Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 047, 40 pages      arXiv:1808.00743      https://doi.org/10.3842/SIGMA.2019.047
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Rational KdV Potentials and Differential Galois Theory

Sonia Jiménez a, Juan J. Morales-Ruiz b, Raquel Sánchez-Cauce c and María-Ángeles Zurro c
a) Junta de Castilla y León, Salamanca, Spain
b) Departamento de Matemática Aplicada, E.T.S. Edificación, Universidad Politécnica de Madrid, Madrid, Spain
c) Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

Received September 11, 2018, in final form May 29, 2019; Published online June 25, 2019

Abstract
In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schrödinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated to the Schrödinger equation. Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.

Key words: differential Galois theory; KdV hierarchy; Schrödinger operator; Darboux transformations; spectral curves; rational solitons.

pdf (564 kb)   tex (39 kb)  

References

  1. Adler M., Moser J., On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys. 61 (1978), 1-30.
  2. Airault H., McKean H.P., Moser J., Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), 95-148.
  3. Beardon A.F., Ng T.W., Parametrizations of algebraic curves, Ann. Acad. Sci. Fenn. Math. 31 (2006), 541-554.
  4. Braverman A., Etingof P., Gaitsgory D., Quantum integrable systems and differential Galois theory, Transform. Groups 2 (1997), 31-56, arXiv:alg-geom/9607012.
  5. Brezhnev Yu.V., What does integrability of finite-gap or soliton potentials mean?, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), 923-945, arXiv:nlin.SI/0505003.
  6. Brezhnev Yu.V., Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials, in Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, Contemp. Math., Vol. 563, Amer. Math. Soc., Providence, RI, 2012, 1-31, arXiv:1011.1642.
  7. Brezhnev Yu.V., Elliptic solitons, Fuchsian equations, and algorithms, St. Petersburg Math. J. 24 (2013), 555-574.
  8. Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators, Proc. R. Soc. London Ser. A 118 (1928), 557-583.
  9. Clarkson P.A., Vortices and polynomials, Stud. Appl. Math. 123 (2009), 37-62, arXiv:0901.0139.
  10. Crum M.M., Associated Sturm-Liouville systems, Quart. J. Math. Oxford 6 (1955), 121-127.
  11. Darboux G., Sur une proposition relative aux équations linéaires, Comptes Rendus Acad. Sci. 94 (1882), 1456-1459.
  12. Darboux G., Leçons sur la théorie générale des surfaces. II, Gauthier-Villars, Paris, 1889.
  13. Ehlers F., Knörrer H., An algebro-geometric interpretation of the Bäcklund-transformation for the Korteweg-de Vries equation, Comment. Math. Helv. 57 (1982), 1-10.
  14. Gel'fand I.M., Dikii L.A., Asymptotic properties of the resolvent of Sturm-Liouville equations and the algebra of Korteweg-de Vries equations, Russian Math. Surveys 30 (1975), no. 5, 77-113.
  15. Gesztesy F., Holden H., Soliton equations and their algebro-geometric solutions. Vol. I. $(1+1)$-dimensional continuous models, Cambridge Studies in Advanced Mathematics, Vol. 79, Cambridge University Press, Cambridge, 2003.
  16. Gu C., Hu H., Zhou Z., Darboux transformations in integrable systems. Theory and their applications to geometry, Mathematical Physics Studies, Vol. 26, Springer, Dordrecht, 2005.
  17. Hermite C., Sur l'équation de Lamé, Oeuvres of Charles Hermite. III, Gauthier-Villars, Paris, 1912.
  18. Jiménez S., Morales-Ruiz J.J., S'anchez-Cauce R., Zurro M.-A., Differential Galois theory and Darboux transformations for integrable systems, J. Geom. Phys. 115 (2017), 75-88.
  19. Marshall I., Semenov-Tian-Shansky M., Poisson groups and differential Galois theory of Schroedinger equation on the circle, Comm. Math. Phys. 284 (2008), 537-552, arXiv:0710.5456.
  20. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991.
  21. Morales-Ruiz J.J., Rueda S.L., Zurro M.-A., Algebro-geometric solitonic solutions and differential Galois theory, arXiv:1708.00431.
  22. Morales-Ruiz J.J., Rueda S.L., Zurro M.-A., Factorization of KdV Schrödinger operators using differential subresultants, arXiv:1902.05443.
  23. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1986.
  24. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  25. Wilson G., On the quasi-Hamiltonian formalism of the KdV equation, Phys. Lett. A 132 (1988), 445-450.

Previous article  Next article  Contents of Volume 15 (2019)