Two-dimensional rational solitons and their blowup via the Moutard transformation

We construct a family of two-dimensional stationary Schrödinger operators with rapidly decaying smooth rational potentials and nontrivial $L_2$ kernels. We show that some of the constructed potentials generate solutions of the Veselov–Novikov equation that decay rapidly at infinity, are nonsingular at $t=0$, and have singularities at finite times $t\ge t_0>0$.