Propagative Sine-Gordon solitons in the spatially forced Kelvin-Helmholtz instability

The spatio-temporal evolution of the vortex sheet separating two finite-depth layers of immiscible fluids is examined in the vicinity of threshold when spatially periodic forcing is imposed at the horizontal boundaries. As a result of the Galilean invariance of the problem, the interface deformation is shown to satisfy a coupled system of evolution equations involving not only the usual “short-wave” at the critical wavenumber but also a shallow-water “long-wave” associated with the mean elevation of the interface. The weakly nonlinear model is further studied in the Boussinesq approximation where it reduces to a forced Klein-Gordon equation. Thus, the secondary Benjamin-Feir instability of nonlinear Stokes wavetrains is analysed in the absence of forcing. When spatial forcing is reintroduced, the competition between the imposed external length scale and the natural length scale of the interface is shown analytically to give rise to one-dimensional propagating Sine-Gordon phase solitons. Numerical simulations of the Klein-Gordon evolution model fully confirm this prediction and also lead to the determination of the range of stability of phase solitons.