On Boussinesq's equation for water waves

A century and a half ago, J. Boussinesq derived an equation for the propagation of water waves in a channel. Despite the fundamental importance of this equation for a number of physical phenomena, mathematical results on it remain scarce. One reason for this is that the equation is ill-posed. In this paper, we establish several results on the Boussinesq equation. First, by solving the direct and inverse problems for an associated third-order spectral problem, we develop an Inverse Scattering Transform (IST) approach to the initial value problem. Using this approach, we establish a number of existence, uniqueness, and blow-up results. For example, the IST approach allows us to identify physically meaningful global solutions and to construct, for each $T > 0$, solutions that blow up exactly at time $T$. Our approach also yields an expression for the solution of the initial value problem for the Boussinesq equation in terms of the solution of a Riemann--Hilbert problem. By analyzing this Riemann--Hilbert problem, we arrive at asymptotic formulas for the solution. We identify ten main asymptotic sectors in the $(x,t)$-plane; in each of these sectors, we compute an exact expression for the leading asymptotic term together with a precise error estimate.