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Journal of Mathematical Sciences

, Volume 136, Issue 6, pp 4411–4418 | Cite as

Symmetry constraints for real dispersionless Veselov-Novikov equation

  • L. V. Bogdanov
  • B. G. Konopelchenko
  • A. Moro
Article

Abstract

Symmetry constraints for dispersionless integrable equations are discussed. It is shown that under symmetry constraints, the dispersionless Veselov-Novikov equation is reduced to the (1+1)-dimensional hydrodynamic-type systems.

Keywords

Jacobi Equation Symmetry Constraint Soliton Equation Integrable Deformation Novikov Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM (1981).Google Scholar
  2. 2.
    L. V. Bogdanov and B. G. Konopelchenko, Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type, Preprint arXiv:nlin.SI/0312013 (2003).Google Scholar
  3. 3.
    Y. Cheng and Y. S. Li, “The constraint of the Kadomtsev-Petviashvili equation and its special solutions,” Phys. Lett. A, 157, 22 (1991).MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. A. Dubrovin and S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices: differential geometry and Hamiltonian theory,” Russ. Math. Surv., 44, 35, (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    N. M. Ercolani et al., Eds., Singular Limits of Dispersive Waves, Nato Adv. Sci. Inst. Ser. B, Phys. 320, Plenum Press, New York (1994).zbMATHGoogle Scholar
  6. 6.
    E. V. Ferapontov, “Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry,” Differ. Geom. Appl., 11, 117 (1999).zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Gibbons and S. P. Tsarev, “Conformal maps and reductions of the Benney equations,” Phys. Lett. A, 258, 263 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Y. Kodama, “A method for solving the dispersionless KP equation and its exact solutions,” Phys. Lett. A, 129, 223 (1988).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Y. Kodama, “Solutions of the dispersionless Toda equation,” Phys. Lett. A, 147, 477 (1990).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Y. Kodama and J. Gibbons, “A method for solving the dispersionless KP hierarchy and its exact solutions, Phys. Lett. A, 135, No. 3, 167 (1989).MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Konopelchenko and L. Martinez Alonso, “\(\overline \partial \)-equations, integrable deformations of quasi-conformal mappings, and Whitham hierarchy,” Phys. Lett. A, 286, 161 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    B. G. Konopelchenko and L. Martinez Alonso, “Nonlinear dynamics on the plane and integrable hierarchies of infinitesimal deformations,” Stud. Appl. Math., 109, 313–336 (2002).MathSciNetCrossRefGoogle Scholar
  13. 13.
    B. G. Konopelchenko, L. Martinez Alonso, and O. Ragnisco, “The \(\overline \partial \)-approach to the dispersionless KP hierarchy,” J. Phys. A: Math. Gen., 34, 10209 (2001).Google Scholar
  14. 14.
    B. G. Konopelchenko and A. Moro, “Geometrical optics in nonlinear media and integrable equations,” J.Phys. A: Math. Gen., 37, L105–L111 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    B. G. Konopelchenko and A. Moro, “Integrable equations in nonlinear geometrical optics,” Stud. Appl. Math. (to appear); Preprint arXiv:nlin.SI/0403051 (2004).Google Scholar
  16. 16.
    B. G. Konopelchenko and U. Pinkall, “Integrable deformations of affine surfaces via Knizhnik-Veselov-Novikov equation,” Phys. Lett. A, 245, 239–245 (1998).CrossRefGoogle Scholar
  17. 17.
    B. Konopelchenko, J. Sidorenko, and W. Strampp, “(1 + 1)-Dimensional integrable systems as symmetry constraints of (2 + 1)-dimensional systems,” Phys. Lett. A, 157, 17 (1991).MathSciNetCrossRefGoogle Scholar
  18. 18.
    I. M. Krichever, “Averaging method for two-dimensional integrable equations,” Funkts. Anal. Prilozh., 22, 37 (1988).zbMATHMathSciNetGoogle Scholar
  19. 19.
    I. M. Krichever, “The τ-function of the universal Whitham hierarchy, matrix models, and topological field theories,” Commun. Pure Appl. Math., 47, 437 (1994).zbMATHMathSciNetGoogle Scholar
  20. 20.
    A. Yu. Orlov, “Vertex operator, \(\overline \partial \)-problem, symmetries, variational identities, and Hamiltonian formalism for (2 + 1) integrable systems,” in: Nonlinear and Turbulent Processes in Physics (V. Baryakhtar, Ed.), World Scientific, Singapore (1988).Google Scholar
  21. 21.
    L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).zbMATHGoogle Scholar
  22. 22.
    A. P. Veselov and S. P. Novikov, “Finite-zone two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations,” Dokl. Akad. Nauk SSSR, 279, 20 (1984).MathSciNetGoogle Scholar
  23. 23.
    V. E. Zakharov, “Benney equations and quasi-classical approximation in the inverse problem method,” Funkts. Anal. Prilozh., 14, 89 (1980).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • L. V. Bogdanov
    • 1
  • B. G. Konopelchenko
    • 2
  • A. Moro
    • 3
  1. 1.Landau Institute for Theoretical PhysicsMoscow
  2. 2.Dipartimento di FisicaUniversità degli Studi di LecceItaly
  3. 3.Dipartimento di FisicaUniversità degli Studi di LecceItaly

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