Nalini Joshi

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In this paper, we find all possible asymptotic behaviours of the solutions of the second Painlevé equation y = 2y + xy+ α as the parameter α → ∞ in the local region x ≪ α. We prove that these are asymptotic behaviours by finding explicit error bounds. Moreover, we show that they are connected and complete in the sense that they correspond to all possible(More)
The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, PII . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé(More)
The relations between the different linear problems for Painlevé equations is an intriguing open problem. Here we consider our second and fourth Painlevé hierarchies given in Publ. Res. Inst. Math. Sci. (Kyoto) 37 327-347 (2001), and show that they could alternatively have been derived using the linear problems of Jimbo and Miwa. That is, we give a gauge(More)
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable equations have the Painlevé property, that is, all solutions are single-valued around all movable singularities. In this(More)
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)
We present a systematic approach to the construction of Miura transformations for discrete Painlevé equations. Our method is based on the bilinear formalism and we start with the expression of the nonlinear discrete equation in terms of τ -functions. Elimination of τ -functions from the resulting system leads to another nonlinear equation, which is a(More)
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differentialdifference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to(More)
Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé (PIV ) hierarchies recently found by the same authors [Publ. Res. Inst.(More)