Soliton, breather and shockwave solutions of the Heisenberg and the $T\bar T$ deformations of scalar field theories in 1+1 dimensions

Abstract: In this note we study soliton, breather and shockwave solutions in certain two dimensional field theories. These include: (i) Heisenberg's model suggested originally to describe the scattering of high energy nucleons (ii) $T\bar T$ deformations of certain canonical scalar field theories with a potential. We find explicit soliton solutions of these models with sine-Gordon and Higgs-type potentials. We prove that the $T\bar T$ deformation of a theory of a given potential does not correct the mass of the soliton of the undeformed one. We further conjecture the form of breather solutions of these models. We show that certain $T\bar T$ deformed actions admit shockwave solutions that generalize those of Heisenberg's Lagrangian.