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Functional Analysis and Its Applications

, Volume 45, Issue 4, pp 278–290 | Cite as

Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations

  • B. A. DubrovinEmail author
  • M. V. Pavlov
  • S. A. Zykov
Article
  • 82 Downloads

Abstract

We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete classification of irreducible solutions.

Key words

Frobenius manifold WDVV associativity equations linearly degenerate PDEs algebraic Riccati equation 

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References

  1. [1]
    G. Darboux, Leçons sur systèmes orthogonaux et les coordonnées curvilignes, Paris, 1910.Google Scholar
  2. [2]
    B. A. Dubrovin, “Completely integrable Hamiltonian systems associated with matrix finitegap operators and Abelian varieties,” Funkts. Anal. Prilozhen., 11:4 (1977), 28–41; English transl.: Functional Anal. Appl., 11: 4 (1977), 265–277.zbMATHMathSciNetGoogle Scholar
  3. [3]
    B. A. Dubrovin, “On differential geometry of strongly integrable systems of hydrodynamic type,” Funkts. Anal. Prilozhen., 24:4 (1990), 25–30; English transl.: Functional Anal. Appl., 24: 4 (1990), 280–285.MathSciNetGoogle Scholar
  4. [4]
    B. Dubrovin, “Integrable systems in topological field theory,” Nucl. Phys. B, 379:3 (1992), 627–689.CrossRefMathSciNetGoogle Scholar
  5. [5]
    B. Dubrovin, “Geometry of 2D topological field theories,” in: Integrable Systems and Quantum Groups, Montecatini, Terme, 1993, Lecture Notes in Math., vol. 1620, Springer-Verlag, Berlin, 1996, 120–348.CrossRefGoogle Scholar
  6. [6]
    B. A. Dubrovin and S.P. Novikov, “The Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogoliubov-Whitham averaging method,” Dokl. Akad. Nauk SSSR, 270 (1983), 781–785; English transl.: Sov. Math. Doklady, 27 (1983), 665–669.MathSciNetGoogle Scholar
  7. [7]
    D. F. Egorov, “A class of orthogonal systems,” Uch. Zap. Moskov. Univ., Sec. Fiz.-Mat., 18 (1901), 1–239.Google Scholar
  8. [8]
    G. A. El, A. M. Kamchatnov, M. V. Pavlov, and S. A. Zykov, “Kinetic equation for a soliton gas and its hydrodynamic reductions,” J. Nonlinear Sci., 21:2 (2011), 151–191.CrossRefMathSciNetGoogle Scholar
  9. [9]
    P. Lancaster, L. Rodman, Algebraic Riccati Equations, Clarendon Press, Oxford University Press, Oxford, 1995.zbMATHGoogle Scholar
  10. [10]
    J. W. van de Leur, R. Martini, “The construction of Frobenius manifolds from KP taufunctions,” Comm. Math. Phys., 205:3 (1999), 587–616.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    S. P. Tsarev, “Geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method.,” Izv. Akad. Nauk SSSR, Ser. Mat., 54:5 (1991), 1048–1068; English transl.: Math. USSR Izv., 37 (1991), 397–419.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • B. A. Dubrovin
    • 1
    • 2
    Email author
  • M. V. Pavlov
    • 3
  • S. A. Zykov
    • 4
    • 5
  1. 1.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly
  2. 2.Laboratory of Geometric methods in Mathematical Physics V. A. Steklov Mathematical InstituteMoscow State UniversityMoscowRussia
  3. 3.Laboratory of Geometric methods in Mathematical Physics P. N. Lebedev Physical Institute of RASMoscow State UniversityMoscowRussia
  4. 4.University of SalentoLecceItaly
  5. 5.Department of PhysicsInstitute of Metal Physics, Ural branch of RASEkaterinburgRussia

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