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Journal of Nonlinear Science

, Volume 28, Issue 5, pp 1843–1864 | Cite as

The WDVV Associativity Equations as a High-Frequency Limit

  • Maxim V. PavlovEmail author
  • Nikola M. Stoilov
Article
  • 119 Downloads

Abstract

In this paper, we present a new “Hamiltonian” approach for construction of integrable systems. We found an intermediate dispersive system of a Camassa–Holm type. This three-component system has simultaneously a high-frequency (short wave) limit equivalent to the remarkable WDVV associativity equations and a dispersionless (long wave) limit coinciding with a dispersionless limit of the Yajima–Oikawa system.

Keywords

Associativity equations Integrable dispersive systems Lax pair Dispersionless limit High-frequency limit Bi-Hamiltonian structure Hydrodynamic type system Conservation law 

Mathematics Subject Classification

37K05 37K10 37K20 37K25 

Notes

Acknowledgements

The authors would like to thank Michal Marvan for independently verifying some computations, as well as Oleksandr Chvartatskyi, Eugene Ferapontov and Folkert Müller-Hoissen for the enlightening discussions and important remarks. We are grateful to the anonymous referees, whose comments helped us to improve the presentation of our results. Both authors would like to thank the Mathematical Institute at the University of Göttingen as well as the Max-Plank Institute for Dynamics and Self-Organization for their hospitality. MVP’s work was partially supported by the grant from the Presidium of RAS “Fundamental Problems of Nonlinear Dynamics” and by the RFBR Grant 16-51-55012 China. NMS’s work was partially supported by the Marie Curie Actions Intra-European Fellowship HYDRON (FP7-PEOPLE-2012-IEF, Project Number 332136) and the EU projects PARI and FEDER.

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Authors and Affiliations

  1. 1.Lebedev Physical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.Institut de Mathématiques de BourgogneUniversité de BourgogneDijon CedexFrance

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