  The Skyrme model, though it admits correctly a wide range of static
properties of the nucleon, does not seem to reproduce properly the scattering
behavior of nucleons at high energies. In this paper we present a $T\bar
T$-like deformation of it, inspired by a 1+1 dimensional model, in which
boosted nucleons behave like shock waves. The scattering of the latter
saturates the Froissart bound. We start by showing that 1+1 dimensional $T\bar
T$ deformations of the free abelian pion action are in fact generalizations of
the old Heisenberg model for nucleon-nucleon scattering, yielding the same
saturation of the Froissart bound. We then deform the strong coupling limit of
the bosonized action of multi-flavor QCD in two dimensions using the $T\bar T$
deformation of the WZW action with a mass term. We derive the classical soliton
solution that corresponds to the nucleon, determine its mass and discuss its
transformation into a shock-wave upon boosting. We uplift this action into a
3+1 dimensional $T\bar T$-like deformation of the Skyrme action. We compare
this deformed action to that of chiral perturbation theory.
  A possible holographic gravity dual interpretation is explored.
