Nonreciprocal 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) has been observed to unidirectionally increase its propagation signals. In this work, the simple addition of nonlinear interactions (as in Fermi and Zabusky) 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 unidirectional active internal 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, both damped and undamped, are analytically obtained and compared to numerical simulations showing a good agreement. The dynamics of two interacting non-reciprocal solitons under different conditions, is also numerically studied.