elements (SNAILs). We show that the dynamics in terms of the phase difference coordinate across a SNAIL is described by the Korteweg-de Vries (KdV) or modified Korteweg-de Vries (mKdV) equations in the continuum approximation, depending on the external magnetic flux bias, which admit solitary wave solutions. We also validate the KdV-derived solitons by numerically solving the original, discrete element circuit equations. Soliton solutions to the (m)KdV equation spatially modulate the velocity for probe microwaves propagating in the TWPA, resulting in the effective realization of analogue black hole-white hole event horizon pairs.