B\"acklund Transformations for the Boussinesq Equation and Merging Solitons

The B\"acklund transformation (BT) for the "good" Boussinesq equation and its superposition principles are presented and applied. Unlike many other standard integrable equations, the Boussinesq equation does not have a strictly algebraic superposition principle for 2 BTs, but it does for 3. We present associated lattice systems. Applying the BT to the trivial solution generates standard solitons but also what we call "merging solitons" --- solutions in which two solitary waves (with related speeds) merge into a single one. We use the superposition principles to generate a variety of interesting solutions, including superpositions of a merging soliton with $1$ or $2$ regular solitons, and solutions that develop a singularity in finite time which then disappears at some later finite time. We prove a Wronskian formula for the solutions obtained by applying a general sequence of BTs on the trivial solution. Finally, we show how to obtain the standard conserved quantities of the Boussinesq equation from the BT, and how the hierarchy of local symmetries follows in a simple manner from the superposition principle for 3 BTs.