By T. Aoki, H. Majima, Y. Takei, N. Tose

This quantity comprises 23 articles on algebraic research of differential equations and comparable issues, such a lot of which have been awarded as papers on the overseas convention ''Algebraic research of Differential Equations вЂ“ from Microlocal research to Exponential Asymptotics'' at Kyoto college in 2005. Microlocal research and exponential asymptotics are in detail attached and supply strong instruments which have been utilized to linear and non-linear differential equations in addition to many similar fields reminiscent of genuine and intricate research, imperative transforms, spectral conception, inverse difficulties, integrable platforms, and mathematical physics. The articles contained the following current many new effects and ideas, supplying researchers and scholars with worthy feedback and instructive tips for his or her paintings. This quantity is devoted to Professor Takahiro Kawai, who's one of many creators of microlocal research and who brought the means of microlocal research into exponential asymptotics. This commitment is made at the party of Professor Kawai's sixtieth birthday as a token of deep appreciation of the real contributions he has made to the sphere. Introductory notes at the medical works of Professor Kawai also are included.

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**Additional resources for Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics**

**Example text**

For example, in the case of the operator P given by (3), the associated bicharacteristic strip passing through (x, y; ξ, η) = (1, 0; −i, 1) is given by ⎧ x(t) = −4(t + 1/2)(t2 + t − 1/2) ⎪ ⎪ ⎪ ⎨ y(t) = −6it2 (t + 1)2 ⎪ ξ(t) = −2it − i ⎪ ⎪ ⎩ η(t) = 1. (4) Hence its projection to the base space forms a (unique) self-intersection point at (x, y) = (0, −3i/2). The situation is schematically illustrated in Fig. 3 with an appropriate labelling of solutions of the following characteristic equation of P : ξ 3 + 3η 2 ξ + 2ixη 3 = 0 with η = 1.

Alk of elements in {a0 , a1 , . . , al }, the element alk is not a zero divisor on R/(al0 , . . , alk−1 ). Note that the notion of tame regular sequences is independent of the ordering of aj ’s and that the second condition in Deﬁnition 3 is not assumed. This notion can be stated in terms of the Koszul complex. Let us denote by K(a0 , a1 , . . , al ; R) the Koszul complex associated with the sequence a0 , a1 , . . , al with coeﬃcients in R. Theorem 6. The following two conditions are equivalent: 1.

After this success, Kawai, Aoki and I tried to extend exact WKB analysis in two diﬀerent directions. First, suggested by M. Jimbo, we began to apply exact WKB analysis to the study of isomonodromic deformations of linear diﬀerential equations and Painlev´e equations associated with them. To explain the results we have obtained, let us consider the simplest case, that is, the case of the following Painlev´e I equation with a large parameter η here: d2 λ = η 2 (6λ2 + t). dt2 (PI ) This non-linear equation, or its Hamiltonian form dλ ∂KI =η , dt ∂ν dν ∂KI = −η dt ∂λ (6) with KI = (ν 2 − 4λ3 − 2tλ)/2, describes the compatibility condition of the following system of linear diﬀerential equations: ∂2 3η −2 η −1 ν 2 3 + ψ = 0 with Q − η Q = 4x + 2tx + 2K − , I I I ∂x2 x − λ 4(x − λ)2 (SLI ) ∂ψ 1 ∂AI ∂ψ 1 = AI − ψ with AI = (DI ) ∂t ∂x 2 ∂x 2(x − λ) (cf.