Solitary wave solutions and modulational instability analysis of the nonlinear Schrödinger equation with higher-order nonlinear terms in the left-handed nonlinear transmission lines

Highlights

• Left-handed nonlinear transmission line is considered with higher order nonlinearity.

• The nonlinear Schrödinger equation with cubic–quintic nonlinearities is introduced.

• The generation of envelope solitons is studied through modulational instability.

Abstract

We report the modulational instability (MI) analysis for the modulation equations governing the propagation of modulated waves in a practical left-handed nonlinear transmission lines with series of nonlinear capacitance. Considering the voltage in the spectral domain and the Taylor series around a certain modulation frequency, we show in the continuum limit, that the dynamics of localized signals is described by a nonlinear Schrödinger equation with a cubic–quintic nonlinear terms. The MI process is then examined and we derive the gain spectra of MI for the generation of solitonlike-object in the transmission line metamaterials. We emphasize on the effect of losses on the MI gain spectra. An exact kink-darklike solutions is derived through the auxiliary equation method. It comes out that the width of the darklike solution decreases as the attenuation constant increases. Our theoretical solution is in good agreement with our numerical observation.

Introduction

Metamaterials (MMs), first known as left-handed materials (LHMs) or negative refractive index materials (NIMs), still attract attention in the scientific communities nowadays. The concept of MMs has a much broader scope than that of LHM or NIM. Due in large part to MMs, the classical subject of electromagnetism and optics have experienced a number of new discoveries and advances in research. First proposed by Veselago theoretically in 1968 [1] for a material whose electric permittivity and magnetic permeability are simultaneously negative, LHM possesses many new features such as negative refraction, backward wave propagation, reversed Doppler shift, and backward Cerenkov radiation. Research in LHM was stagnant for more than 30 years due to the lack of experimental verification. The revolution dealing with LHM occurred in 1996 [2] when Sir Pendry discovered the wire medium whose permittivity is negative, followed by the discovery of negative permeability by Sir Pendry et al. in 1999 [3] and LHM by Smith et al. in 2000 [4]. LHM has the unavoidable disadvantage of big loss and narrow bandwidth, and such disadvantages restrict the applications of LHM.

Transmission line (TL) theory, which is a well-known method for the analysis and design of electromagnetism (EM) materials and applications [5], can also be used for the study of the EM properties of the LHMs. LH nonlinear transmission lines (NLTLs) are often employed as the nonlinear media, which incorporate loaded nonlinear capacitance or inductance elements with the ordinary LH TL structures [6], [7], [8], [9]. Such NLTLs provide a simple and easy-to-realize system to investigate the EM wave propagation in nonlinear LH media [10]. In this sense, several researchers have further studied the characteristics and applications of split-ring resonator (SRR)-based LHMs. However, since the resonant structures such as SRRs are lossy, narrow-banded, and most importantly, anisotropic, they have not performed as expected, even at microwave frequency, let alone at higher frequencies. They have prompted several researchers to investigate a TL approach to realizing LHMs [11]. This approach has, in turn, led to nonresonant structures that have lower losses and wider bandwidths. In particular, MMs with right-handed (RH) and LH properties known as composite right/left handed (CRLH) MMs have led to the development of several novel microwave devices [12], [13], [14]. Several novel applications may emerge from the exploitation of their nonlinear behavior. The excitation of nonlinearities in LH NL TL leads to some effects and therefore further applications. Solitary wave and soliton propagation is one of such nonlinear phenomena [15]. Nonlinearity typically originates from the sensitivity of the refractive index of the medium to the intensity of the propagating signal. Nonlinearity leads to the self-phase modulation (SPM) phenomenon, resulting in spectral broadening and instantaneous frequency variation along the temporal profile of the pulse. On the other hand, dispersion results in the temporal broadening featuring a temporal chirp. In the balanced condition where the instantaneous chirp frequency from SPM is counterbalanced by the temporal chirp from dispersion, the pulse preserves its shape during propagation. In optical fibers, solitons are typically Schrödinger-type solitons, which are modulated solitons [15]. Recently, Schrödinger solitons have also been demonstrated in LH MM TL [7], [9], [16], [17], [19].

In physics problems, the quintic nonlinearity can be equal to or even more important than the cubic one [20], [21] as it is responsible for stability of localized solutions. So, in this work, we derive through the Taylor series around a certain modulation frequency approach, a nonlinear Schrödinger (NLS) equation with higher-order nonlinearity for the CRLH TL. More specifically, we are interested in how the presence of the losses affects the MI of the background and the possibility of creation of waves localized in space and time in such systems. The rest of the paper is organized as follows. The model and equation of motion is introduced in Section 2. Then, we revisit the MI criteria in the cubic- quintic (CQ) NLS equation in Section 3. Exact analytical solutions and numerical proof-of this result reported in Section 4. The last section concludes the work.