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- N. Joshi, P. A. Treharne
- 2007

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… (More)

- N. Joshi, P. A. Treharne
- 2008

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… (More)

- P. A. Treharne, A. S. Fokas
- J. Nonlinear Science
- 2008

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… (More)

- N. Joshi, P. A. Treharne
- 2007

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 and fifth Painlevé equations, together with the previously known Fuchs– Garnier pair for the fourth and sixth Painlevé equations. These Fuchs–Garnier pairs have an… (More)

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