Singularity formation in the generalized Benjamin-Ono equation
The paper aims to clarify the problem of the wave collapse, i.e., formation of a singularity in finite time, in the generalized Benjamin-Ono (BO) equation, \[ u_t + u^pu_x - \frac{1}{\pi}\int_{-\infty}^{+\infty}u_{xx}(x-y)\frac{dy}{y}=0, \] where the integral is realized in the sense of its principal value, and \(p\) is an integer. The BO equation with \(p=1\) is one of classical model equations for nonlinear small-amplitude long surface waves in inviscid liquids; it describes propagation of such waves in a finite layer of a fluid placed on the top of a heavier infinitely deep fluid. This equation, with \(p=1\), is an exactly integrable one. Much less is known about the generalized BO equations with \(p\geq2\). A conjecture is that such extended equations will, in the generic case, lead the wave collapse. In order to check this hypothesis directly, the authors perform many runs of numerical simulations of the equation with \(p=3\) and periodic boundary conditions. An important issue is to develop an adequate numerical scheme, as a straightforward one develops an intrinsic (artificial) instability due to the aliasing phenomenon. To overcome this problem, the authors devise a special adaptive scheme, which is demonstrated to be stable. Using this scheme, they demonstrate that the evolution of an initial pulse with a Gaussian shape indeed leads to a local singularity (blow-up of the solution) in finite time. It is conjectured that this should be a generic feature of the generalized BO equation for the case with \(p \geq2\).
scientific article; zbMATH DE number 2113146