Non-reciprocal solitons in an active elastic solid

Non-reciprocity has been observed in several physical scenarios such as in optics, quantum mechanics, colloidal spinners, sea urchin embryos until active metamaterials. A metamaterial can be modeled as an interacting spring-mass system. In fact, a linearly interacting spring-system but made non-reciprocal due to the incorporation of sensors and motors (that is, due to the incorporation of active forces) (Brandenbourger<i>et al</i>2019<i>Nat. Commun.</i><b>117</b>1-8) has been observed to unidirectionally increase its propagation signals. In this work, the simple addition of nonlinear interactions (as in Fermi (1955<i>Los Alamos Report</i>LA-1940) and Zabusky (1967<i>Nonlinear Partial Differential Equations</i>(Academic) pp 223-58)) to the previous active metamaterial is seen to generate non-reciprocal solitons, that is, single-wave solutions but with unidirectional growing amplitude and accelerated motion in time. Starting with a discrete spring-mass model and after coarse-graining it, one extends the well-known KdV and modified KdV (mKdV) equations to the non-reciprocal scenario due to the addition of internal active forces, thus highlighting that active forces can also be used to add non-reciprocity in a system. As a result of the KdV and mKdV treatable mathematical structure, their solutions are analytically obtained and compared to numerical simulations showing a good agreement. Finally, the dynamics of two interacting non-reciprocal solitons under different conditions, is also studied.