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The Skyrme model, though it admits correctly a wide range of staticproperties of the nucleon, does not seem to reproduce properly the scatteringbehavior of nucleons at high energies. In this paper we present a $T\barT$-like deformation of it, inspired by a 1+1 dimensional model, in whichboosted nucleons behave like shock waves. The scattering of the lattersaturates the Froissart bound. We start by showing that 1+1 dimensional $T\barT$ deformations of the free abelian pion action are in fact generalizations ofthe old Heisenberg model for nucleon-nucleon scattering, yielding the samesaturation of the Froissart bound. We then deform the strong coupling limit ofthe 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 solitonsolution that corresponds to the nucleon, determine its mass and discuss itstransformation into a shock-wave upon boosting. We uplift this action into a3+1 dimensional $T\bar T$-like deformation of the Skyrme action. We comparethis deformed action to that of chiral perturbation theory.  A possible holographic gravity dual interpretation is explored.

A $T\bar T$-like deformation of the Skyrme model and the Heisenberg model of nucleon-nucleon scattering