Solitary Waves on Graphene Superlattices

Fractal structure of the soliton scattering for the graphene superlattice equation

“…Figure 1 shows the kink solution for b = 0, 10, 50, 100, and 150 (notice that in Ref. [13] it has been plotted for b ≤ 1). It has been numerically calculated by applying a Gauss-Konrod quadrature formula for the left-hand side of Eq.…”

“…The graphene superlattice equation (GSLeq) is expected to be non-integrable since it is not in the list of nonlinear Klein-Gordon equations passing the Painlevé test [11]; hence, its solitary waves are not true solitons, as suggested in Ref. [12] and in the authors' review paper [13]. The scattering of kinks and antikinks in non-integrable equations is generally characterized by the existence of a critical initial speed v cr for its initial head-on speed v. For v > v cr the solitons either pass through or bounce off each other reappearing after collisions with a phase shift in their positions.…”

Padé schemes with Richardson extrapolation for the sine-Gordon equation

“…The general solution for its initial-value problem can be obtained by using the inverse scattering method, a kind of nonlinear Fourier method [6,7]. The numerical study of the sGE with small perturbations still attract the attention of mathematicians and physicists [3], including new applications, like the propagation of nonlinear electromagnetic waves in graphene superlattices [8].…”

Является Ли Кинковое Решение Нелинейного Уравнения Клейна--Гордона Солитоном?

“…В работе [10] исследована возможность существования уединенных электромагнитных волн в графеновой сверхрешетке, выведено уравнение, описывающее распространение таких волн в рассматриваемой структуре. В литературе это уравнение получило название уравнение Крючкова-Кухаря (KKeq) [11]. Это уравнение является обобщением уравнения синус-Гордона, в явном виде учитывающим взаимную зависимость движений носителей тока в перпендикулярных друг другу направлениях.…”