コロキウム

In studying the blow-up of a nonlinear wave equation, one often uses a so-called self-similar solution in the asymptotic expansion. If such a solution is radially symmetric, then it satisfies an ordinary differential equation called a profile equation. In this talk, we study the case where the profile equations are given by either a generalized Emden-Fowler equation or a nonlinear Heun equation. The linear part of Heun equation has four regular singular points on the Riemann sphere. After elementary transformations these profile equations are written in an autonomous Hamiltonian form with two degrees of freedom. We first study the nonintegrability and the singularity of solutions around the moving singularity. In the study of nonintegrability around the fixed singular point, these equations have close relation with the Hamiltonian studied by Taimanov and Bolsinov ([cf. Inv. Math. 2000]) related the geodesic flow on a Riemannian manifold.They showed the nonintegrability, entropy, and some noncummutativity of the monodromy group. The nonintegrability phenomenon in their result has a local counterpart to some Hamiltonian system with a singular point, which was noted by Gorni and Zampieri [cf. Diff. Geom. Appl. 2005]. We study the latter type equation and show the semi-global nonintegrability of Hamiltonians which has more than 2 singular points. We finally discuss the relation of these results to the nonintegrability of the profile equation.