The "good" Boussinesq equation: a Riemann-Hilbert approach

We develop an inverse scattering transform formalism for the good Boussinesq equation on line. Assuming that solution exists, we show it can be expressed in terms of a $3 \times 3$ matrix Riemann-Hilbert problem. The problem is formulated two reflection coefficients whose definitions involve only initial data, and has form which makes suitable evaluation long-time asymptotics via Deift-Zhou steepest descent arguments.