  In systems with linear electron-phonon interaction (EPI), bound states of
polarons, or bipolarons, form by gaining energy from the lattice deformation.
The quadratic EPI case is fundamentally different: bipolarons form because
electrons lose less energy when the total charge density is "compacted". As the
coupling constant is increased, the bipolarons first appear as extended (but
finite radius) soliton-type states. They subsequently decrease in radius until
their size reaches the inter-atomic scale. We present the first numerically
exact solution of the bipolaron problem from quadratic EPI in the presence of
both on-site Hubbard and long-range Coulomb repulsion, and compute estimates of
the largest superconducting transition temperature within the bipolaron
mechanism. We find that $T_c/\Omega$ ratios, where $\Omega$ is the optical
phonon frequency, can be several times larger than what one may expect from the
linear EPI provided the phonon frequency is increased by orders of magnitude on
occupied sites. Electron-electron repulsion can be tolerated at the expense of
stronger EPI and the most detrimental effect comes from the Coulomb potential
because it easily eliminates extended soliton states.
