A class of nonlinear integrable PDEs admit non-smooth solitary wave solutions as certain special weak solutions. These solutions can be used to capture main attributes of solutions of these PDEs, e.g. the breakdown of regularity and the nature of long time asymptotics. In fact, under appropriate assumptions, these solutions can be formulated in terms of forward/inverse spectral problems involving approximation problems so that multiple integral formulae arise, which can motivate new types of random matrix models or orthogonal functions. This talk is devoted to an introduction on these aspects with focus on our recent results.