  Let $F:\mathbb Z^2\to \mathbb Z$ be the pointwise minimum of several linear
functions. The theory of smoothing of integer-valued superharmonic function
allows us to prove that under certain conditions there exists the pointwise
minimal superharmonic function which coincides with $F$ "at infinity".
  We develop such a theory to prove the existence of so-called solitons (or
strings) in a certain sandpile model, studied by S. Caracciolo, G. Paoletti,
and A. Sportiello. Thus we made a step towards understanding the phenomena of
the identity in the sandpile group for a square where solitons appear according
to experiments. We prove that sandpile states, defined using our smoothing
procedure, move changeless when we send waves (that is why we call them
solitons), and can interact, forming triads and nodes.
