Title: Bäcklund Transformations for the Boussinesq Equation and Merging Solitons,
Abstract: The Bäcklund 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.