Traveling waves of the regularized short pulse equation

Abstract

Abstract. In the present work, we revisit the so-called regularized short pulse equation (RSPE) and, in particular, explore the traveling wave solutions of this model. We theoretically analyze and numerically evolve two sets of such solutions. First, using a fixed point iteration scheme, we numerically integrate the equation to find solitary waves. It is found that these solutions are well approximated by a truncated series of hyperbolic secants. The dependence of the soliton's parameters (height, width, etc) to the parameters of the equation is also investigated. Second, by developing a multiple scale reduction of the RSPE to the nonlinear Schrödinger equation, we are able to construct (both standing and traveling) envelope wave breather type solutions of the former, based on the solitary wave structures of the latter. Both the regular and the breathing traveling wave solutions identified are found to be robust and should thus be amenable to observations in the form of few optical cycle pulses.

Modeling of ultrashort pulse propagation in lossy nonlinear metamaterials

“…Apart from these exact solutions, approximate peakon-like and breather-like solitary waves, which can be regarded as ultrashort gap solitons in nonlinear MMs, were also presented [29]. Furthermore, in [37], both localized and extended waveforms that arise in the SPE were considered, and direct simulations showed that the most robust solution is the breathertype one (although some of the single-hump variants of the periodic solutions may be preserved during evolution as well) -see also relevant work in Reference [32]. Notice that, in Reference [37], multi-peakon, multi-breather, and multi-hump waveforms were found to be subject to symmetry breaking instabilities, thus, being less robust during propagation.…”

“…Such corrections, play an important regularizing role on the SPE, which, otherwise, admits multi-valued (so-called "loop") solutions [31]. The RSPE, on the other hand, admits smooth traveling-wave solutions [30], which have structure similar to solitary waves of the NLS model [32]. In our system, however, such a 4th-order derivative term can not be obtained, because of the assumed form of the permittivity and permeability (cf.…”

Superoscillations and leaky spectra

“…It is now well understood how band-limted functions can possess local oscillations arbitrarily faster than their fastest Fourier components [1][2][3][4][5][6]. This is the mathematical phenomenon of superoscillation, with applications to physics [7]; examples are superresolution microscopy [8][9][10], and a possible mechanism for heating the solar corona [11].…”

Μαθηματικές Μοντελοποιήσεις Προβλημάτων Κυματικής Διάδοσης Σε Μη Γραμμικά Ηλεκτρομαγνητικά Μεταϋλικά

“…Τέτοιες διορθώσεις, παίζουν σημαντικό ρόλο στην SPE, η οποία, διαφορετικά, δέχεται λύσεις πολλαπλών τιμών (αποκαλούμενες «loop») [71]. Η RSPE έχει λύσεις ομαλούς παλμούς οδεύοντος κύματος [68], που έχουν παρόμοια δομή με τα μοναδικά κύματα NLS [77]. Ωστόσο, στο υπό εξέταση υλικό Drude-Lorentz δεν μπορεί να ενσωματωθεί ένας τέτοιος όρος παραγώγου 4-ης τάξης, λόγω της θεωρούμενης συναρτησιακής μορφής της επιτρεπτότητας και διαπερατότητας [βλ.…”