Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg–de Vries equations

A numerical approach to blow-up issues for dispersive perturbations of Burgers’ equation

“…This would correspond to p ≥ 4 for the generalized KdV equation. Recall that this type of blow-up for (1) is supported by numerical simulations in the super critical case p > 4 ( [8,34], and in the critical case p = 4 [34] and rigorously proven in the critical case p = 4 ( [44,45]). The dynamics of the blow-up is different in the L 2 -critical and in L 2 -supercritical cases.…”

“…In [34] it was shown that an implicit Runge-Kutta method of fourth order (IRK4), a two-stage Gauss scheme, is both very efficient and accurate up to blow-up for generalized KdV equations. For the initial value problem y = f (y, t), y(t 0 ) = y 0 and constant time steps t m , m = 0, 1, .…”

“…Accuracy of the numerical solution is controlled as discussed in [33,35,34] via the numerically computed energies (10) and (14) which will depend on time due to unavoidable numerical errors. We use the quantity…”

“…Dynamic rescaling. To study blow-up in the solutions to fKdV, the scale invariance (11) can be used as for gKdV and NLS equations in the form of a dynamic rescaling, see for instance [54,34] and references therein. To this end the constant λ in (11) is replaced with a t-dependent function L(t)…”

“…As in [34] for gKdV, we take into account the fact that the peak at x = x m (t) developing eventually into a blow-up is travelling with increasing speed. To address this, we choose a commoving frame.…”