Numerical Study of the Generalized Korteweg–de Vries Equations with Oscillating Nonlinearities and Boundary Conditions

The focus here is upon the generalized Korteweg–de Vries equation,ut+ ux+1pupx+ uxxx= 0,where p = 2, 3,.... When p ≥ 5, it is thought that the equation is not globally wellposed in time for L2-based Sobolev class data. Various numerical simulations carriedout by multiple research groups indicate that solutions can blowup in finite time forlarge, smooth initial data. This is known to be the case in the critical case p = 5,but remains a conjecture for supercritical values of p. Studied here are methods forcontrolling this potential blow up. Several candidates are put forward; the additionof dissipation or of higher order dispersion are two obvious candidates. However,these apparently can only work for a limited range of nonlinearities. However, theintroduction of high frequency temporal oscillations appear to be more effective. Bothtemporal oscillation of the nonlinearity and of the boundary condition in an initial-boundary-value configuration are considered. The bulk of the discussion will turnaround this prospect in fact.