Communications in Mathematical Physics

, Volume 267, Issue 1, pp 117–139 | Cite as

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour

  • Boris DubrovinEmail author


Hamiltonian perturbations of the simplest hyperbolic equation u t + a(u) u x = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.


Soliton Conservation Laws Poisson Bracket Hyperbolic System Approximate Symmetry 
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  1. 1.
    Arnold V.I., Gusein-Zade S.M., Varchenko A.N. (1985) Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts. Monographs in Mathematics 82. Birkhäuser Boston, Inc., Boston, MAGoogle Scholar
  2. 2.
    Baikov, V.A., Gazizov, R.K., Ibragimov, N.Kh. Approximate symmetries and formal linearization. PMTF 2, 40–49 (1989) (In Russian)Google Scholar
  3. 3.
    Bressan, A. One dimensional hyperbolic systems of conservation laws. In: Current developments in mathematics, 2002, Somerville, MA: Int. Press, 2003, pp. 1–37Google Scholar
  4. 4.
    Brézin É. Marinari E., Parisi G. (1990) A nonperturbative ambiguity free solution of a string model. Phys. Lett. B 242, 35–38CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Camassa R., Holm D.D. (1993) An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664CrossRefADSMathSciNetzbMATHGoogle Scholar
  6. 6.
    Degiovanni, L., Magri, F., Sciacca,V. On deformation of Poisson manifolds of hydrodynamic type. Commun. Math. Phys. 253, no. 1, 1–24 (2005)Google Scholar
  7. 7.
    Dickey L.A. (2003) Soliton equations and Hamiltonian systems. Second edition. Advanced Series in Mathematical Physics 26. World Scientific Publishing Co., Inc., River Edge NJzbMATHGoogle Scholar
  8. 8.
    Dobrokhotov, S., Pankrashkin, K., Semenov, E. On Maslov’s conjecture on the structure of weak point singularities of the shallow water equations. Dokl. Akad. Nauk 379, no. 2, 173–176 (2001); English translation: Doklady Math. 64, 127–130 (2001)Google Scholar
  9. 9.
    Dubrovin, B., Zhang, Y. Normal forms of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants., 2001Google Scholar
  10. 10.
    Dubrovin, B., Liu, S.-Q., Zhang, Y. On hamiltonian perturbations of hyperbolic systems of conservation laws, I: quasitriviality of bihamiltonian perturbations. Comm. Pure and Appl. Math. 59, 559–615 (2006)Google Scholar
  11. 11.
    Dubrovin, B., Novikov, S.P. Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method. Dokl. Akad. Nauk SSSR 270, no. 4, 781–785 (1983); English translation: Soviet Math. Dokl. 27, 665–669 (1983)Google Scholar
  12. 12.
    El G.A. (2005) Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos 15, 037103CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Faddeev L.D., Takhtajan L.A. (1987) Hamiltonian methods in the theory of solitons. Springer Series in Soviet Mathematics, Springer-Verlag, BerlinzbMATHGoogle Scholar
  14. 14.
    Fokas A.S. (1995) On a class of physically important integrable equations. Physica D 87, 145–150CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Getzler E. (2002) A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111, 535–560CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Grava, T., Klein, C. Numerical solution of the small disperion limit of the KdV equation and Whitham equations., 2005Google Scholar
  17. 17.
    Gurevich A., Meshcherkin A. (1984) Expanding self-similar discontinuities and shock waves in dispersive hydrodynamics. Sov. Phys. JETP 60, 732–740Google Scholar
  18. 18.
    Gurevich A., Pitaevski L. (1974) Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP Lett. 38, 291–297Google Scholar
  19. 19.
    Hou T.Y., Lax P.D. (1991) Dispersive approximations in fluid dynamics. Comm. Pure Appl. Math. 44, 1–40CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Kapaev, A.A.Weakly nonlinear solutions of the equation \({\rm P}_1^2\). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187 (1991), Differentsialnaya Geom. Gruppy Li i Mekh. 12, 88–109, 172–173, 175; translation in J. Math. Sci. 73, no. 4, 468–481 (1975)Google Scholar
  21. 21.
    Khesin B., Misiołek G. (2003) Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176, 116–144CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Kodama, Y., Mikhailov, A. Obstacles to asymptotic integrability. In: Algebraic aspects of integrable systems, Progr. Nonlinear Differential Equations Appl. 26, Boston, MA: Birkhäuser, 1997, pp. 173–204Google Scholar
  23. 23.
    Kudashev V., Suleimanov B. (1996) A soft mechanism for the generation of dissipationless shock waves. Phys. Lett. A 221, 204–208CrossRefADSGoogle Scholar
  24. 24.
    Lax, P., Levermore, D. The small dispersion limit of the Korteweg-de Vries equation. I, II, III. Comm. Pure Appl. Math. 36, 253–290, 571–593, 809–829 (1983)Google Scholar
  25. 25.
    Lax, P. D., Levermore, C.D., Venakides, S. The generation and propagation of oscillations in dispersive initial value problems and their limiting behavior. In: Important developments in soliton theory, Springer Ser. Nonlinear Dynam., Berlin: Springer, 1993, pp. 205–241Google Scholar
  26. 26.
    Liu S.Q., Zhang Y. (2005) Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 54, 427–453CrossRefADSMathSciNetzbMATHGoogle Scholar
  27. 27.
    Liu, S.Q., Zhang, Y. On quasitriviality of a class of scalar evolutionary PDEs. J. Geom. Phys., 2006, to appear., 2005Google Scholar
  28. 28.
    Lorenzoni P. (2002) Deformations of bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 44, 331–375CrossRefADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Potëmin, G. Algebro-geometric construction of self-similar solutions of the Whitham equations. Uspekhi Mat. Nauk 43, no. 5(263), 211–212 (1988); translation in Russ. Math. Surv. 43, 252–253 (1988)Google Scholar
  30. 30.
    Strachan I.A.B. (2003) Deformations of the Monge/Riemann hierarchy and approximately integrable systems. J. Math. Phys. 44, 251–262CrossRefADSMathSciNetzbMATHGoogle Scholar
  31. 31.
    Tsarëv, S.P. The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Izv. Akad. Nauk SSSR Ser. Mat. 54, no. 5, 1048–1068 (1990); English translation in Math. USSR-Izv. 37, 397–419 (1991)Google Scholar
  32. 32.
    Zabusky N.J., Kruskal M.D. (1965) Interaction of “solitons" in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243CrossRefADSGoogle Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.SISSATriesteItaly

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