We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3 × 3 matrix Fuchs-Garnier pairs for the third, fourth, and fifth Painlevé equations, together with the previously known Fuchs-Garnier pair for the sixth Painlevé equation. These Fuchs-Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2 × 2 matrix Fuchs-Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3 × 3 matrix pairs, we found an integral auto-transformation for the standard Fuchs-Garnier pair for the fifth Painlevé equation. It generates an Okamoto-like Bäcklund transformation for the fifth Painlevé equation. Another application is an integral transformation relating two different 2 × 2 matrix Fuchs-Garnier pairs for the third Painlevé equation.

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q t and q xxx have the same sign (KdVI) or two boundary conditions if q t and q xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neu-mann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x, 0), q(0, t)} and {q(x, 0), q(0, t), q x (0, t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q x (0, t), q xx (0, t)} and {q xx (0, t)}, respectively. We show that this can be achieved without solving for q(x, t) by analysing a certain "global relation" which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ (t) (t, k), where Φ (t) satisifies the t-part of the associated Lax pair evaluated at x = 0. Indeed, by employing a Gelfand-Levitan-Marchenko triangular representation for Φ (t) , the global relation can be solved explicitly for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ (t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation.

We found Fuchs-Garnier pairs in 3×3 matrices for the first and second Painlevé equations which are linear in the spectral parameter. As an application of our pairs for the second Painlevé equation we use the generalized Laplace transform to derive an invertible integral transformation relating two its Fuchs-Garnier pairs in 2×2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and the one found by Harnad, Tracy, and Widom. Together with the certain other transformations it allows us to relate all known 2 × 2 matrix Fuchs-Garnier pairs for the second Painlevé equation with the original Garnier pair.

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In this paper, the simultaneous identification of damping or anti-damping coefficient and initial value for some PDEs is considered. An identification algorithm is proposed based on the fact that the output of system happens to be decomposed into a product of an exponential function and a periodic function. The former contains information of the damping coefficient, while the latter does not. The convergence and error analysis are also developed. Three examples, namely an anti-stable wave equation with boundary anti-damping, the Schrödinger equation with internal anti-damping, and two connected strings with middle joint anti-damping, are investigated and demonstrated by numerical simulations to show the effectiveness of the proposed algorithm.