Abstract
We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow, 1978.
P. J. Olver, Application of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.
L. M. Alonso, “Schrödinger spectral problems with energy dependent potentials as sources of nonlinear Hamiltonian evolution equations,” J. Math. Phys., 21 (1980), 2342–2349.
D. J. Kaup, “Finding eigenvalue problems for solving nonlinear evolution equations,” Progr. Theor. Phys., 54 (1975), 72–78.
V. B. Matveev and M. I. Yavor, “Solutions presque périodiques et à N-solutions de l'équation hydrodynamique nonlinéaire de Kaup,” Ann. Inst. H. Poincaré, 31 (1979), 25–41.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Svinin, A.K. Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies. Mathematical Notes 74, 91–99 (2003). https://doi.org/10.1023/A:1025023317616
Issue Date:
DOI: https://doi.org/10.1023/A:1025023317616