Integrable Systems and Isomonodromy Deformations

We analyze in detail three classes of isomondromy deformation problems associated with integrable systems. The first two are related to the scaling invariance of the n× n AKNS hierarchies and the Gel’fand-Dikii hierarchies. The third arises in string theory as the representation of the Heisenberg group by [(L)+, L] = I where L is an nth order scalar differential operator. The monodromy data is constructed in each case; the inverse monodromy problem is solved as a Riemann-Hilbert problem; and a simple proof of the Painlevé property is given for the general case. Introduction 1. Overdetermined systems and isomonodromy equations 2. The forward monodromy problem at z = ∞ 3. The forward monodromy problem at z = 0 4. The inverse problem and the Painlevé property 5. Rational solutions 6. Bäcklund transformations 7. Scaling, self-similarity, and construction of isomonodromy deformations 8. Gel’fand-Dikii equations and isomonodromy 9. Isomonodromy deformations and string equations Introduction It has been known since work of Ablowitz and Segur [AS1] that there is an intimate relationship between equations like KdV which are integrable by the inverse scattering method and Painlevé equations; see also [AS2]. It was observed by Flaschka and Newell [FN] that just as KdV gives an isospectral flow for the Schrödinger operator, Painlevé equations are monodromy preserving flows for linear systems with irregular singular points. Certain of these problems have been investigated in detail, for 2 × 2 systems and second order scalar problems: [FN], [FZ], [IN], [JMU], [JM1], [JM2]. Research of the authors was supported by National Science Foundation grants DMS-8916968 and DMS9123844 1 2 In this paper we discuss general isomonodromy problems associated to n×n systems and higher order equations. 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; cf. [Ma], [Mi]. We hope, among other things, to simplify and clarify the treatment of isomonodromy deformations and their relation to Riemann-Hilbert problems, on the principle that the more general the case the less reliance on special features. The paper is organized as follows. In §1 we describe the formal connection between matrix isomonodromy equations and overdetermined n× n systems in two variables. This connection is made rigorous in the next three sections, which describe the forward problem at the singular points z = ∞ and z = 0, connect it to a Riemann-Hilbert problem, and prove the Painlevé property. A few examples of equations and systems which occur in this context are: the Painlevé II equation 4(xu)x + uxxx − 6uux = 0; the system of three equations of order one (xui)x = aiu+ biujuk, {i, j, k} = {1, 2, 3}; the system of two equations of order two with cubic nonlinearity (xu1)x + 1 2u1xx − u 2 1u2 = 0 = (xu2)x − 12u2xx + u 2 2u1; and the system of two equations of order two with quadratic nonlinearity (xu1)x + i √ 3 u1xx + 2u2u2x = 0 = (xu2)x − i 3u2xx + 2u1u1x. In §5 the rational solutions of the isomonodromy equations are constructed by solving finite linear systems. This extends results of Airault [Ai], who found Bäcklund transformations giving rational solutions of some Painlevé equations. We develop the gauge theory of Bäcklund transforms in §6. The gauge transformations take the wave functions for one solution to those of a new solution and thus transform solutions to solutions. Isomonodromy deformations arise from integrable systems in two ways. Some can be obtained as self-similar solutions of the given nonlinear evolution equations; see [AS2]. This construction is given in §7 for isospectral deformations of n× n first-order operators 3 (AKNS-ZS systems); each of the examples above is of this type. In §7 we also treat the isospectral deformations of an n-th order scalar differential operator (the Gel’fand-Dikii hierarchy). Examples include the equation (xu)x + u+ 1 4uxxx + 3 2uux = 0 which corresponds to self-similar solutions of the KdV equation and the system (xu1)x = u1xx − 2u0x, (xu0)x + u0 = 2 3u1xx + 23u1u1x − u0xx which corresponds to self-similar solutions of the Boussinesq system. In all these cases the Lax pair for the integrable system can be rescaled to obtain a Lax pair for the corresponding isomonodromy deformation problem. This was done in detail in the 2×2 case by Flaschka and Newell [FN] for the modified KdV equation and its associated isomonodromy problem, the Painlevé II equation, as well as the sine-Gordon equation and the Painlevé III equation (cf. also Its and Novokshenov [IN]). In section 8 we show that all self-similar solutions of Gel’fand-Dikii flows are in fact solutions of isomonodromy equations, and we treat the direct and inverse problems for these equations. The results are analogous to those of §§2, 3, 4. A second class of isomonodromy problems was obtained by M. Douglas [Do] in recent two-dimensional theories of quantum gravity. These problems are obtained by replacing L̇ by ~I in the Lax equation L̇ = [ (L)+ , L ] for the Gel’fand-Dikii flows. This device leads to a representation of the Heisenberg group and to a class of isomonodromy problems different from those obtained by the scaling invariance. Two of the simplest examples are 1 4uxxx + 3 2uux + ~ = 0; 1 16u (5) + 5 8uuxxx + 5 4uxuxx + 15 8 u ux + ~ = 0. This class of isomonodromy problems was treated by Moore [Mo1], [Mo2]. We thank him for useful discussions of the topic. In §9 we focus on some aspects of the mathematical analysis of these problems and give a different, more complete and self-contained treatment. 1. Overdetermined systems and isomonodromy equations In this section we outline the formal connection between certain overdetermined linear n× n systems and monodromy preserving equations. Let J and μ be diagonal matrices belonging to the space Md(C) of d × d complex matrices; we assume that each has distinct diagonal entries and has trace zero. Suppose