Boundary layer collapses described by the two-dimensional intermediate long-wave equation

We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter $D$. In the limits of large and small $D$, the 2d-ILW equation respectively tends to the 2d Benjamin–Ono and 2d Zakharov–Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude $a$ and width $\sigma$, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane $(a,\sigma)$ for various values of $D$. The amplitude threshold $a$ increases as $D$ and $\sigma$ decrease and tends to infinity at $D\to0$. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all $D$ except $D=0$. But the equation itself has not been proved for small $D$. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.