Abstract
We find the inverse and direct recursion operator for the intrinsic generalized sine-Gordon equation in any number n > 2 of independent variables. Among the flows generated by the direct operator we identify a higher-dimensional analogue of the pmKdV equation.
Article PDF
References
Yu. Aminov, “On immersions of regions of the n-dimensional Lobachevsky space into (2n − 1)-dimensional Euclidean space,” Dokl. Akad. Nauk SSSR, 236, 521–524 (1977).
H. Baran and M. Marvan, “A conjecture concerning nonlocal terms of recursion operators,” in this issue.
J. L. Barbosa, W. Ferreira, and K. Tenenblat, “Submanifolds of constant sectional curvature in pseudo-Riemannian manifolds,” Ann. Global. Anal. Geom., 14, 381–401 (1996).
R. Beals and K. Tenenblat, “An intrinsic generalization for the wave and sine-Gordon equations,” in: Differential Geometry (B. Lawson et al., eds.), Pitman Monogr., 52, Longman, (1991), pp. 25–46.
M. Boiti, J. J.-P. Léon, and F. Pempinelli, “Canonical and noncanonical recursion operators in multidimensions,” Stud. Appl. Math., 78, 1–19 (1988).
P. T. Campos and K. Tenenblat, “Bäcklund transformations for a class of systems of differential equations,” Geom. Funct. Anal., 4, 270–287 (1994).
J. Cieśliński and Yu. A. Aminov, “A geometric interpretation of the spectral problem for the generalized sine-Gordon system,” J. Phys. A: Math. Gen., 34, L153–L159 (2001).
J. Douglas, “Solution of the inverse problem of the calculus of variations,” Trans. Amer. Math. Soc., 50, 71–128 (1941).
W. Ferreira, “On metrics of constant sectional curvature,” Mat. Contemp., 9, 91–110 (1995).
D. Ferus and F. Pedit, “Isometric immersions of space forms and soliton theory,” Math. Ann., 305, 329–342 (1996).
A. S. Fokas and P. M. Santini, “Recursion operators and bi-Hamiltonian structure in multidimensions, I,” Commun. Math. Phys., 115, 375–419 (1988).
A. S. Fokas and P. M. Santini, “Recursion operators and bi-Hamiltonian structure in multidimensions, II,” Commun. Math. Phys., 116, 449–474 (1988).
C. H. Gu and H. S. Hu, “Explicit solutions to the intrinsic generalization for the wave and sine-Gordon equations,” Lett. Math. Phys., 29, 1–11 (1993).
G. A. Guthrie, “Recursion operators and nonlocal symmetries,” Proc. Roy. Soc. London A, 446, 107–114 (1994).
N. G. Khor’kova, “Conservation laws and nonlocal symmetries,” Mat. Zametki, 44, 134–144 (1988).
I. S. Krasil’shchik and A. M. Vinogradov, eds., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Transl. Math. Monogr., 182, Amer. Math. Soc., Providence, Rhode Island (1999).
I. S. Krasilshchik and A. M. Vinogradov, “Nonlocal symmetries and the theory of coverings: An addendum to A. M. Vinogradov’s ‘Local symmetries and conservation laws’,” Acta Appl. Math., 2, 79–96 (1984).
I. S. Krasil’shchik and A. M. Vinogradov, “Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations,” Acta Appl. Math., 15, 161–209 (1989).
S. Kumei, “Invariance transformations, invariance group transformations, and invariance groups of the sine-Gordon equations,” J. Math. Phys., 16, 2461–2468 (1975).
M. Marvan, “Another look on recursion operators,” in: Differential Geometry and Applications, Proc. Conf. Brno, 1995, Masaryk Univ., Brno (1987), pp. 393–402; http://www.emis.de/proceedings.
M. Marvan, “Some local properties of Bäcklund relations,” Acta Appl. Math., 54, 1–25 (1998).
M. Marvan, “Scalar second-order evolution equations possessing an irreducible sl 2-valued zero-curvature representation,” J. Phys. A: Math. Gen., 35, 9431–9439 (2002).
M. Marvan and A. Sergyeyev, “Recursion operator for the Nizhnik-Veselov-Novikov equation,” J. Phys. A: Math. Gen., 36, L87–L92 (2003).
P. J. Olver, “Evolution equations possessing infinitely many symmetries,” J. Math. Phys., 18, 1212–1215 (1977).
A. Sergyeyev, “Why nonlocal recursion operators produce local symmetries: new results and applications,” J. Phys. A: Math. Gen., 38, 3397–3407 (2005).
K. Tenenblat, “A note on solutions for the intrinsic generalized wave and sine-Gordon equation,” J. Math. Anal. Appl., 166, 288–301 (1992).
K. Tenenblat and P. Winternitz, On the symmetry groups of the intrinsic generalized wave and sine-Gordon equations,juvyd J. Math. Phys., 34, 3527–3542 (1993).
K. Tenenblat and C. L. Terng, “A higher dimension generalization of the sine-Gordon equation and its Bäcklund transformation,” Bull. Amer. Math. Soc., 1, 589–593 (1979).
C. L. Terng, “A higher dimension generalization of the sine-Gordon equation and its soliton theory,” Ann. Math., 111, 491–510 (1980).
C. L. Terng, “Soliton equations and differential geometry,” J. Differ. Geom., 45, 407–445 (1997).
V. E. Zakharov and B. G. Konopelchenko, “On the theory of recursion operator,” Commun. Math. Phys., 94, 483–509 (1984).
Z. X. Zhou, “Darboux transformations for the twisted so(p, q) system and local isometric immersions of space forms,” Inverse Probl., 14, 1353–1370 (1998).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 117–128, 2006.
Rights and permissions
About this article
Cite this article
Marvan, M., Pobořil, M. Recursion operator for the intrinsic generalized sine-Gordon equation. J Math Sci 151, 3151–3158 (2008). https://doi.org/10.1007/s10958-008-9024-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-008-9024-4