Abstract
We study a certain family of Schrödinger operators whose eigenfunctions ϕ(χ, λ) satisfy a differential equation in the spectral parameter λ of the formB(λ,∂ λ)ϕ=Θ(x)ϕ. We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class ofbispectral potentials. This extends and complements a result of Duistermaat and Grünbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler, M., Moser, J.: On a class of polynomials connected with the KdV equations. Commun. Math. Phys.61, 1–30 (1978)
Airault, H., Mc Kean, H., Moser, J.: Rational and elliptic solutions of the KdV equation and a related many body problem. Commun. Pure Appl. Math.30, 95–148 (1977)
Bishop, R., Goldberg, S.: Tensor analysis on manifolds. New York: Dover 1980
Chen, H., Lee, Y., Lin, J.: Preprint PL83-002, University of Maryland 1982
Crum, M.: Associated Sturm-Liouville systems. Quart. J. Math., Ser. 2,6, 121–127 (1955)
Darboux, G.: Leçons sur la Théorie Générale de Surface et les Applications Géométriques du Calcul Infinitésimal, Deuxième Partie. Paris, France: Gauthiers-Villars 1889
Deift, P., Trubowitz, E.: Inverse scattering on the line. CPAM32, 121–251 (1977)
Duistermaat, J.J., Grünbaum, F.A.: Differential equations in the spectral parameter. Commun. Math. Phys.103, 177–240 (1986)
Fröhlicher, A., Nijenhuis, A.: Theory of vector-valued differential forms. Part I. Indag. Math., t.18, 338–350 (1956).
Fuchssteiner, B.: Mastersymmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations. Prog. Theor. Phys.70 (6), 1508–1522 (1983)
Fuchssteiner, B.: Mastersymmetries for completely integrable systems in statistical mechanics. In: Garrido, L. (ed.), Proc. Sitges Conference. Lecture Notes in Physics, vol.216, pp. 305–315. Berlin, Heidelberg, New York: Springer 1984
Gel'fand, I.M., Dikii, L.A.: Fractional powers of operators and Hamiltonian systems. Funkt. Anal. Pril.10 (4), 13–29 (1976)
Gel'fand, I.M., Dikii, L.A.: Resolvents and Hamiltonian systems. Funkt. Anal. Pril.11(2), 11–27 (1977)
Grünbaum, F.A.: Differential equations in the spectral parameter: the higher order case. In: Proceedings of the conference on Tomographic Inverse Problems, Montpellier, pp. 307–322 (1986)
Grünbaum, F.A.: Some nonlinear evolution equations and related topics arising in medical imaging. Physica D18, 308–311 (1986)
Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962
Ince, E.L.: Ordinary differential equations. New York: Dover 1956
Lang, S.: Differential manifolds. Reading, Mass.: Addison-Wesley 1972
Lax, P.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.31, 467–490 (1968)
Levi, D.: Hierarchies of nonlinear integrable evolution equations with variable coefficients. In: Leon, Jérôme J.P. (ed.), Workshop on Nonlinear Evolution Equations and Dynamical Systems, Montpellier, p. 75–85. Singapore: World Scientific 1986
Levi, D., Ranisco, O., Sym, A.: Dressing method vs. classical Darboux transformations. Il Nuovo Cim.83B(1), 34–42 (1984)
Magri, F.: Equivalence transformations for nonlinear evolution equations. J. Math. Phys.18 (7), 1405–1411 (1977)
Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys.19, 1156–1162 (1978)
Magri, F., Kosmann-Schwarzbach, Y.: Poisson-Nijenhuis structures. To appear
Matveev, V.B., Salle, M.A., Rybin, A.V.: Darboux transformations and coherent interaction of the light pulse with two level media. Inverse Problems4, 173–183 (1988)
Oevel, G., Fuchssteiner, B., Blaszak, M.: Action-Angle representation of multisolitons by potentials of mastersymmetries. Prog. Theor. Phys.83(3), 395–413 (1990)
Oevel, W., Falck, M.: Master symmetries for finite dimensional integrable systems. Prog. Theor. Phys.75(6), 1328–1341 (1986)
Oevel, W., Fuchssteiner, B.: Explicit formulas for symmetries and conservation laws of the Kadomtsev-Petviashvili equation. Phys. Lett.88A(7), 323–327 (1982)
Olver, P.: Applications of Lie groups to differential equations. Berlin, Heidelberg, New York: Springer 1986
Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. IHES61, 5–65 (1985)
Wasow, W.: Asymptotic expansions for ordinary differential equations. New York: Dover 1987
Wright, P.E.: Darboux transformations, algebraic varieties of Grassmann manifolds, commuting flows and bispectrality. PhD thesis, University of California, Berkeley 1987
Zubelli, J.P.: Differential equations in the spectral parameter for matrix differential operators of AKNS type. PhD thesis, University of California, Berkeley, 1989
Zubelli, J.P.: Differential equations in the spectral parameter for matrix differential operators. Physica D43, 269–287 (1990)
Zubelli, J.P.: Rational solutions of nonlinear evolution equations, vertex operators and bispectrality. J. Differ. Eqs. (to appear)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
Zubelli, J.P., Magri, F. Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV. Commun.Math. Phys. 141, 329–351 (1991). https://doi.org/10.1007/BF02101509
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02101509