Search

The focus here is upon the generalized Korteweg–de Vries equation, ut+ux+1pupx+uxxx=0,where p=2,3,…. When p≥5, it is thought that the equation is not globally well posed in time for L2-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.

Graphene has transformed the fields of plasmonics and photonics, and become an indispensable component for devices operating in the terahertz to mid-infrared range. Here, for instance, graphene surface plasmons can be excited, and their extreme interfacial confinement makes them vastly effective for sensing and detection. The rapid, robust, and accurate numerical simulation of optical devices featuring graphene is of paramount importance and many groups appeal to Black-Box Finite Element solvers. While accurate, these are quite computationally expensive for problems with simplifying geometrical features such as multiple homogeneous layers, which can be recast in terms of interfacial (rather than volumetric) unknowns. In either case, an important modeling consideration is whether to treat the graphene as a material of small (but non-zero) thickness with an effective permittivity, or as a vanishingly thin sheet of current with an effective conductivity. In this contribution we ponder the correct relationship between the effective conductivity and permittivity of graphene, and propose a new relation which is based upon a concrete mathematical calculation that appears to be missing in the literature. We then test our new model both in the case in which the interface deformation is non-trivial, and when there are two layers of graphene with non-flat interfacial deformation.

Second-order correct versions of the usual KdV–BBM models for unidirectional propagation of long-crested, surface water waves are considered here. The class of models studied here has a Hamiltonian structure and, in certain circumstances, is globally well posed. A fully discrete, numerical algorithm based on the Fourier-spectral method is developed and its convergence tested. We then use this algorithm to generate solitary-wave solutions to the model. While such waves are known to exist, exact formulas for them are not available. The heart of the paper is a sequence of numerical experiments aimed at understanding the stability of individual solitary waves, their interaction, and whether or not the model exhibits resolution of general initial data into solitary waves. A comparison is made between the first-order correct KdV–BBM models and the associated second-order correct equations. A number of tentative conjectures pertaining to the models are put forward on the basis of these experiments.