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).