Non-linear shallow water dynamics with odd viscosity

In this work, we derive the Korteweg-de Vries (KdV) equation corresponding to the surface dynamics of a shallow depth (h) two-dimensional fluid with odd viscosity (ν_o) subject to gravity (g) in the long-wavelength weakly nonlinear limit. In the long-wavelength limit, the odd viscosity term plays the role of surface tension albeit with opposite signs for the right and left movers. We show that there exist two regimes with a sharp transition point within the applicability of the KdV dynamics, which we refer to as weak (|ν_o|< 1/6 (gh^3)^1/2) and strong (|ν_o|>1/6 (gh^3)^1/2) parity-breaking regimes. While the `weak' parity breaking regime results in minor qualitative differences in the soliton amplitude and velocity between the right and left movers, the `strong' parity breaking regime on the contrary results in solitons of depression (negative amplitude) in one of the chiral sectors.