コロキウム

I talk about my recent results on the solvability of representative integrable partial differential equations including the Korteweg-de Vries (KdV) equation and nonlinear Schr\"odinger equation under meromorphic initial conditions by quadrature when the inverse scattering transform (IST) is applied. It is a key to solve the Schr\"odinger equation or two-dimensional Zakharov-Shabat systems appearing in the Lax pairs in application of the IST. We can prove that these linear systems are always integrable in the sense of differential Galois theory if and only if the meromporphic potentials corresponding to the initial condition are reflectionless, under the condition that they are absolutely integrable on \mathbb{R} \ (−R_0, R_0) for some R_0 > 0. We will mainly concentrate on the case of the KdV equation from time constraints.