In this Letter, we derive the Korteweg–de Vries (KdV) equation corresponding to the surface dynamics of a shallow depth ($h$) two-dimensional fluid with odd viscosity (${\nu }_o$) 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 $\left(\right|{\nu }_o|<\sqrt{g{h}^3}/6)$ and strong $\left(\right|{\nu }_o|>\sqrt{g{h}^3}/6)$ 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.

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• Boundary layers
• Surface gravity waves
• Surface tension effects
• Thin fluid films
• Solitons
• Viscosity