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... asymptotic behavior is —— exp(f x3/2) be added before 7T X argx reaches π. At which value of argx the extra portion is first added was the subject of debate, a debate only recently resolved by Olver =-=[22]-=-, who pointed out that uniform bounds on the error were only obtainable for |argx| < —, that is, up to the last Stokes line before argx = π. In fact, Stokes himself knew by direct calculation that the...

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...e limiting procedures. Yet, the geometric content of this statement is still very obscure. Much more surprising is connection 3, again due in large part to Gamier [18]. It is fairly well known (Davis =-=[19]-=-) that the change of variables converts Painleve II into 1 1 g 2v a" — 2q —2q a + - -~- — υs* * ty 9 t2 3ί which, for |ί|->oo in certain regions of the t plane reduces to g" = 2g3-2g. This is solvable...

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...his reproduces the solution, due to Ablowitz and Segur, of (1.2) by a Marchenko integral equation [2]. Other special cases yield the rational or Airy-function solutions of (1.2) discovered by Airault =-=[3]-=-, but this time by procedures very familiar from the inverse-scattering derivation of multisoliton formulae. The second and somewhat more complicated example, whose study we begin in this paper, is af...

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... of the Airy function. This method produces the unique solution of (2.11) which satisfies Monodromy- and Spectrum-Preserving Deformations 71 (Hastings and McLeod [11]). More recently, Ablowitz et al. =-=[12]-=- have shown how to bypass the f-evolution in linearizing (2.11) and similar equations for solutions which decay at + oo. This is more satisfactory, since scattering theory is not really applicable for...

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...ys that the eigenvalues of the matrix are unchanged when the Aj solve (2.27) - a typical isospectral flow. One special case singled out (and solved) by Garnier is the system (2.28) which is now known =-=[20, 21]-=- to contain the finite-gap KdV theory as special case. It was, in fact, discovered some 60 years earlier by C. Neumann as describing uncoupled harmonic oscillators constrained to move on a sphere. Gar...

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...rties hold, and in particular that the scattering matrix A(ζ) = Φ~1Ψ is defined with its diagonal entries admitting analytic extension. The ζ behavior of A(ζ) (or the (ζ, ί) behavior of A(ζ, t) - see =-=[33]-=-) is inferred from (3.2) For the class of equations and solutions we wish to discuss, condition (3.4) does not hold. Therefore, we propose a new method, in which one focuses central attention on Eq. (...

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...t ζ = 0 and oo are unchanged (when x is varied, and q, r change as functions of x). For this it is necessary and sufficient that (as function of x) r = q x , and (1.2) is the second Painleve equation =-=[1]-=-. By posing an irregular-singular RiemannHilbert problem, we exploit the connection with (1.1) to reduce (1.2) to an equivalent system of linear singular integral equations. In a special case, this re...

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...rization via the Riemann-Hilbert problem for (1.3) provide a link between the inverse scattering transform and the extensive work of Barouch-McCoy-Tracy-Wu [4-6] on the Ising model and of Sato et al. =-=[7]-=- on monodromy-preserving deformations and quantum field theory. Our principal aims in this first paper are: 1. To point out relations between mondromy - and spectrum - preserving deformations. 2. To i...

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...s many further problems about singular points and deformation theory. After completing a first draft of this paper, we learned that K. Ueno of the Kyoto University RIMS had carried out investigations =-=[27]-=- which overlap ours to some extent. He derives deformation equations for nxn systems with singular points of various ranks at C = 0, oo, and establishes results of the kind contained in Sect. 3C above...

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...ct of-(O) and exp8zζ3/3. Appropriate scaling of the solutions φ and ψ with the factors exp±4iζ3/3 then shows the jump (6.4) to be the product of a Stokes multiplier and a recessive solution. Zakharov =-=[30]-=- has recently propounded an extension of inverse-scattering ideas, based on Riemann-Hilbert problems. This generalizes the Zakharov-Shabat method sketched above, in that the eigenfunctions are reconst...