Major research activity
I study linear and nonlinear partial differential equations by using techniques of complex analysis.
1. A useful tool in complex analysis is the method of majorants. It makes it possible to simplify some estimates and I have obtained an integral representation of solutions to some equations by using majorants. Moreover, majorants enable one to construct good function spaces in which multiplication is well-defined. These spaces are useful in nonlinear problems and I have solved some nonlinear initial value problems in them: I have proven that solutions exist for a long time if the initial data are small in a certain sense.
2. Ordinary differential equations in the complex domain is a well-known, classical subject. In particular, ODEs with regular singularities are the most important topic in this field. They have a higher dimensional analogue, namely Fuchsian partial differential equations. I have studied Fuchsian PDEs by using majorants, integral representations and successive approximation. Moreover, I have constructed singular solutions to some non-Fuchsian nonlinear PDEs by reducing them to Fuchsian ones. This constuction has been motivated by WTC expansions in mathematical physics.
3. My interest is now moving to integrable systems. These equations appear in mathematical physics and have rich mathematical structure. In particular, they can be solved by using the inverse scattering method. I am trying to apply this method to the asymptotic study of some equations.
Major relevant publications
- Yamane, Hideshi, Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. J. Math. Soc. Japan 66 (2014), no. 3, 765–803.
- Tahara, T. and Yamane, H., Logarithmic singularities of solutions to nonlinear partial differential equations, with Hidetoshi Tahara, Journal of the Mathematical Society of Japan, 60(2), 603-630 (2008).
- Yamane, H, Local Existence for Nonlinear Cauchy Problems with Small Analytic Data, Journal of Mathematical Sciences, University of Tokyo, 18, 51-65 (2011).
- Yamane, H, Ramified Cauchy problem for a class of Fuchsian operators with tangent characteristics, , Journal de Mathematiques Pures et Appliquées, 79(3), 271-294 (2000)
- Yamane, H, Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation II, SIGMA 11 (2015), 020