By Takashi Aoki, Hideyuki Majima, Yoshitsugu Takei, Nobuyuki Tose (eds.)

This quantity comprises 23 articles on algebraic research of differential equations and similar subject matters, so much of that have been awarded as papers on the overseas convention "Algebraic research of Differential Equations – from Microlocal research to Exponential Asymptotics" at Kyoto collage in 2005.

Microlocal research and exponential asymptotics are in detail attached and supply robust instruments which have been utilized to linear and non-linear differential equations in addition to many similar fields corresponding to actual and complicated research, crucial transforms, spectral concept, inverse difficulties, integrable platforms, and mathematical physics. The articles contained right here current many new effects and concepts.

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 $64000 contributions he has made to the sector. Introductory notes at the clinical works of Professor Kawai also are included.

**Read Online or Download Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai PDF**

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**Extra info for Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai**

**Example text**

The bifurcation of a Stokes curve observed in Fig. 5 (ii) is due to the singularity that the direction ﬁeld (1) acquires at a simple turning point. Impressively enough, the smooth transition between Fig. 5 (i) and Fig. 5 (iii) via Fig. 5 (ii) is attained with the addition of Stokes curves emanating from the virtual turning point x = 0. One should observe some clumsy transition if they were not added. A subtle and interesting fact is that Fig. 5 (ii) switches the relative location of a Stokes curve emanating from a traditional turning point and that from a virtual turning point.

F2m is a regular sequence in C[u0 , . . , u2m , t]. Remark 1. It is clear that the sequence F0 , . . , F2m is a regular sequence in C[u0 , . . , u2m ] for every ﬁxed t. Hence (N Y )02m has a ﬁnite number of solutions for every ﬁxed t. Under suitable generic condition, the number is 22m and the number of ramiﬁcation points of solutions over t is 2m22m . These will be proved in our forthcoming paper. Remark 2. Of course we can consider (N Y )02m+1 for (N Y )2m+1 . The sequence of polynomials corresponding to (N Y )02m+1 is, however, not a regular sequence in C[u0 , .

36 Takashi Aoki et al. , a family of higher order non-linear equations whose ﬁrst member coincides with one of the second order Painlev´e equations. The ﬁrst member of the NoumiYamada hierarchy is the following (N Y )2 , which is a symmetric form of the fourth Painlev´e equation (PIV ); (N Y )2 : η −1 dfj = fj (fj+1 − fj+2 ) + αj dt (j = 0, 1, 2), (10) where fj = fj−3 (j = 3, 4) and αj (j = 0, 1, 2) are constants that satisfy α0 + α1 + α2 = η −1 . , an overdetermined system of linear diﬀerential equations whose compatibility conditions are given by (10), is as follows: ⎞⎛ ⎞ ⎛ ⎞ ⎛ f1 1 ψ0 (2α1 + α2 )/3 ψ0 ∂ ⎠ ⎝ψ1 ⎠ , ⎝ψ1 ⎠ = ⎝ x (−α1 + α2 )/3 f2 − η −1 x ∂x ψ2 xf0 x −(α1 + 2α2 )/3 ψ2 (12) ⎞⎛ ⎞ ⎛ ⎞ ⎛ 0 ψ0 f2 − t/2 −1 ψ0 ∂ 0 f0 − t/2 −1 ⎠ ⎝ψ1 ⎠ .