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Hamiltonian Perturbations at the Second-Order Approximation


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.

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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.

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Correspondence to Di Yang.

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Communicated by Nikolai Kitanine.

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Yang, D. Hamiltonian Perturbations at the Second-Order Approximation. Ann. Henri Poincaré 21, 3919–3937 (2020).

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Mathematics Subject Classification

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