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A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann-Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine-Gordon equation in light-cone coordinates, the nonlinear Schrödinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrödinger equation is derived.

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We discuss an algorithmic approach for both deriving discrete analogues of Painleve equations as well as using such equations to characterize "similarity" reductions of spatially discrete integrable evolution equations. As a concrete example we show that a discrete analogue of Painleve I can be used to characterize "similarity" solutions of the Kac-Moerbeke equation. It turns out that these similarity solutions also satisfy a special case of Painleve IV equation. In addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions. As an example we use the WKB method in the presence of two turning points of the third order to parametrize (at the continuous limit) the solution of Painleve I in terms of the solution of discrete Painleve I. Finally we show that these results are useful for investigating the partition function of the matrix model in 2D quantum gravity associated with the measure exp[-t^z 2 -t 2 z 4 -ί 3 z 6 ].

Let q(x, t) satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation on the finite interval, 0 < x < L, with q 0 (x) = q(x, 0), g 0 (t) = q(0, t), f 0 (t) = q(L, t). Let g 1 (t) and f 1 (t) denote the unknown boundary values q x (0, t) and q x (L, t), respectively. We first show that these unknown functions can be expressed in terms of the given initial and boundary conditions through the solution of a system of nonlinear ODEs. Although the question of the global existence of solution of this system remains open, it appears that this is the first time in the literature that such a characterization is explicitely described for a nonlinear evolution PDE defined on the interval; this result is the extension of the analogous result of [4] and [6] from the halfline to the interval. We then show that q(x, t) can be expressed in terms of the solution of a 2 × 2 matrix Riemann-Hilbert problem formulated in the complex k -plane. This problem has explicit (x, t) dependence in the form exp[2ikx + 4ik 2 t], and it has jumps across the real and imaginary axes. The relevant jump matrices are explicitely given in terms of the spectral data {a(k), b(k)}, {A(k), B(k)}, and {A(k), B(k)}, which in turn are defined in terms of q 0 (x), {g 0 (t), g 1 (t)}, and {f 0 (t), f 1 (t)}, respectively.

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