| Title | EULER-POINCARE FORMALISM OF COUPLED KDV TYPE SYSTEMS AND DIFFEOMORPHISM GROUP ON S1 |
| Publication Type | Journal Article |
| Year of Publication | 2005 |
| Authors | Guha, P |
| Secondary Title | Communications in Applied Analysis |
| Volume | 9 |
| Issue | 1 |
| Start Page | 131 |
| Pagination | 145 |
| Date Published | 01/2005 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 53A07, 53B50 |
| Abstract | In this paper we show that almost all the coupled KdV equations follow from the geodesic flows of L2 metric on the semidirect product space
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| URL | http://www.acadsol.eu/en/articles/9/1/9.pdf |
| Short Title | Euler-Poincar ́e Formalism |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] M. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, 149, Cambridge University Press, Cambridge (1991).
[2] M. Alber, R. Camassa, Yu. Fedorov, D. Holm and J.E. Marsden, The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE’s of shallow water and Dym type, Comm. Math. Phys., 221 (2001), no. 1, 197–227. [3] M. Antonowicz and A. Fordy, Coupled KdV equation with multi-Hamiltonian structures, Physica, 28D (1987), 345-357. [4] E. Arbarello, C. De Concini, V.G. Kac and C. Procesi, Moduli space of curves and representation theory, Comm. Math. Phys., 117 (1988), 1-36. [5] V.I. Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Graduate Texts in Mathematics, Volume 60, Springer-Verlag, 1989. [6] S. Baker, V.Z. Enolskii and A.P. Fordy, Integrable quartic potentials and coupled KdV equations, Phys. Lett. A, 201 (1995), no. 2-3, 167–174. [7] R. Camassa and D. Holm, A completely integrable dispersiveshallow water equation with peaked solutions, Phys. Rev. Lett., 71 (1993), 1661-1664. [8] H. Cendra, D. Holm, J. Marsden and T. Ratiu, Lagrangian reduction, the Euler-Poincare equations and semidirect products, To appear in: AMS Arnold Volume II, and all other references therein. [9] D. Ebin and J. Marsden, Groups of diffeomorphisms and themotion of an incompressible fluid, Ann. Math., 92 (1970), 102-163. [10] A.S. Fokas and R.L. Anderson, On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems, J. Math. Phys., 23 (1982), 1066–1073. [11] B. Fuchssteiner and A.S. Fokas, Symplectic structures, their B ̈acklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47–66. [12] P. Guha, Diffeomorphism with some Sobolev metric, geodesic flow and integrable systems, Journal of Dynamical Systems and Control Theory, 8 (2002), 529. [13] P. Guha, Integrable Geodesic Flows on the (Super)extension of Bott-Virasoro Group, Letters in Mathematical Physics, 52 (2000), 311-328. [14] P. Guha, Geometry of Kaup-Newell equation, Reports in Mathematical Physics, 50 (2002), 1-13. [15] J. Harnad and B.A. Kupershmidt, Symplectic geometries on T ∗ G, Hamiltonian group actions and integrable systems, J. Geom. Phys., 16 (1995), 168-206. [16] A.A. Kirillov, Infinite dimensional Lie groups: their orbits, invariants and representations. The geometry of moments, Lect. Notes in Math., 970, Springer-Verlag (1982), 101-123. [17] A.A. Kirillov, Merits and demerits of the orbit method, Bull. Amer. Math. Soc. (N.S.), 36 (1999), no. 4, 433–488. [18] B.A. Kupershmidt, Mathematics of dispersive water waves, Comm. Math. Phys., 99 (1985), no. 1, 51–73. [19] V.F. Lazutkin and T.F. Penkratova, Normal forms and versal deformations for Hill’s equation, Funkcional. Anal. i Prilovzen., 9 (1975), no. 4, 41–48, In Russian. [20] P. Marcel, V. Ovsienko and C. Roger, Extension of the Virasoro and Neveu-Schwartz algebras and generalized Sturm-Liouvilleoperators, Lett. Math. Phys., 40 (1997), 31-39. [21] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, Springer-Verlag, 1994; Second Edition, 1999.
[22] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. [23] Y. Nutku and O. Oguz, Bi-Hamiltonian structure of a pair of coupled KdV equations, Nuovo Cimento B, 11 105 (1990), no. 12, 1381–1383. [24] V. Yu. Ovsienko and B.A. Khesin, KdV super equation as an Euler equation, Funct. Anal. Appl., 21 (1987), 329-331. [25] V. Yu. Ovsienko and C. Rogers, Extension of Virasoro group and Virasoro algebra by modules of tensor densities on S 1 , Funktsional. Anal. i Prilozhen., 30 (1996), no. 4, 86–88; Translation in: Funct. Anal. Appl., 30 (1996), no. 4, 290–291. [26] S. Shkoller, Geometry and the curvature of diffeomorphismgroups with H 1 metric and mean hydrodynamics, Math. AP/9807078. [27] G. Segal, Unitary representations of some infinite dimensional groups, Comm. Math. Phys., 80 (1981), 301-342. [28] M. Wadati, K. Konno and I. Ichikawa, A generalization of inverse scattering method, J. Phys. Soc. Japan, 46 (1979), no. 6, 1965–1966 [29] J.P. Wang, A list of 1+1 dimensional integrable equations and their properties. Recent advances in integrable systems (Kowloon, 2000), J. Nonlinear Math. Phys., 9 (2002), suppl. 1, 213–233. [30] E. Witten, Coadjoint orbits of the Virasoro group, Comm. Math. Phys., 114 (1988), no. 1, 1–53. |