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

Abstract

The focus here is upon the generalized Korteweg–de Vries equation, $$\begin{aligned} u_t + u_x + \frac{1}{p} \left( u^p \right) _x +u_{xxx} \, = \, 0, \end{aligned}$$where \(p = 2, 3, \ldots \). When \(p \ge 5\), it is thought that the equation is not globally well posed in time for \(L_2\)-based Sobolev class data. Various numerical simulations carried out by multiple research groups indicate that solutions can blowup in finite time for large, 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 for controlling this potential blow up. Several candidates are put forward; the addition of dissipation or of higher order dispersion are two obvious candidates. However, these apparently can only work for a limited range of nonlinearities. However, the introduction of high frequency temporal oscillations appear to be more effective. Both temporal oscillation of the nonlinearity and of the boundary condition in an initial-boundary-value configuration are considered. The bulk of the discussion will turn around this prospect in fact.