Skip to main content

Hamiltonian Perturbations at the Second-Order Approximation

Abstract

Integrability condition of Hamiltonian perturbations of integrable Hamiltonian PDEs of hydrodynamic type up to the second-order approximation is considered. Under a nondegeneracy assumption, we show that the Hamiltonian perturbation at the first-order approximation is integrable if and only if it is trivial, and that under a further assumption, the Hamiltonian perturbation at the second-order approximation is integrable if and only if it is quasi-trivial.

This is a preview of subscription content, access via your institution.

References

  1. Ablowitz, M.J., Fokas, A.S., Musslimani, Z.H.: On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  2. Arsie, A., Lorenzoni, P., Moro, A.: On integrable conservation laws. Proc. R. Soc. A Math. Phys. Eng. Sci. 471, 20140124 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Ashton, A.C., Fokas, A.S.: A non-local formulation of rotational water waves. J Fluid Mech. 689, 129–148 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  4. Degiovanni, L., Magri, F., Sciacca, V.: On deformation of Poisson manifolds of hydrodynamic type. Commun. Math. Phys. 253, 1–24 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  5. Dubrovin, B.: Geometry of 2D Topological Field Theories. In: Francaviglia, M., Greco, S. (eds.) Integrable Systems and Quantum Groups (Montecatini Terme, 1993). Springer Lecture Notes in Math, vol. 1620, pp. 120–348. Springer, Berlin (1996)

    Chapter  Google Scholar 

  6. Dubrovin, B.: Hamiltonian perturbations of hyperbolic systems of conservation laws, II: Universality of critical behaviour. Comm. Math. Phys. 267, 117–139 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  7. Dubrovin, B.: On universality of critical behaviour in Hamiltonian PDEs. Amer. Math. Soc. Transl. 224, 59–109 (2008)

    MathSciNet  Article  Google Scholar 

  8. Dubrovin, B.: Hamiltonian Perturbations of Hyperbolic PDEs and Applications. Lecture notes in School/Workshop on Integrable Systems and Scientific Computing, ICTP (2009)

  9. Dubrovin, B.: Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions. In: Sidoravicius, V. (eds) New Trends in Mathematical Physics. Springer, Dordrecht, pp. 231–276 (2009)

  10. Dubrovin, B.: Hamiltonian PDEs: deformations, integrability, solutions. J. Phys. A Math. Theor. 43, 434002 (2010)

    ADS  MathSciNet  Article  Google Scholar 

  11. Dubrovin, B.: Gromov–Witten invariants and integrable hierarchies of topological type. Topology, Geometry, Integrable Systems, and Mathematical Physics. Amer. Math. Soc. Transl. Ser 2, 141–171 (2014)

    MATH  Google Scholar 

  12. Dubrovin, B., Liu, S.-Q., Zhang, Y.: On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasi-Triviality of bi-Hamiltonian perturbations. Commun. Pure. Appl. Math. 59, 559–615 (2006)

    MathSciNet  Article  Google Scholar 

  13. Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs. Adv. Math. 293, 382–435 (2016)

    MathSciNet  Article  Google Scholar 

  14. Dubrovin, B., Novikov, S.P.: The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method. Soviet Math. Dokl. 270, 665–669 (1983)

    MathSciNet  MATH  Google Scholar 

  15. Dubrovin, B., Novikov, S.P.: On Poisson brackets of hydrodynamic type. Soviet Math. Dokl. 279, 294–297 (1984)

    MathSciNet  MATH  Google Scholar 

  16. Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. Preprint arXiv:math/0108160 (2001)

  17. Dyachenko, A.I., Kachulin, D.I., Zakharov, V.E.: Collisions of two breathers at the surface of deep water. Nat. Hazards Earth Syst. Sci. 13, 3205 (2013)

    ADS  Article  Google Scholar 

  18. Getzler, E.: A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111, 535–560 (2002)

    MathSciNet  Article  Google Scholar 

  19. Haantjes, J.: On Xm-forming sets of eigenvectors. Indagat. Math. 17, 158–162 (1955)

    MathSciNet  Article  Google Scholar 

  20. Liu, S.-Q., Zhang, Y.: On quasi-triviality and integrability of a class of scalar evolutionary PDEs. J. Geom. Phys. 57, 101–119 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  21. Liu, S.-Q., Wu, C.-Z., Zhang, Y.: On properties of Hamiltonian structures for a class of evolutionary PDEs. Lett. Math. Phys. 84, 47–63 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  22. Serre, D.: Systems of Conservation Laws 2: Geometric Structures, Oscilliations, and Initial-Boundary Value Problems. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  23. Tsarev, S.: The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method. Math. USSR Izv. 37, 397–419 (1991)

    MathSciNet  Article  Google Scholar 

  24. Whitham, G.B.: Linear and Nonlinear Waves. Wiley Interscience, New York (1974)

    MATH  Google Scholar 

  25. Zakharov, V.E.: Stability of periodic waves of finte amplitude on the surface of a deep fluid. Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki 9, 86–94 (1968)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank Youjin Zhang, Si-Qi Liu and Boris Dubrovin for their advisings and helpful discussions. He is grateful to Youjin Zhang for suggesting the question about two-component water wave equations, and to Giordano Cotti and Mao Sheng for helpful discussions; he also wishes to thank Boris Dubrovin for suggesting the general question. Part of the work was done at Tsinghua University and at SISSA; we thank Tsinghua University and SISSA for excellent working conditions and financial supports. The work was partially supported by PRIN 2010-11 Grant “Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions” of Italian Ministry of Universities and Researches, by the Marie Curie IRSES project RIMMP, and by a starting research grant from University of Science and Technology of China.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Di Yang.

Additional information

Communicated by Nikolai Kitanine.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yang, D. Hamiltonian Perturbations at the Second-Order Approximation. Ann. Henri Poincaré 21, 3919–3937 (2020). https://doi.org/10.1007/s00023-020-00962-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-020-00962-w

Mathematics Subject Classification

  • 37K10
  • 37J30
  • 35Q53
  • 37L50