zbMATH — the first resource for mathematics

Integrable systems and isomonodromy deformations. (English) Zbl 0795.35077
The authors consider three classes of deformation problems associated with integrable systems. The first two are related to the scaling invariance of the \(n\times n\) AKNS hierarchies and Gel’fand-Dikij hierarchies. The corresponding isomonodromy equations are generally of order greater than 2, so they are not the classical Painlevé transcendents. However they do have the Painlevé property, in fact the stronger property that any solution has a single-valued meromorphic extension to the entire plane.
The third class of monodromy problems considered in the paper is connected with the equation of M. R. Douglas [Phys. Lett. B 238, 176-180 (1990)] in two-dimensional quantum gravity. These problems are obtained by replacing \(\dot L\) by \(\hbar I\) in the Lax equation for the Gel’fand-Dikij hierarchies. The monodromy data is constructed in each case.
Reviewer: P.Holod (Kiev)

35Q15 Riemann-Hilbert problems in context of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
PDF BibTeX Cite
Full Text: DOI
[1] Ablowitz, M.J.; Segur, H., Exact linearization of a Painlevé transcendent, Phys. rev. lett., 38, 1103-1106, (1977)
[2] Ablowitz, M.J.; Segur, H., Solitons and the inverse scattering transform, (1981), SIAM Philadelphia · Zbl 0299.35076
[3] Airault, H., Rational solutions of Painlevé equations, Studies in applied mathematics, 61, 31-53, (1979) · Zbl 0496.58012
[4] Beals, R.; Coifman, R.R., Scattering and inverse scattering for first order systems, Commun. pure appl. math., 87, 39-90, (1984) · Zbl 0514.34021
[5] Coddington, E.A.; Levinson, N., Theory of ordinary differential equations, (1955), McGraw-Hill New York · Zbl 0042.32602
[6] Douglas, M., Strings in less than one dimension and the generalized KdV hierarchies, Phys. lett. B, 238, 176-180, (1990) · Zbl 1332.81211
[7] Flaschka, H.; Newell, A.C., Monodromy and spectrum preserving deformations, I, Commun. math. phys., 76, 65-116, (1980) · Zbl 0439.34005
[8] Flaschka, H.; Newell, A.C., Multi-phase similarity solutions of integrable evolution equations, Physica D, 3, 203-221, (1981) · Zbl 1194.37102
[9] A.S. Fokas and X. Zhou, Integrability of Painlevé transcendents, to appear.
[10] Gambier, E., Sur LES équations differentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, Acta math., 33, 1-55, (1910) · JFM 40.0377.02
[11] Its, A.R.; Novokshenov, V.Y., The isomonodromic deformation method in the theory of Painlevé equations, () · Zbl 0592.34001
[12] Jimbo, M.; Miwa, T.; Ueno, K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, Physica D, 2, 306-352, (1981) · Zbl 1194.34167
[13] Jimbo, M.; Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Physica D, 2, 407-448, (1981) · Zbl 1194.34166
[14] Jimbo, M.; Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III, Physica D, 4, 26-46, (1983) · Zbl 1194.34169
[15] Malgrange, B., Équations differentielles à coefficients polynomiaux, (1991), Birkhaüser Boston · Zbl 0764.32001
[16] Miwa, T., Painlevé property of monodromy preserving equations and the analyticity of the τ function, Publ. RIMS Kyoto university, 17, 703-721, (1981) · Zbl 0605.34005
[17] Moore, G., Geometry of the string equations, Commun. math. phys., 133, 261-304, (1990) · Zbl 0727.35134
[18] Moore, G., Matrix models of 2D gravity and isomonodromic deformation, Prog. theor. phys. suppl., 102, 255-285, (1990) · Zbl 0875.33006
[19] Moutard, T.F., Note sur LES équations differentielles linéaires du second ordre, Compt. rend. acad. sci. Paris, 80, 729, (1876) · JFM 07.0189.01
[20] Sattinger, D.H., Hamiltonian hierarchies on semisimple Lie algebras, Studies in applied math., 72, 65-86, (1985) · Zbl 0584.58022
[21] Beals, R.; Deift, P.; Tomei, C., Direct and inverse scattering on the line, (1988), AMS Providence · Zbl 0699.34016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.