvilli equation, Phys. D 9 (1983), 439-445.
  16. Orlov A.Yu., Shulman E.I., Additional symmetries for integral and conformal algebra representation, Lett. Math. Phys. 12 (1986), 171-179.
  17. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer, Berlin, 1987.

Previous article   Next article   Contents of Volume 3 (2007)