This project is concerned with analysis of nonlinear wave propagation in several model equations arising in fluid mechanics, which have higher-order nonlocal nonlinearity compared to the local quadratic nonlinearity of the KdV equation, but still capture some of the interesting properties of the Euler equation and retain some integrable features. Due to the nonlocal nonlinearity, their dynamical features are different from those of the KdV equation in at least two ways: they allow wave breaking of strong solutions and at the same time they permit peaked waves which can be stable enough to persist in the time evolution of weak solutions.
The group plans to work with the following model equations: the rotation-Camassa-Holm equation, the reduced Ostrovsky equation, the modified Camassa-Holm equation and the Novikov equation. New mathematical studies of wave breaking and stability of peaked waves are developed, which do not rely on integrability and conserved quantities of the underlying equations. These techniques will help to shed new light on the modeling, characterization, and prediction of waves of large amplitudes in oceanographic applications.
Research Team: Anna Geyer ( Delft U of Technology), Guilong Gui (Northwest University, Xian), Yue Liu (U of Texas at Arlington), Dmitry Pelinovsky (McMaster U, Hamilton)