Skip to main content

Symmetry constraints for real dispersionless Veselov-Novikov equation

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.

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

References

  1. 1.

    M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM (1981).

  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).

  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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  MATH  Google 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).

    MATH  Google Scholar 

  6. 6.

    E. V. Ferapontov, “Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry,” Differ. Geom. Appl., 11, 117 (1999).

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    J. Gibbons and S. P. Tsarev, “Conformal maps and reductions of the Benney equations,” Phys. Lett. A, 258, 263 (1999).

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Y. Kodama, “A method for solving the dispersionless KP equation and its exact solutions,” Phys. Lett. A, 129, 223 (1988).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Y. Kodama, “Solutions of the dispersionless Toda equation,” Phys. Lett. A, 147, 477 (1990).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  MATH  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  MATH  Google 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).

  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).

    Article  Google 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).

    MathSciNet  Article  Google Scholar 

  18. 18.

    I. M. Krichever, “Averaging method for two-dimensional integrable equations,” Funkts. Anal. Prilozh., 22, 37 (1988).

    MATH  MathSciNet  Google 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).

    MATH  MathSciNet  Google 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).

    MATH  Google 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).

    MathSciNet  Google Scholar 

  23. 23.

    V. E. Zakharov, “Benney equations and quasi-classical approximation in the inverse problem method,” Funkts. Anal. Prilozh., 14, 89 (1980).

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

__________

Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bogdanov, L.V., Konopelchenko, B.G. & Moro, A. Symmetry constraints for real dispersionless Veselov-Novikov equation. J Math Sci 136, 4411–4418 (2006). https://doi.org/10.1007/s10958-006-0234-3

Download citation

Keywords

  • Jacobi Equation
  • Symmetry Constraint
  • Soliton Equation
  • Integrable Deformation
  • Novikov Equation