- Home
- Programs & Courses
- Graduate School of Science and Engineering : Science of Environment and Mathematical Modeling (Laboratories : Analysis Laboratory)

## Analysis Laboratory

#### Staff

Yoshitsugu TAKEI

[Professor]

Acceptable course | |
---|---|

Master's degree course | ○ |

Doctoral degree course | × |

ytakei@mail.doshisha.ac.jp

Office : HS-310

Database of Researchers

#### <Research Topics>

- Algebraic analysis of singular perturbation theory
- Exact WKB analysis of differential equations

#### <Research Contents>

The main subject of my research is algebraic analysis of singular perturbation theory, in particular, the exact WKB analysis of linear differential equations and nonlinear differential equations.

It is well known that a one-dimensional Schrödinger equation containing Planck's constant as a small parameter has asymptotic formal solutions called WKB solutions. In the exact WKB analysis we give an analytic meaning to WKB solutions through the Borel resummation method. Using the exact WKB analysis, we can analyze the global behavior of solutions in a very explicit manner. For example, together with T. Kawai (RIMS, Kyoto Univ.) and T. Aoki (Kindai Univ.) I have succeeded in providing a recipe for computing the monodromy group of Fuchsian equations, i.e., second order ordinary differential equations with regular singularities, in terms of the Borel sum of WKB solutions. Recently I have been studying higher order equations and systems of differential equations from the viewpoint of the exact WKB analysis.

My research interest lies also in the analysis of nonlinear differential equations and difference equations. As a matter of fact, the exact WKB analysis gives a powerful tool for the global analysis of Painlevé equations, typical second order nonlinear ordinary differential equations that have close relationship with the soliton-type partial differential equations (i.e., the so-called integrable systems), and their discrete analogues. I am now trying to establish the fundamental theory of the exact WKB analysis for Painlevé equations and discrete Painlevé equations.

It is well known that a one-dimensional Schrödinger equation containing Planck's constant as a small parameter has asymptotic formal solutions called WKB solutions. In the exact WKB analysis we give an analytic meaning to WKB solutions through the Borel resummation method. Using the exact WKB analysis, we can analyze the global behavior of solutions in a very explicit manner. For example, together with T. Kawai (RIMS, Kyoto Univ.) and T. Aoki (Kindai Univ.) I have succeeded in providing a recipe for computing the monodromy group of Fuchsian equations, i.e., second order ordinary differential equations with regular singularities, in terms of the Borel sum of WKB solutions. Recently I have been studying higher order equations and systems of differential equations from the viewpoint of the exact WKB analysis.

My research interest lies also in the analysis of nonlinear differential equations and difference equations. As a matter of fact, the exact WKB analysis gives a powerful tool for the global analysis of Painlevé equations, typical second order nonlinear ordinary differential equations that have close relationship with the soliton-type partial differential equations (i.e., the so-called integrable systems), and their discrete analogues. I am now trying to establish the fundamental theory of the exact WKB analysis for Painlevé equations and discrete Painlevé equations.

#### <Keywords>

- Differential equations
- Exact WKB analysis

- Theory of resurgent functions
- Painlevé equations