Integrable evolution equations on spaces of tensor densities and their peakon solutions

Communications in Mathematical Physics

Volumn 299, Issue 1, 2010, Pages 129-161

  Lenells, Jonatan ^a   Misiołek, Gerard ^b   Tiǧlay, Feride ^c,d  

^a LEIBNIZ UNIVERSITY HANNOVER   (Germany)

^b UNIVERSITY OF NOTRE DAME   (United States)

^c UNIVERSITY OF NEW ORLEANS   (United States)

^d EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77955550646     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-010-1069-9     Document Type: Article

References (29)

1. Arnold V.: Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses application à l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16, 319-361 (1966).
2. Beals R., Sattinger D., Szmigielski J.: Multipeakons and the classical moment problem. Adv. Math. 154, 229-257 (2000).
3. Camassa R., Holm D. D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661-1664 (1993).
4. Constantin A., McKean H. P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949-982 (1999).
5. Degasperis, A., Procesi, M.: Asymptotic integrability. In: Symmetry and Perturbation Theory (Rome 1998), River Edge, NJ: World Scientific Publishers, 1999.
6. Degasperis A., Holm D. D., Hone A. N. W.: A new integrable equation with peakon solutions. Teoret. Mat. Fiz. 133, 1463-1474 (2002).
7. Ebin D., Marsden J. E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92, 102-163 (1970).
8. Fuchssteiner B., Fokas A. S.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 47-66 (1981).
9. Guieu L., Roger C.: L'Algebre et le Groupe de Virasoro: aspects geometriques et algebriques, generalisations. Publications CRM, Montreal (2007).
10. Holm, D. D., Marsden, J. E.: Momentum maps and measure valued solutions (peakons, filaments, and sheets) of the Euler-Poincaré equations for the diffeomorphism group. In: The Breadth of Symplectic and Poisson Geometry, Progr. Math. 232, Boston, MA: Birkhäuser, 2005, pp. 203-235.
11. Hone A. N. W., Wang J. P.: Prolongation algebras and Hamiltonian operators for peakon equations. Inverse Problems 19, 129-145 (2003).
12. Holm D. D., Staley M. F.: Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dynam. Syst. 2, 323-380 (2003) (electronic).
13. Hunter J. K., Saxton R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498-1521 (1991).
14. Khesin B., Lenells J., Misiołek G.: Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann. 342, 617-656 (2008).
15. Khesin B., Misiołek G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116-144 (2003).
16. Khesin, B., Wendt, R.: The Geometry of Infinite-Dimensional Groups. Ergebnisse der Mathematik Vol. 51, New York: Springer, 2008.
17. Kirillov, A.: Infinite dimensional Lie groups: their orbits, invariants and representations. The geometry of moments. Lect. Notes in Math. 970, New York: Springer-Verlag, 1982, pp. 101-123.
18. 1). Funkt. Anal. Prilozh. 21, 35-46 (1987).
19. Lang S.: Differential Manifolds. Springer, New York (1972).
20. Lenells J.: Traveling wave solutions of the Camassa-Holm equation. J. Diff. Eq. 217, 393-430 (2005).
21. Lundmark H.: Formation and dynamics of shock waves in the Degasperis-Procesi equation. J. Nonlinear Sci. 17, 169-198 (2007).
22. Lundmark H., Szmigielski J.: Degasperis-Procesi peakons and the discrete cubic string. Int. Math. Res. Pap. 2, 53-116 (2005).
23. Misiołek G.: A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24, 203-208 (1998).
24. Misiołek G.: Classical solutions of the periodic Camassa-Holm equation. Geom. Funct. Anal. 12, 1080-1104 (2002).
25. O'Neill B.: Submersions and geodesics. Duke Math. J. 34, 363-373 (1967).
26. Ovsienko, V.: Coadjoint representation of Virasoro-type Lie algebras and differential operators on tensor-densities. In: Infinite dimensional Kähler manifolds (Oberwolfach 1995), DMV Sem. 31, Basel: Birkhäuser, 2001, pp. 231-255.
27. Ovsienko V., Tabachnikov S.: Projective Differential Geometry. Cambridge Univ. Press, Cambridge (2005).
28. Saxton R., Ti{dotless}uglay F.: Global existence of some infinite energy solutions for a perfect incompressible fluid. SIAM J. Math. Anal. 40, 1499-1515 (2008).
29. Taylor M.: Pseudodifferential Operators and Nonlinear PDE. Birkhäuser, Boston, Boston, MA (1991).