1.1 Background and literature review
To study of optical soliton is a dynamic research area in the fields of mathematical physics [1] and fiber optic communication systems [2]. It can play a vital role in the telecommunication industry to explain the soliton propagation effect in optical fibers and their impact on optical fiber communication systems. In the context of fiber optics, the relevant models can explain the propagation of soliton pulses through intercontinental distances. The dynamics of soliton propagation through nonlinear optics, optical fibers, metamaterials, and crystals are described by several model equations, such as the nonlinear unstable Schrödinger’s equation [3], Sasa–Satsuma equation [4], complex Ginzburg–Landau equation [5], perturbed Gerdjikov–Ivanov equation [6], Lakshmanan–Posezian–Daniel equation [7, 8], Chen–Lee–Liu equation [9–11], Liquid crystals equation [12] and several others. It is worth mentioning that some researchers have studied a number of various known models and investigated their corresponding soliton dynamics via diverse analytical methods, viz. the Kudryashov method [13–15], the generalized Kudryashov method [16], the extended Kudryashov method [17], the trial solution method [18], the extended trial equation method [19], the modified simple equation method [20], the sine-Gordon expansion equation method [21, 22], the extended sinh-Gordon equation expansion method [23–25], simplest equation method [26], the extended simplest equation method [27], new extended direct algebraic method [28], new auxiliary equation expansion method [29] and so on. This paper deals with one of such models viz. the Kundu–Mukherjee–Naskar (KMN) equation, which can be applied to address optical wave propagation through coherently excited resonant waveguides in particular in the phenomena of bending of light beams [19]. It is also used to address the problems of hole waves and oceanic rogue waves [30]. The model can further find to be applicable to the study of soliton pulses occurring in (2+1)-dimensional equations [31]. The most important feature of this model is that it has been given as a new extension of nonlinear Schrödinger (NLS) equation with the inclusion of different forms of nonlinearity with regard to Kerr and non-Kerr law nonlinearities to study soliton pulses in (2+1)-dimensions [31, 32]. Recently, solitons in KMN equation have been addressed by several researchers to recover some optical solitons using trial equation technique [33], extended trial function method [19], ansatz approach and sine Gordon expansion method [34], F-expansion and functional variable principle [35], new extended algebraic method [36], the method of undetermined coefficients and Lie symmetry [37], modified simple equation approach [20, 38] and first integral method [39]. As a result, investigators have reported some new optical solutions such as dark, bright, singular type soliton solutions. However, no studies have been found to investigate the optical solutions to the KMN equation by using the generalized Kudryashov method (gKM) and new auxiliary equation method (NAEM).