Non-reciprocal solitary waves in the parity-broken Ablowitz-Ladik model

The Nelson-Ninomiya (NN) theorem precludes the existence of states with unidirectional propagation in 1D crystals. In this work, we show that, upon breaking parity symmetry, certain types of nonlinear lattices circumvent the limitations of the NN theorem, enabling self-induced non-reciprocal dynamics. Specifically, we study the Ablowitz-Ladic (AL) model, an integrable discretization of the nonlinear Schrodinger equation. In its standard form, the AL equation supports stable nonlinear localized eigenstates, i.e., solitons. We demonstrate that breaking parity in this model can either generate non-reciprocal linear instabilities on its static soliton states or drive them into a fully non-reciprocal regime, in which the static soliton solutions cease to exist but, although still localized, nidirectionally accelerate and amplify towards one end of the lattice.