arXiv subjects
Publications and source records attributed to Hidetoshi Tahara.
In this paper, we consider a nonlinear Fuchsian type partial differential equation of the second order in the complex domain. Under a very weak assumption, we show the uniqueness of the solution. The result is applied to the problem of analytic continuation of local holomorphic solutions of this equation.
In this paper, we consider a class of singular nonlinear first order partial differential equations $t(\partial u/\partial t)=F(t,x,u, \partial u/\partial x)$ with $(t,x) \in \mathbb{R} \times \mathbb{C}$ under the assumption that $F(t,x,z_1,z_2)$ is a function which is continuous in $t$ and holomorphic in the other variables. Under suitable conditions, we determine all the solutions of this equation in a neighborhood of the origin.
This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t=F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. Under a very weak assumption we show the uniqueness of the solution of this equation. The results are applied to the problem of analytic continuation of local holomorphic solutions of equations of this type.
The paper discusses the summability of formal solutions of some linear q-difference-differential equations, and improves the previous result in [Tahara-Yamazawa, Opsucula Math. 35 (2015), 713-738].
The paper discusses a holomorphic nonlinear singular partial differential equation $(t \partial_t)^mu=F(t,x,\{(t \partial_t)^j \partial_x^αu \}_{j+α\leq m, j<m})$ under the assumption that the equation is of nonlinear totally characteristic type. By using the Newton Polygon at $x=0$, the notion of the irregularity at $x=0$ of the equation is defined. In the case where the irregularity is greater than one, it is proved that every formal power series solution belongs to a suitable formal Gevrey class. The precise bound of the order of the formal Gevrey class is given, and the optimality of this bound is also proved in a generic case.
The paper shows the summability of formal solutions of some linear q-difference-differential equations by using q-Laplace and q-Borel summation method.
We construct a family of singular solutions to some nonlinear partial differential equations which have resonances in the sense of a paper due to T. Kobayashi. The leading term of a solution in our family contains a logarithm, possibly multiplied by a monomial. As an application, we study nonlinear wave equations with quadratic nonlinearities. The proof is by the reduction to a Fuchsian equation with singular coefficients.