  Theoretical studies on linear shear instabilities often use simple velocity
and density profiles (e.g. constant, piecewise) for obtaining good qualitative
and quantitative predictions of the initial disturbances. Furthermore, such
simple profiles provide a minimal model for obtaining a mechanistic
understanding of otherwise elusive shear instabilities. However, except a few
specific cases, the efficacy of simple profiles has remained limited to the
linear stability paradigm. In this work we have proposed a general framework
that can simulate the fully nonlinear evolution of a variety of stratified
shear instabilities as well as wave-wave and wave-topography interaction
problems having simple profiles. To this effect, we have modified the classical
vortex method by extending the Birkhoff-Rott equation to multiple interfaces,
and furthermore, have incorporated background shear across a density interface.
The latter is more subtle, and originates from the understanding that
Bernoulli's equation is not just limited to irrotational flows, but can be
modified to make it applicable for piecewise velocity profiles. We have solved
diverse problems that can be essentially reduced to the multiple interacting
interfaces paradigm, e.g. spilling and plunging breakers, stratified shear
instabilities like Holmboe and Taylor-Caulfield, jet flows, and even
wave-topography interaction problem like Bragg resonance. Free-slip boundary
being a vortex sheet, its effect can also be effectively captured using vortex
method. We found that the minimal models capture key nonlinear features, e.g.
wave breaking features like cusp formation and roll-ups, which are observed in
experiments and/or extensive simulations with smooth, realistic profiles.
