Skip to main content
SpringerLink
Log in

On the classification of conditionally integrable evolution systems in (1 + 1) dimensions

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We generalize earlier results of Fokas and Liu and find all locally analytic (1 + 1)-dimensional evolution equations of order n that admit an N-shock-type solution with N ≤ n + 1. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all (1+1)-dimensional evolution systems u t = F (x, t, u, ∂u/∂x,..., ∂n u/∂ x n) that are conditionally invariant under a given generalized (Lie-Bäcklund) vector field Q(x, t, u, ∂u/∂x,..., ∂k u/∂x k)∂/∂u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. E. Adler, B. Gürel, M. Gürses, and I. Habibullin, “Boundary conditions for integrable equations,” J. Phys. A: Math. Gen., 30, 3505–3513 (1997).

    Article  Google Scholar 

  2. A. Andreytsev, “Classification of systems of nonlinear evolution equations admitting higher-order conditional symmetries,” in: Proc. 4th Int. Conf. Symmetry in Nonlinear Mathematical Physics (Kyiv, 2001), Part 1, Institute of Mathematics, Kyiv (2002), pp. 72–79.

    Google Scholar 

  3. P. Basarab-Horwath and R. Z. Zhdanov, “Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries,” J. Math. Phys., 42, No. 1, 376–389 (2001).

    Article  MathSciNet  Google Scholar 

  4. A. V. Bocharov et al., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (I. S. Krasil’shchik and A. M. Vinogradov, Eds.), Amer. Math. Soc., Providence, Rhode Island (1998).

    Google Scholar 

  5. A. S. Fokas and Q. M. Liu, “Generalized conditional symmetries and exact solutions of non-integrable equations,” Theor. Math. Phys., 99, 571–582 (1994).

    Article  MathSciNet  Google Scholar 

  6. A. S. Fokas and Q. M. Liu, “Nonlinear interaction of traveling waves of nonintegrable equations,” Phys. Rev. Lett., 72, 3293–3296 (1994).

    Article  MathSciNet  Google Scholar 

  7. N. Kamran, R. Milson, and P. Olver, “Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems,” Adv. Math., 156, 286–319 (2000).

    Article  MathSciNet  Google Scholar 

  8. A. V. Mikhailov, A. B. Shabat, and V. V. Sokolov, “The symmetry approach to classification of integrable equations,” in: What is integrability?, Springer-Verlag, Berlin (1991), pp. 115–184.

    Google Scholar 

  9. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1986).

    Google Scholar 

  10. J. Rubin and P. Winternitz, “Point symmetries of conditionally integrable nonlinear evolution equations,” J. Math. Phys., 31, 2085–2090 (1990).

    Article  MathSciNet  Google Scholar 

  11. A. V. Samokhin, “Full symmetry algebra for ODEs and control systems,” Acta Appl. Math., 72, Nos. 1–2, 87–99 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. V. Samokhin, “Symmetries of linear and linearizable systems of differential equations,” Acta Appl. Math., 56, 253–300 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Sergyeyev, “Constructing conditionally integrable evolution systems in (1 + 1) dimensions: a generalization of invariant modules approach,” J. Phys. A: Math. Gen., 35, 7653–7660 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  14. S. R. Svirshchevskii, “Lie-Bäcklund symmetries of linear ODEs and invariant linear spaces,” in: Modern Group Analysis, MFTI, Moscow (1993), pp. 75–83.

    Google Scholar 

  15. R. Z. Zhdanov, “Conditional Lie-Bäcklund symmetry and reduction of evolution equations,” J. Phys. A: Math. Gen., 28, 3841–3850 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Z. Zhdanov, “Higher conditional symmetries and reduction of initial value problems for nonlinear evolution equations,” in: Proc. Int. Conf. Symmetry in Nonlinear Mathematical Physics (Kyiv, 1999), Part 1, Institute of Mathematics, Kyiv (2000), pp. 255–263.

    Google Scholar 

  17. R. Z. Zhdanov, “Higher conditional symmetry and reduction of initial value problems,” Nonlinear Dynamics, 28, 17–27 (2002).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Silesian University in Opava, Opava, Czech Republic

    A. Sergyeyev

Authors
  1. A. Sergyeyev
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Sergyeyev, A. On the classification of conditionally integrable evolution systems in (1 + 1) dimensions. J Math Sci 136, 4392–4400 (2006). https://doi.org/10.1007/s10958-006-0232-5

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-006-0232-5

Keywords

Search

Navigation