5BAn extended tanh-function method is proposed for constructing multiple traveling wave solutions of nonlinear partial differential equations in a unified way. The key idea of this method is to take full advantage of a generalized Riccati equation involving two parameters and use its solutions to replace the tanh function in the tanh-function method. It is quite interesting that the sign of the parameters can be used to exactly judge the numbers and types of these traveling wave solutions.

Extended Tanh-function Method and its Applications to Nonlinear Equations

Dispersive optical solitons with Schrödinger–Hirota equation using the $$P^{6}$$-model expansion approach

We present an analytical study of modulation instability (MI) in metamaterial (MM) with first-order and second-order nonlinear dispersion for three types of saturable nonlinearity. Apart from the conventional MI gain spectrum, we find two additional MI gain spectra that arise from the second-order nonlinear dispersion in the MM (provided that the group-velocity dispersion does not vanish): one showing a low cutoff frequency but no high cutoff frequency and the other showing no cutoff frequency. Under certain conditions, a threshold power may be required for MI to occur. These MI gain spectra rarely exist in conventional materials and suggest the possibility of observing MI at arbitrarily high perturbation frequencies, which could be explored for the generation of high-repetition-rate ultrashort electromagnetic pulse trains.

Modulation instability with arbitrarily high perturbation frequencies in metamaterials with nonlinear dispersion and saturable nonlinearity

Considering the theory of electromagnetic waves from the Maxwell's equations, we introduce a (3+1)-dimensionsal cubic-quintic complex Ginzburg-Landau equation describing the dynamics of dissipative light bullets in nonlinear metamaterials. The model equation, which is derived beyond the slowly varying envelope approximation, includes the effects of diffraction, dispersion, loss, gain, cubic, and quintic nonlinearities, as well as cubic and quintic self-steepening effects. The modulational instability of the plane waves is studied both theoretically, using the linear stability analysis, and numerically, using direct simulations of the Fourier space of the proposed nonlinear wave equation, based on the Drude model. The linear theory predicts instability for any amplitude of the primary wave. Also, in the linear stability analysis, self-steepening effects of different orders are confronted and one discusses their effects on the behavior of the gain spectrum under both normal and anomalous group-velocity dispersion regimes. Analytical results are equally confronted to direct numerical simulations and fully agree with the predictions from the gain spectra. Modulational instability is manifested by clusters of solitons and multihump and dromion-like structures, whose emergence and features depend not only on system parameters, such as the cubic and quintic self-steepening coefficients, but also on the propagation distance under a suitable balance between nonlinear and dispersive effects.

Modulation instability in nonlinear metamaterials modeled by a cubic-quintic complex Ginzburg-Landau equation beyond the slowly varying envelope approximation.

Taking into account relaxing Kerr nonlinearity and walk-off effects, the conditions and gain spectra of cross-phase modulation-induced modulational instability (XPM-MI) of two incoherently copropagating optical waves of different frequencies and same polarization are investigated. We devote particular attention to the mixed case in which one pulse propagates under the normal group-velocity dispersion (GVD) regime, while the second one is under an anomalous GVD regime. We unveil that in the limit of an instantaneuous nonlinear response, the typical frequency with maximum gain converges to a finite value in the mixed GDV regime, while it continuously grows with the group-velocity mismatch in the normal GVD regime. As a result, the maximum gain typically decreases with the group-velocity mismatch in the mixed regime, contrasting with the opposite trend in the normal GVD regime. Further, we show that besides the mode having maximum gain at a frequency decaying with $1/{\ensuremath{\tau}}^{1/3}$ in the slow response limit, there is a second mode having maximum gain with a distinct scaling behavior ${\mathrm{\ensuremath{\Omega}}}_{\text{max}}\ensuremath{\propto}1/\ensuremath{\tau}$ in the absence of group-velocity mismatch. The associated maximum gains scale, respectively, as $1/{\ensuremath{\tau}}^{2/3}$ and $1/\ensuremath{\tau}$, thus signaling the corresponding quadratic and linear dispersion relation of these modes in the low-frequency limit. A detailed analysis of the influence of the nonlinear response time and group-velocity dispersion on the MI gain spectrum is also provided.

Modulation instability in noninstantaneous Kerr media with walk-off and cross-phase modulation for mixed group-velocity-dispersion regimes

Copropagation of two pulses of different frequencies and modulation instabilities induced by cross-phase modulation in metamaterials

Metamaterials can be designed to have desirable linear and nonlinear electromagnetic responses to light waves, thus become a new candidate for generating and manipulating solitons. Here, we present a review of optical solitons in metamaterials, focusing on the new properties of temporal, spatial and spatiotemporal solitons relating to the unique electromagnetic property of metamaterials.

Optical solitons in metamaterials

Solitons have been found in many physical branches. Previous investigations on solitons mainly focused on various
nonlinear dispersive media in nature. It is well known that in the nonlinear dispersive medium, soliton formation is due
to the exact balance between the nonlinearity and the group velocity dispersion (GVD). When the signs of nonlinearity
and GVD are the same, that is, the positive (focusing) nonlinearity vs. positive (normal) GVD, or the negative
(defocusing) nonlinearity vs. negative (anomalous) GVD, the interaction between nonlinearity and GVD may lead to the
generation of dark solitons. In this paper, we study the formation and propagation of dark electromagnetic solitons in the
metamaterials, artificial structures that display properties beyond those available in naturally occurring materials. We use
an extended Tanh-function expansion method to solve the nonlinear equation for ultrashort electromagnetic pulse
propagation in metamaterial with a nonlinear polarization, and get dark solitary solutions under various conditions. It is
found that, due to the role of the second-order nonlinear dispersion resulted from the dispersive magnetic permeability,
dark solitons can be formed in the absence of linear dispersion, or even in the case of anomalous linear GVD for a
focusing nonlinearity, challenging the traditional conditions for dark soliton formation.

Dark electromagnetic solitons in the anomalous group-velocity dispersion regime in metamaterials

We study the propagation property of soliton pulses in negative-index metamaterials with a nonlinear polarization, and
especially analyze the influence of the controllable self-steepening effect, which is resulted from the dispersive magnetic
permeability in negative-index metamaterials, on soliton formation and propagation. The results indicate that the
negative self-steepening effect also leads to the asymmetry of soliton pulse, the center shift and the decay of higher-order
soliton. In addition, the controllable self-steepening effect can be used to counteract the shift of soliton pulse resulted
from the third order dispersion effect to make the soliton pulse propagation without center shift to some extent.

Property of electromagnetic solitons in negative-index metamaterials

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Papers

Copropagation of two pulses of different frequencies and modulation instabilities induced by cross-phase modulation in metamaterials

Optical solitons in metamaterials

Dark electromagnetic solitons in the anomalous group-velocity dispersion regime in metamaterials

Property of electromagnetic solitons in negative-index metamaterials