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Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

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Abstract

We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.

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References

  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)

    Google Scholar 

  2. Beals R., Sattinger D., Szmigielski J.: Multipeakons and the classical moment problem. Adv. Math. 154, 229–257 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Camassa R., Holm D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Constantin A., McKean H.P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  7. Ebin D., Marsden J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92, 102–163 (1970)

    Article  MathSciNet  Google Scholar 

  8. Fuchssteiner B., Fokas A.S.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 47–66 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  9. Guieu L., Roger C.: L’Algebre et le Groupe de Virasoro: aspects geometriques et algebriques, generalisations. Publications CRM, Montreal (2007)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Hunter J.K., Saxton R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Khesin B., Misiołek G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116–144 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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. Kirillov A., Yuriev D.: Kähler geometry of the infinite dimensional homogeneous space M = Diff+ (S 1)/Rot(S 1). Funkt. Anal. Prilozh. 21, 35–46 (1987)

    Google Scholar 

  19. Lang S.: Differential Manifolds. Springer, New York (1972)

    MATH  Google Scholar 

  20. Lenells J.: Traveling wave solutions of the Camassa-Holm equation. J. Diff. Eq. 217, 393–430 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lundmark H.: Formation and dynamics of shock waves in the Degasperis-Procesi equation. J. Nonlinear Sci. 17, 169–198 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Lundmark H., Szmigielski J.: Degasperis-Procesi peakons and the discrete cubic string. Int. Math. Res. Pap. 2, 53–116 (2005)

    Article  MathSciNet  Google Scholar 

  23. Misiołek G.: A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24, 203–208 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Misiołek G.: Classical solutions of the periodic Camassa-Holm equation. Geom. Funct. Anal. 12, 1080–1104 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. O’Neill B.: Submersions and geodesics. Duke Math. J. 34, 363–373 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  28. Saxton R., Tıuglay F.: Global existence of some infinite energy solutions for a perfect incompressible fluid. SIAM J. Math. Anal. 40, 1499–1515 (2008)

    Article  MathSciNet  Google Scholar 

  29. Taylor M.: Pseudodifferential Operators and Nonlinear PDE. Birkhäuser, Boston, Boston, MA (1991)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Leibniz Universität Hannover, D-30167, Hannover, Germany

    Jonatan Lenells

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Gerard Misiołek

  3. Department of Mathematics, University of New Orleans, Lake Front, New Orleans, LA, 70148, USA

    Feride Tiğlay

  4. Section de Mathématiques, École Polytechnique Fédérale de Lausanne, CH–1015, Lausanne, Switzerland

    Feride Tiğlay

Authors
  1. Jonatan Lenells
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  2. Gerard Misiołek
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  3. Feride Tiğlay
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Correspondence to Jonatan Lenells.

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Communicated by P. Constantin

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Lenells, J., Misiołek, G. & Tiğlay, F. Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions. Commun. Math. Phys. 299, 129–161 (2010). https://doi.org/10.1007/s00220-010-1069-9

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  • DOI: https://doi.org/10.1007/s00220-010-1069-9

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