Skip to main content

A differential-geometric criterion of the kinematic integrability of nonlinear differential equations

Abstract

A theorem on the existence of a G-representation and a differential-geometric criterion of the kinematic integrability for nonlinear differential equations from the Λ2-G-classes is proved. Examples of zero-curvature representations and metrics for some equations of mathematical physics are presented.

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

References

  1. 1.

    E. G. Poznyak and A. G. Popov, “Lobachevsky geometry and equations of mathematical physics,” Dokl. Ross. Akad. Nauk, 332, No. 4, 418–421 (1993).

    Google Scholar 

  2. 2.

    L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in the Soliton Theory [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  3. 3.

    S. A. Zadadaev, Λ 2-Representations of equations of mathematical physics and their applications, Thesis, Moscow State Univ. (1998).

  4. 4.

    A. Cartan, Differential Calculus. Differential Forms [Russian translation], Mir, Moscow (1971).

    MATH  Google Scholar 

  5. 5.

    V. S. Malakhovsky, An Introduction to the Theory of Exterior Forms [in Russian], Kaliningrad (1980).

Download references

Author information

Affiliations

Authors

Additional information

__________

Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tikhomirov, D.V., Zadadaev, S.A. A differential-geometric criterion of the kinematic integrability of nonlinear differential equations. J Math Sci 141, 1075–1080 (2007). https://doi.org/10.1007/s10958-007-0035-3

Download citation

Keywords

  • Soliton
  • Laplace Equation
  • Gaussian Curvature
  • Nonlinear Differential Equation
  • Heat Conduction Equation