A quasi-linear Schrödinger equation for large amplitude inertial oscillations in a rotating shallow fluid

This paper derives a 2D, quasi-linear Schrödinger equation in Lagrangian coordinates that describes the effects of weak pressure gradients on large amplitude inertial oscillations in a rotating shallow fluid. The coefficients of the equation are singular at values of the gradient of the wave amplitude that correspond to the vanishing of the Jacobian of the transformation from Lagrangian to Eulerian coordinates, but solutions do not appear to form singularities dynamically. Two regimes of high and moderate nonlinearity are identified, depending on whether or not phase differences in the components of the amplitude gradient are required to maintain a non-zero Jacobian. Numerical simulations show that moderately nonlinear solutions of the quasi-linear Schrödinger equation behave in a qualitatively similar way to solutions of a linear Schrödinger equation, whereas highly nonlinear solutions generate rapidly oscillating, small-scale waves.