Short pulse equations and localized structures in frequency band gaps of nonlinear metamaterials

Abstract

We consider short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. In the frequency "band gaps" (where linear electromagnetic waves are evanescent) with linear effective permittivity ǫ < 0 and permeability µ > 0, we derive two short-pulse equations (SPEs) for the high-and lowfrequency band gaps. The structure of the solutions of the SPEs is also briefly discussed, and connections with the soliton solutions of the nonlinear Schrödinger equation are presented. . The frequency dependence of the effective permittivity ǫ and permeability µ of these media is such that there exist frequency bands where the medium displays either a right-handed (RH) behavior (ǫ > 0, µ > 0) or a left-handed (LH) behavior (ǫ < 0, µ < 0), thus exhibiting negative refraction at microwave [2,3,4] or optical frequencies [5]. Frequency band gaps, i.e., frequency domains where linear EM waves are evanescent (e.g., for ǫ < 0 and µ > 0), also exist in metamaterials. Hence, when a nonlinearity occurs, say in the dielectric response of the medium (a physically relevant situation in nonlinear metamaterials [6,7,8,9,10,11,12]), then nonlinearity-induced localization of EM waves is possible. Such localization is indicated by the formation of gap solitons, which occur mainly in nonlinear optics [13] and Bose-Einstein condensates (BECs) [14], by means of the nonlinear Schrödinger (NLS) equation with a periodic potential. Gap solitons were also predicted to occur in nonlinear metamaterials [15]. There, the approximation of slowly varying electric and magnetic field envelopes, led to a nonlinear Klein-Gordon (NKG) equation supporting gap solitons.Nonlinear models describing localization of wave packets in periodic media, e.g., the NLS equation in optics [13] and NKG equation in metamaterials [15] are usually derived in the framework of the slowly-varying envelope approximation. However, as far as ultra-short pulse propagation is concerned, i.e, for pulse widths of the order of a few cycles of the carrier

A direct method for solving the generalized sine-Gordon equation II

“…was shown to model the evolution of ultra-short pulses in the band gap of nonlinear metamaterials [12]. Here, we demonstrate that the generalized sG equation ( …”

“…We can confirm this fact by considering the simplest 1-soliton solution. In fact, it follows from (4.9) and (4.10) that the corresponding tau functions are given by 12) where the bar appended to the variables is omitted for simplicity. The parametric solution (4.11) now becomes…”

Ultrashort pulses and short-pulse equations in 2+1 dimensions

“…Equation (5) is the (2 + 1)-dimensional generalization of the 1D Klein-Gordon type model derived in the context of nonlinear fiber optics [12,23] (in this case, μ = const) and nonlinear metamaterials [13,[24][25][26] (in this case, ∂ ωμ = 0). Below, we will analyze the latter (more general) case, and assume that both permittivity and permeability are frequency dependent.…”

“…Furthermore, this model is also interesting from a mathematical point of view, due to the existence of an infinite hierarchy of conserved quantities [15], its connection to the sG model and, thus, to its complete integrability [16]. The SPE admits various types of solutions, including singular soliton solutions-the so-called loop solitons [17]-as well as other nonsingular solutions, such as peakons, breather-and periodic-type wave forms [13,[17][18][19]. Note that, recently, wave-breaking phenomena [20], as well as the global well-posedness question [21] of the SPE, were also investigated.…”

Well-posedness and small data scattering for the generalized Ostrovsky equation

“…On the other hand, the short-pulse equation (SPE) proved to be extremely interesting from a mathematical point of view due to the existence of an infinite hierarchy of conserved quantities [10], an ingenious transformation that related it to the integrable sine-Gordon equation and illustrated its complete integrability [11] and which, in turn, allowed the calculation of explicit analytical solutions of loop-and of breather-form for this model [12]. More recently, on the analysis side, the global well-posedness question [13] and wave-breaking phenomena in this equation were studied [16], while interesting generalizations such as the regularized version of the SPE [17] and applications including the emergence of SPE in descriptions of nonlinear metamaterials [18] have also emerged. Notice that while we are not aware of applications presently of this equation for p 4, we will consider the equation in its generalized form presented above, keeping our results as general as possible in what follows.…”