Instability of second sound under heat flow near the superfluid transition

In heat-carrying superfluids, second sounds become unstable when the difference between the superfluid velocity and the normal fluid velocity exceeds a critical value. We derive a nonlinear evolution equation of the phase of the complex order parameter in this unstable region near the superfluid transition. It indicates emergence of trains of soliton-like heat pulses, which will be nearly periodic in a weakly unstable regime and chaotic in a strongly unstable regime. Experimentally, this instability cannot be observed in usual one-dimensional geometries due to appearance of a normal fluid region at large heat currents. However, we predict that it can be realized in a cell with a constricted part.