1. Introduction
Nonlinear Schrödinger equations cover a wide range of complex-values equations, consisting of both the linear and nonlinear partial differential equations, which have much relevance in quantum physics, optics, nonlinear waves, and the design of optical fiber devices to mention but a few. These equations are precisely used to describe the movement of electrons concerning energies and locations at every prescribed space–time position. In addition, modern communication and networking industries working with optical fiber cables and devices enormously utilize various nonlinear Schrödinger equations for perfect modeling of the burning intricate state of affairs, see [1–3] and the references therein. Moreover, among special classes of nonlinear Schrödinger equations is the class of the Gerdjikov–Ivanov equations [(4 )-(7 )] that model several complex-valued dynamical procedures in optical fibers and the field of nonlinear complex-valued evolution equations; more specifically, the Gerdjikov–Ivanov model, which is equally referred to as derivative nonlinear Schrodinger equation 3 , is relevant in pulse propagation in photonic crystal fibers, optical fibers, and metamaterials to mention a few. In addition, various analytical procedures have been used in recent times to examine this subclass of equations, including the use of the famous sine-Gordon method [5], and many other methods, see [6, 7] and references therein. Additionally, the enormous relevance of these nonlinear evolution equations in describing contemporary processes cannot be over-emphasized; as such, various researchers embarked on a searching race for optimal analytical and computational techniques. Analytical solutions are very important in unearthing some of the physical characteristics associated with the governing model while serving as estimators for computational examinations. Thus, among the notable analytical methods for the study of nonlinear evolution equations include Kudryashov's technique [8, 9], ansatz and sub-ODE method [10], mode-matching approach for acoustic waves in waveguide [11], commutative hyper-complex approach [12], collective variables methods [13], bilinear Hirota approach [14], Lie's symmetry approach [15], exponential solution ansatz [16], Riccati equation approach [17], exponential expansion method [18], $(G/G^{\prime} )$-expansion approach [19], auxiliary equation mapping method [20], tanh expansion methods [21, 22] and the decomposition method by Adomian [23, 24] among others; read also the recent finding in [25–27] and references therein for a more updated submission on the methodology, analysis as well as implementation of some efficient analytical approaches on various evolution equations.
However, this study further intends to deploy the classical Kudryashov's method [8, 9] together with some of its variants, including, for instance, the modified Kudryashov's method [28] and the enhanced Kudryashov's method [29] to examine the class of Gerdjikov–Ivanov equations [4 –7 ] in favor of its vast applications. Certainly, Kudryashov's method is among the famous analytical methods for the study of nonlinear evolution equations that give various exact solutions in terms of exponential functions; by the way, these exponential solutions can be transformed into some hyperbolic solutions, which have more applications in the study of pulse propagations in optical media. In addition, the importance and great efficiency of Kudryashov's method have led several authors to propose several modifications, including the generalized Kudryashov's method [30, 31] improved general Kudryashov's approach [32] and the generalized extended Kudryashov's method [33] to mention but a few. Undeniably, these variants of Kudryashov's method outlined in [28–33] provide more analytical solutions due to the presence of additional parameters either in the related Riccati equation or in the predicted solution form, which ultimately give room for multiple exact solutions.
Furthermore, motivated by the immense applications of the Kudryashov method [8, 9] and its variants, especially the modified and enhanced Kudryashov methods in [28] and [9], respectively, for their easy accomplishment and the fact their solution can be transformed into various solitonic equations; in addition to the lack of substantial literature on the class of Gerdjikov–Ivanov equations with regards to availability of assorted solitonic solutions, this study is thus aimed at deploying these very Kudryashov-based methods to construct diverse exponential solitonic solutions for the class Gerdjikov–Ivanov equations [(4 )–(7 )]. In addition, both the implementation of the methods as well as the analysis of the acquired solutions that are put forward by the methods will be carried out. In the same vein, the significance of the present study is enormous, looking at its importance in nonlinear sciences, in addition to its additional value to various numerical and experimental studies. Lastly, some solutions to be acquired will be graphically portrayed and discussed, concerning the evolutional influences of the involving parameters, including the relevance of the Kudryashov-index d in the propagation profiles. Finally, we arrange the present manuscript as follows: section 2 describes the governing model; section 3 gives the steps for the beseeched methods; sections 4 and 5 give the application of the adopted methods on the standard Gerdjikov–Ivanov equation and perturbed Gerdjikov–Ivanov equation, respectively; section 6 gives the graphical illustrations and discussion of results, while section 7 gives the concluding remarks.
2. Governing models
This section gives the governing models, the Gerdjikov–Ivanov equations [4 –7 ] that model several complex-valued dynamical procedures in optical fibers and the field of nonlinear complex-valued evolution equations. Certainly, the Gerdjikov–Ivanov model, which is equally referred to as derivative nonlinear Schrodinger equation III, is relevant in pulse propagation in photonic crystal fibers, optical fibers, and metamaterials to mention a few. The two models are nonlinear complex-valued partial differential equations that would later be transformed into their corresponding nonlinear ODEs via the application of wave transformation for a smooth implementation of the sought analytical procedures.
2.1. Standard Gerdjikov–Ivanov equation
The one-dimensional standard Gerdjikov–Ivanov equation to be examined in the present study is given by the following non-dimensional complex-valued partial differential equations as follows [4–7]
$\begin{eqnarray}\iota {q}_{t}+{{aq}}_{{xx}}+b| q{| }^{4}q+\iota {{cq}}^{2}{q}_{x}^{* }=0,\end{eqnarray}$
where $\iota =\sqrt{-1}$ is the imaginary number, q = q(x, t) is a complex-valued wave profile of the spatial variable x and temporal variable t. In addition, a is the coefficient of the Group-Velocity Dispersion (GVD), b is the coefficient of the quintic nonlinearity term, while the parameter c is the coefficient of the nonlinear dispersive term. Moreover, one can express ∣q∣2 as qq*, that is, ∣q∣2 = qq*, with ∣q∣ as a modulus of a complex function q, while q* is a conjugate of a complex function q.2.2. Perturbed Gerdjikov–Ivanov equation
The one-dimensional perturbed Gerdjikov–Ivanov equation to be examined in the present study is given by the following non-dimensional complex-valued partial differential equations as follows [4–7]
$\begin{eqnarray}\begin{array}{rcl} & & \iota {q}_{t}+{{aq}}_{{xx}}+b| q{| }^{4}q+\iota {{cq}}^{2}{q}_{x}^{* }\\ & & \quad =\iota \left[\alpha {q}_{x}+\beta {\left(| q{| }^{2n}q\right)}_{x}+\gamma {\left(| q{| }^{2n}\right)}_{x}q\right],\end{array}\end{eqnarray}$
where $\iota =\sqrt{-1}$ is the imaginary number, q = q(x, t) is a complex-valued wave profile of the spatial variable x and temporal variable t. In addition, a is the coefficient of the GVD, b is the coefficient of the quintic nonlinearity term, c is the coefficient of the nonlinear dispersive term, while q* is a conjugate of a complex function q. Moreover, on the right-hand side, α represents the inter-modal dispersion, β and γ denote the self-steepening and higher-order dispersion effects, respectively, while n( ≥ 1) denotes the full nonlinearity.2.3. Model analysis
This subsection gives the analytical analysis of the two models prior to the implementation of the three adopted methods.
| I. Standard Gerdjikov–Ivanov equation To begin with, the governing complex-valued standard Gerdjikov–Ivanov equation in equation ( $\begin{eqnarray}q(x,t)=P(\xi ){{\rm{e}}}^{\iota Q(x,t)},\end{eqnarray}$ where $\iota =\sqrt{-1},$ ξ is the wave transformation variable, while Q(x, t) is a real-valued function explicitly expressed as follows $\begin{eqnarray}\xi =\eta (x-\upsilon t),\qquad \mathrm{and}\ \qquad Q(x,t)=-{kx}+\omega t+\vartheta ,\end{eqnarray}$ where k, ω, and ϑ are the soliton's frequency, wavenumber, and the phase constant, respectively. Moreover, substituting equation ( $\begin{eqnarray}\left({{ak}}^{2}+\omega \right)P+{{ckP}}^{3}-{{bP}}^{5}-a{\eta }^{2}P^{\prime\prime} =0,\end{eqnarray}$ where $P^{\prime\prime} =\tfrac{{{\rm{d}}}^{2}P}{{\rm{d}}{\xi }^{2}},$ while the imaginary component reveals $\begin{eqnarray}\upsilon =-2{ak}+{{cP}}^{2},\end{eqnarray}$ where P = P(ξ). Indeed, the above equation in equation ( | |
| II. Perturbed Gerdjikov–Ivanov equation In the same way, the complex-valued perturbed Gerdjikov–Ivanov equation in equation ( $\begin{eqnarray}\left({{ak}}^{2}+\alpha k+\omega \right)P+{{ckP}}^{3}-{{bP}}^{5}+\beta {{kP}}^{2n+1}-a{\eta }^{2}P^{\prime\prime} =0,\end{eqnarray}$ where $P^{\prime\prime} =\tfrac{{{\rm{d}}}^{2}P}{{\rm{d}}{\xi }^{2}},$ while the imaginary component reveals $\begin{eqnarray}\upsilon =-2{ak}-\alpha +{{cP}}^{2}-(\beta +2\beta n+2\gamma n){P}^{2n},\end{eqnarray}$ where P = P(ξ). Further, the above equation in equation ( |
In addition, the present study is set to examine the reduced nonlinear ODE determined in equation (5 ) and equation (7 ) for the standard Gerdjikov–Ivanov equation and perturbed Gerdjikov–Ivanov equation, respectively, considering some interesting Kudryashov-based methods outlined in the next section.
3. Kudryashov-based methods
Let us refer to the following generalized nonlinear partial differential equation10 ) into equation (9 ), one gets a reduced ODE of the following generalized form
Moreover, upon substituting the presumed solution of the three Kudryashov-based methods coupled with the respective related differential equations into the acquired reduced ODE in (11 ), one obtains a polynomial in φ(ξ). Further, the resulting polynomial is then examined by setting every coefficient of φj(ξ) for j = 0, 1, 2,…,M to zero to obtain an algebraic system of equations. Next, upon solving these equations, one obtains the various solution sets for Aj, for j = 0, 1, 2,…,M, (not all equal to zero) which eventually yield the solutions for equation (9 ) through the back substitution. Moreover, it is very relevant to state here that this tedious work is carried out in this study with the assistance of Mathematica software.
$\begin{eqnarray}{K}_{1}(q,{q}_{x},{u}_{t},{q}_{{xx}},{q}_{{tt}},{q}_{x}{q}_{t},{q}_{x}{q}_{{tt}},{q}_{{xx}}{q}_{t},{q}_{{xt}}{q}_{t},\ldots )=0.\end{eqnarray}$
Then, if ξ is the wave variable, we seek the following wave transformation $\begin{eqnarray}q(x,t)=P(\xi ),\qquad \mathrm{with}\qquad \xi =\eta (x-\upsilon t),\end{eqnarray}$
with k and $\upsilon$ as non-zero wave constants. Next, substituting transformation expressed in equation ( $\begin{eqnarray}{K}_{2}(P(\xi ),P^{\prime} (\xi ),P^{\prime\prime} (\xi ),P\prime\prime\prime (\xi ),P\unicode{x02057}(\xi ),...)=0,\end{eqnarray}$
where the primes denote derivatization with respect to ξ; that is, $P^{\prime} =\tfrac{{\rm{d}}P}{{\rm{d}}\xi },P^{\prime\prime} =\tfrac{{{\rm{d}}}^{2}P}{{\rm{d}}{\xi }^{2}},P\prime\prime\prime =\tfrac{{{\rm{d}}}^{3}P}{{\rm{d}}{\xi }^{3}},\ldots $. Further, in what follows, we present three methods that are based on the classical Kudryashov's method. | 3.1 Classical Kudryashov's method Considering the ODE delineated in equation ( $\begin{eqnarray}P(\xi )=\displaystyle \sum _{j=0}^{M}{A}_{j}{\phi }^{j}(\xi ),\end{eqnarray}$ where Aj, for j = 1, 2, …, M are constants that are not all equal to zero—to be explicitly obtained, while M is a whole number to be acquired via homogeneous balancing. In addition, the function φ(ξ) in the above equation is said to satisfy the following differential equation $\begin{eqnarray}\phi ^{\prime} (\xi )=\phi (\xi )(\phi (\xi )-1),\end{eqnarray}$ that has an exact analytical solution as follows $\begin{eqnarray}\phi (\xi )=\displaystyle \frac{1}{{{d}{\rm{e}}}^{\xi }+1},\end{eqnarray}$ where d is a non-zero arbitrary constant. | |
| 3.2 Modified Kudryashov's method The modified Kudryashov's method [28] associates the predicted solution of equation ( $\begin{eqnarray}P(\xi )=\displaystyle \sum _{j=0}^{M}{A}_{j}{\phi }^{j}(\xi ),\end{eqnarray}$ where Aj, for j = 1, 2, …, M are equally constants that are not all equal to zero that are to be explicitly obtained, while M is a whole number to be acquired via homogeneous balancing. Further, the function φ(ξ) in equation ( $\begin{eqnarray}\phi ^{\prime} (\xi )=\phi (\xi )(\phi (\xi )-1)\mathrm{ln}(r);\end{eqnarray}$ having the following exact analytical solution $\begin{eqnarray}\phi (\xi )=\displaystyle \frac{1}{{{dr}}^{\xi }+1},\end{eqnarray}$ where d is a non-zero arbitrary constant; while r ≠ 1 is a non-zero positive integer. | |
| 3.3 Enhanced Kudryashov's method The enhanced Kudryashov's method [9] begins with the presumption of the following solution for equation ( $\begin{eqnarray}P(\xi )=\displaystyle \sum _{j=0}^{M}{A}_{j}{\phi }^{j}(\xi ),\end{eqnarray}$ where Aj, for j = 1, 2, …, M are equally constants that are not all equal to zero that are to be explicitly obtained, while M is a whole number to be acquired via homogeneous balancing. In this case, the function φ(ξ) in equation ( $\begin{eqnarray}\phi {{\prime} }^{2}(\xi )={\phi }^{2}(\xi )(1-r{\phi }^{2}(\xi )),\end{eqnarray}$ which the latter admits the following exact solution $\begin{eqnarray}\phi (\xi )=\displaystyle \frac{4d}{4{d}^{2}{e}^{\xi }+{{r}{\rm{e}}}^{-\xi }},\end{eqnarray}$ where d and r are non-zero arbitrary constants. |
4. Standard Gerdjikov–Ivanov equation
In this regard, we re-consider the reduced model in equation (5 ) as follows21 ) becomes
$\begin{eqnarray}\left({{ak}}^{2}+\omega \right)P+{{ckP}}^{3}-{{bP}}^{5}-a{\eta }^{2}P^{\prime\prime} =0.\end{eqnarray}$
Therefore, upon balancing P5 and P″(ξ) in the above equation, one obtains $\begin{eqnarray}5M=M+2,\Longrightarrow M=\displaystyle \frac{1}{2},\end{eqnarray}$
which further necessitates yet a new transformation of the following form $\begin{eqnarray}P{(\xi )=R(\xi )}^{\tfrac{1}{2}},\end{eqnarray}$
upon which equation ( $\begin{eqnarray}4\left({{ak}}^{2}+\omega \right){R}^{2}+4{{ckR}}^{3}-4{{bR}}^{4}+a{\eta }^{2}R{{\prime} }^{2}-2a{\eta }^{2}{RR}^{\prime\prime} =0,\end{eqnarray}$
where R = R(ξ). Equally, we homogenously balance the above equation, utilizing the terms R4 and RR″(ξ) to obtain $\begin{eqnarray*}4M=M+(M+2),\Longrightarrow M=1.\end{eqnarray*}$
Furthermore, with M = 1, the predicted solution takes the following form (in all the three deployed methods) $\begin{eqnarray}R(\xi )={A}_{0}+{A}_{1}\phi (\xi ).\end{eqnarray}$
4.1. Classical Kudryashov's method
In this method, the predicted solution form in equation (25 ) is substituted into equation (24 ) alongside fully implementing the classical Kudryashov's method to obtain the following system of algebraic equations
$\begin{eqnarray*}4{{aA}}_{0}^{2}{k}^{2}-4{A}_{0}^{4}b+4{A}_{0}^{3}{ck}+4{A}_{0}^{2}\omega =0,\end{eqnarray*}$
$\begin{eqnarray*}8{{aA}}_{1}{A}_{0}{k}^{2}-2{{aA}}_{1}{A}_{0}{\eta }^{2}-16{A}_{1}{A}_{0}^{3}b+12{A}_{1}{A}_{0}^{2}{ck}+8{A}_{1}{A}_{0}\omega =0,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl} & & 6{{aA}}_{0}{A}_{1}{\eta }^{2}-{{aA}}_{1}^{2}{\eta }^{2}+4{{aA}}_{1}^{2}{k}^{2}-24{A}_{0}^{2}{A}_{1}^{2}b\\ +12{A}_{0}{A}_{1}^{2}{ck}+4{A}_{1}^{2}\omega & = & 0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}4{{aA}}_{1}^{2}{\eta }^{2}-4{{aA}}_{0}{A}_{1}{\eta }^{2}-16{A}_{0}{A}_{1}^{3}b+4{A}_{1}^{3}{ck} & = & 0,\\ -3{{aA}}_{1}^{2}{\eta }^{2}-4{A}_{1}^{4}b & = & 0,\end{array}\end{eqnarray*}$
which when solved yields the following solution sets:Set-I
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & 0,\qquad {A}_{1}={A}_{1},\qquad \omega =\displaystyle \frac{1}{4}\left(a{\eta }^{2}-4{{ak}}^{2}\right),\\ b & = & -\displaystyle \frac{3a{\eta }^{2}}{4{A}_{1}^{2}},\qquad c=-\displaystyle \frac{a{\eta }^{2}}{{A}_{1}k}.\end{array}\end{eqnarray}$
Thus, with the above solution set, the governing model admits the following exact analytical solution $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{\displaystyle \frac{{A}_{1}}{{{d}{\rm{e}}}^{\eta (x-\upsilon t)}+1}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(a{\eta }^{2}-4{{ak}}^{2}\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A1 and d are any non-zero arbitrary constants.Set-II
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & -{A}_{1},\qquad {A}_{1}={A}_{1},\qquad \omega =\displaystyle \frac{1}{4}\left(a{\eta }^{2}-4{{ak}}^{2}\right),\\ b & = & -\displaystyle \frac{3a{\eta }^{2}}{4{A}_{1}^{2}},\qquad c=\displaystyle \frac{a{\eta }^{2}}{{A}_{1}k}.\end{array}\end{eqnarray}$
Thus, with the above solution set, the governing model admits the following analytical solution $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{\displaystyle \frac{{A}_{1}}{{{d}{\rm{e}}}^{\eta (x-\upsilon t)}+1}-{A}_{1}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(a{\eta }^{2}-4{{ak}}^{2}\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A1 and d are any non-zero arbitrary constants.4.2. Modified Kudryashov's method
Accordingly, when fully implementing the modified Kudryashov's method, the following algebraic equations are revealed
$\begin{eqnarray*}4{{aA}}_{0}^{2}{k}^{2}-4{A}_{0}^{4}b+4{A}_{0}^{3}{ck}+4{A}_{0}^{2}\omega =0,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}8{{aA}}_{1}{A}_{0}{k}^{2}-2{{aA}}_{1}{A}_{0}{\eta }^{2}{\mathrm{ln}}^{2}(r)-16{A}_{1}{A}_{0}^{3}b\\ \quad +\,12{A}_{1}{A}_{0}^{2}{ck}+8{A}_{1}{A}_{0}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}4{{aA}}_{1}^{2}{k}^{2}-{{aA}}_{1}^{2}{\eta }^{2}{\mathrm{ln}}^{2}(r)+6{{aA}}_{0}{A}_{1}{\eta }^{2}{\mathrm{ln}}^{2}(r)\\ -\,24{A}_{0}^{2}{A}_{1}^{2}b+12{A}_{0}{A}_{1}^{2}{ck}+4{A}_{1}^{2}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}4{{aA}}_{1}^{2}{\eta }^{2}{\mathrm{ln}}^{2}(r)-4{{aA}}_{0}{A}_{1}{\eta }^{2}{\mathrm{ln}}^{2}(r)-16{A}_{0}{A}_{1}^{3}b\\ +\,4{A}_{1}^{3}{ck}=0,-3{{aA}}_{1}^{2}{\eta }^{2}{\mathrm{ln}}^{2}(r)-4{A}_{1}^{4}b=0.\end{array}\end{eqnarray*}$
Next, on solving the above algebraic system of equations, one acquires the following solution sets:Set-I
$\begin{eqnarray}\begin{array}{l}{A}_{0}=0,\quad {A}_{1}={A}_{1},\quad \omega =\displaystyle \frac{1}{4}\left(a{\eta }^{2}{\mathrm{ln}}^{2}(r)-4{{ak}}^{2}\right),\\ b=-\displaystyle \frac{3a{\eta }^{2}{\mathrm{ln}}^{2}(r)}{4{A}_{1}^{2}},\\ c=-\displaystyle \frac{a{\eta }^{2}{\mathrm{ln}}^{2}(r)}{{A}_{1}k},\end{array}\end{eqnarray}$
that leads to the resulting exact solution as follows $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{\displaystyle \frac{{A}_{1}}{{{dr}}^{\eta (x-\upsilon t)}+1}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(a{\eta }^{2}{\mathrm{ln}}^{2}(r)-4{{ak}}^{2}\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A1, d and r are any non-zero arbitrary constants.Set-II
$\begin{eqnarray}{A}_{0}=-{A}_{1},\quad {A}_{1}={A}_{1},\quad \omega =\displaystyle \frac{1}{4}\left(a{\eta }^{2}{\mathrm{ln}}^{2}(r)-4{{ak}}^{2}\right),\quad b=-\displaystyle \frac{3a{\eta }^{2}{\mathrm{ln}}^{2}(r)}{4{A}_{1}^{2}},\quad c=\displaystyle \frac{a{\eta }^{2}{\mathrm{ln}}^{2}(r)}{{A}_{1}k},\end{eqnarray}$
that leads to the resulting exact solution as follows $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{\displaystyle \frac{{A}_{1}}{{{dr}}^{\eta (x-\upsilon t)}+1}-{A}_{1}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(a{\eta }^{2}{\mathrm{ln}}^{2}(r)-4{{ak}}^{2}\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A1, d and r are any non-zero arbitrary constants.4.3. Enhanced Kudryashov's method
As proceeds, implementing the enhanced Kudryashov's method on the governing model yields the following system of algebraic equations
$\begin{eqnarray*}4{{aA}}_{0}^{2}{k}^{2}-4{A}_{0}^{4}b+4{A}_{0}^{3}{ck}+4{A}_{0}^{2}\omega =0,\end{eqnarray*}$
$\begin{eqnarray*}-\,2{{aA}}_{1}{A}_{0}{\eta }^{2}+8{{aA}}_{1}{A}_{0}{k}^{2}-16{A}_{1}{A}_{0}^{3}b+12{A}_{1}{A}_{0}^{2}{ck}+8{A}_{1}{A}_{0}\omega =0,\end{eqnarray*}$
$\begin{eqnarray*}-\,{{aA}}_{1}^{2}{\eta }^{2}+4{{aA}}_{1}^{2}{k}^{2}-24{A}_{0}^{2}{A}_{1}^{2}b+12{A}_{0}{A}_{1}^{2}{ck}+4{A}_{1}^{2}\omega =0,\end{eqnarray*}$
$\begin{eqnarray*}4{{aA}}_{0}{A}_{1}{\eta }^{2}r-16{A}_{0}{A}_{1}^{3}b+4{A}_{1}^{3}{ck}=0,3{{aA}}_{1}^{2}{\eta }^{2}r-4{A}_{1}^{4}b=0,\end{eqnarray*}$
that solves to the following solution sets:Set-I
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & {A}_{0},\quad {A}_{1}=-{A}_{0}\sqrt{r},\quad \omega =\displaystyle \frac{1}{4}\left(-5a{\eta }^{2}-4{{ak}}^{2}\right),\\ b & = & \displaystyle \frac{3a{\eta }^{2}}{4{A}_{0}^{2}},\quad c=\displaystyle \frac{2a{\eta }^{2}}{{A}_{0}k},\end{array}\end{eqnarray}$
and eventually leads to the exact exponential solution for the model as follows $\begin{eqnarray}\begin{array}{rcl}q(x,t) & & =\sqrt{{A}_{0}-\displaystyle \frac{4{A}_{0}d\sqrt{r}}{4{d}^{2}{{\rm{e}}}^{\xi }+{{r}{\rm{e}}}^{-\xi }}}\\ & & \times \,\exp \left(\iota \left(-{kx}-\displaystyle \frac{1}{4}t\left(5a{\eta }^{2}+4{{ak}}^{2}\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A0, d and r are any non-zero arbitrary constants.Set-II
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & {A}_{0},\quad {A}_{1}={A}_{0}\sqrt{r},\quad \omega =\displaystyle \frac{1}{4}\left(-5a{\eta }^{2}-4{{ak}}^{2}\right),\\ b & = & \displaystyle \frac{3a{\eta }^{2}}{4{A}_{0}^{2}},\quad c=\displaystyle \frac{2a{\eta }^{2}}{{A}_{0}k},\end{array}\end{eqnarray}$
and eventually leads to the exact exponential solution for the model as follows $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{{A}_{0}+\displaystyle \frac{4{A}_{0}d\sqrt{r}}{4{d}^{2}{{\rm{e}}}^{\xi }+{{r}{\rm{e}}}^{-\xi }}}\\ & & \times \,\exp \left(\iota \left(-{kx}-\displaystyle \frac{1}{4}t\left(5a{\eta }^{2}+4{{ak}}^{2}\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A0, d and r are any non-zero arbitrary constants. Moreover, upon setting the arbitrary constant r to take the form r = 4d2, the obtained solutions in this subsection yield singular and bright solitonic solutions, see [29] for more explanation on this development.5. Perturbed Gerdjikov–Ivanov equation
The governing nonlinear perturbed Gerdjikov–Ivanov equation in equation (2 ) is considered here. As we proceed, we further re-consider the reduced ODE in equation (7 ) as follows38 ) becomes
$\begin{eqnarray}\left({{ak}}^{2}+\alpha k+\omega \right)P+{{ckP}}^{3}-{{bP}}^{5}+\beta {{kP}}^{2n+1}-a{\eta }^{2}P^{\prime\prime} =0.\end{eqnarray}$
In the above equation, upon balancing P2n+1 and P″(ξ), one obtains $\begin{eqnarray}(2n+1)M=M+2,\Longrightarrow M=\displaystyle \frac{1}{n},\end{eqnarray}$
which further necessitates yet a new transformation of the following form $\begin{eqnarray}P{(\xi )=R(\xi )}^{\tfrac{1}{n}}.\end{eqnarray}$
In particular, when n is assumed to take the well-known form, that is, when n = 2, one gets $P{(\xi )=R(\xi )}^{\tfrac{1}{2}}$ upon which equation ( $\begin{eqnarray}\begin{array}{rcl} & & 4(k({ak}+\alpha )+\omega ){R}^{2}+4k(\beta +c){R}^{3}-4{{bR}}^{4}\\ & & \quad +a{\eta }^{2}R{{\prime} }^{2}-2a{\eta }^{2}{RR}^{\prime\prime} =0,\end{array}\end{eqnarray}$
where R = R(ξ). In addition, we homogenously balance the above equation, utilizing the terms R4 and RR″(ξ) to equally obtain 4M = M + (M + 2), ⇒ M = 1. Hence, with M = 1, the predicted solution form is as follows (from all the three deployed approaches) $\begin{eqnarray}R(\xi )={A}_{0}+{A}_{1}\phi (\xi ).\end{eqnarray}$
5.1. Classical Kudryashov's method
As explained in the methodology, the classical Kudryashov's method reveals the following algebraic system of equations
$\begin{eqnarray*}4{{aA}}_{0}^{2}{k}^{2}-4{A}_{0}^{4}b+4{A}_{0}^{3}{ck}+4\alpha {A}_{0}^{2}k+4{A}_{0}^{3}\beta k+4{A}_{0}^{2}\omega =0,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}-\ 2{{aA}}_{1}{A}_{0}{\eta }^{2}+8{{aA}}_{1}{A}_{0}{k}^{2}-16{A}_{1}{A}_{0}^{3}b\\ \quad +\ 12{A}_{1}{A}_{0}^{2}{ck}+8\alpha {A}_{1}{A}_{0}k+12{A}_{1}{A}_{0}^{2}\beta k+8{A}_{1}{A}_{0}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl} & & -{{aA}}_{1}^{2}{\eta }^{2}+6{{aA}}_{0}{A}_{1}{\eta }^{2}+4{{aA}}_{1}^{2}{k}^{2}-24{A}_{0}^{2}{A}_{1}^{2}b\\ & & \quad +12{A}_{0}{A}_{1}^{2}{ck}+4\alpha {A}_{1}^{2}k+12{A}_{0}{A}_{1}^{2}\beta k+4{A}_{1}^{2}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl} & & 4{{aA}}_{1}^{2}{\eta }^{2}-4{{aA}}_{0}{A}_{1}{\eta }^{2}-16{A}_{0}{A}_{1}^{3}b+4{A}_{1}^{3}{ck}\\ & & \quad +\,4{A}_{1}^{3}\beta k=0,-3{{aA}}_{1}^{2}{\eta }^{2}-4{A}_{1}^{4}b=0.\end{array}\end{eqnarray*}$
Further, when the above system is solved, the governing model reveals the following solution sets:
Set-I
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & 0,\quad {A}_{1}={A}_{1},\quad \omega =\displaystyle \frac{1}{4}\left(a{\eta }^{2}-4{{ak}}^{2}-4\alpha k\right),\\ b & = & -\displaystyle \frac{3a{\eta }^{2}}{4{A}_{1}^{2}},\quad c=\displaystyle \frac{-a{\eta }^{2}-{A}_{1}\beta k}{{A}_{1}k},\end{array}\end{eqnarray}$
that yields the following exact solution $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{\displaystyle \frac{{A}_{1}}{{{d}{\rm{e}}}^{\eta (x-\upsilon t)}+1}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(a{\eta }^{2}-4{{ak}}^{2}-4\alpha k\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A1 and d are any non-zero arbitrary constants.Set-II
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & -{A}_{1},\quad {A}_{1}={A}_{1},\quad \omega =\displaystyle \frac{1}{4}\left(a{\eta }^{2}-4{{ak}}^{2}-4\alpha k\right),\\ b & = & -\displaystyle \frac{3a{\eta }^{2}}{4{A}_{1}^{2}},\quad c=\displaystyle \frac{a{\eta }^{2}-{A}_{1}\beta k}{{A}_{1}k},\end{array}\end{eqnarray}$
that yields the following exact solution $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{\displaystyle \frac{{A}_{1}}{{{d}{\rm{e}}}^{\eta (x-\upsilon t)}+1}-{A}_{1}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(a{\eta }^{2}-4{{ak}}^{2}-4\alpha k\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A1 and d are any non-zero arbitrary constants.5.2. Modified Kudryashov's method
As demonstrated, the modified Kudryashov's method yields the following system of algebraic equations
$\begin{eqnarray*}4{{aA}}_{0}^{2}{k}^{2}-4{A}_{0}^{4}b+4{A}_{0}^{3}{ck}+4\alpha {A}_{0}^{2}k+4{A}_{0}^{3}\beta k+4{A}_{0}^{2}\omega =0,\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}8{{aA}}_{1}{A}_{0}{k}^{2}-2{{aA}}_{1}{A}_{0}{\eta }^{2}{\mathrm{ln}}^{2}(r)-16{A}_{1}{A}_{0}^{3}b+12{A}_{1}{A}_{0}^{2}{ck}\\ \quad +\,8\alpha {A}_{1}{A}_{0}k+12{A}_{1}{A}_{0}^{2}\beta k+8{A}_{1}{A}_{0}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}4{{aA}}_{1}^{2}{k}^{2}-{{aA}}_{1}^{2}{\eta }^{2}{\mathrm{ln}}^{2}(r)+6{{aA}}_{0}{A}_{1}{\eta }^{2}{\mathrm{ln}}^{2}(r)-24{A}_{0}^{2}{A}_{1}^{2}b\\ \quad +\,12{A}_{0}{A}_{1}^{2}{ck}+4\alpha {A}_{1}^{2}k+12{A}_{0}{A}_{1}^{2}\beta k+4{A}_{1}^{2}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}4{{aA}}_{1}^{2}{\eta }^{2}{\mathrm{ln}}^{2}(r)-4{{aA}}_{0}{A}_{1}{\eta }^{2}{\mathrm{log}}^{2}(r)-16{A}_{0}{A}_{1}^{3}b+4{A}_{1}^{3}{ck}\\ \quad +\,4{A}_{1}^{3}\beta k=0,-3{{aA}}_{1}^{2}{\eta }^{2}{\mathrm{ln}}^{2}(r)-4{A}_{1}^{4}b=0.\end{array}\end{eqnarray*}$
In addition, when the above system is solved analytically with the aid of the computer package, the following solution sets are thus obtained:Set-I
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & 0,\quad {A}_{1}={A}_{1},\quad \omega =\displaystyle \frac{1}{4}\left(-4{{ak}}^{2}+a{\eta }^{2}{\mathrm{ln}}^{2}(r)-4\alpha k\right),\\ b & = & -\displaystyle \frac{3a{\eta }^{2}{\mathrm{ln}}^{2}(r)}{4{A}_{1}^{2}},\quad c=\displaystyle \frac{-a{\eta }^{2}{\mathrm{ln}}^{2}(r)-{A}_{1}\beta k}{{A}_{1}k},\end{array}\end{eqnarray}$
that leads to the exact exponential solution for the model as follows $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{\displaystyle \frac{{A}_{1}}{{{dr}}^{\eta (x-\upsilon t)}+1}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(-4{{ak}}^{2}+a{\eta }^{2}{\mathrm{ln}}^{2}(r)-4\alpha k\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A1, d and r are any non-zero arbitrary constants.Set-II
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & -{A}_{1},\quad {A}_{1}={A}_{1},\\ \omega & = & \displaystyle \frac{1}{4}\left(-4{{ak}}^{2}+a{\eta }^{2}{\mathrm{ln}}^{2}(r)-4\alpha k\right),\\ b & = & -\displaystyle \frac{3a{\eta }^{2}{\mathrm{ln}}^{2}(r)}{4{A}_{1}^{2}},\quad c=\displaystyle \frac{a{\eta }^{2}{\mathrm{ln}}^{2}(r)-{A}_{1}\beta k}{{A}_{1}k},\end{array}\end{eqnarray}$
that leads to the exact exponential solution for the model as follows $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & \sqrt{\displaystyle \frac{{A}_{1}}{{{dr}}^{\eta (x-\upsilon t)}+1}-{A}_{1}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(-4{{ak}}^{2}+a{\eta }^{2}{\mathrm{ln}}^{2}(r)-4\alpha k\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A1, d and r are any non-zero arbitrary constants.5.3. Enhanced Kudryashov's method
In the same manner, when the enhanced Kudryashov's method is deployed as enshrined in the method's steps, the following algebraic system is thus portrayed
$\begin{eqnarray*}\begin{array}{rcl} & & 4{{aA}}_{0}^{2}{k}^{2}-4{A}_{0}^{4}b+4{A}_{0}^{3}{ck}+4\alpha {A}_{0}^{2}k\\ & & \quad +\,4{A}_{0}^{3}\beta k+4{A}_{0}^{2}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}-2{{aA}}_{1}{A}_{0}{\eta }^{2}+8{{aA}}_{1}{A}_{0}{k}^{2}-16{A}_{1}{A}_{0}^{3}b+12{A}_{1}{A}_{0}^{2}{ck}\\ \quad +\,8\alpha {A}_{1}{A}_{0}k+12{A}_{1}{A}_{0}^{2}\beta k+8{A}_{1}{A}_{0}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}-{{aA}}_{1}^{2}{\eta }^{2}+4{{aA}}_{1}^{2}{k}^{2}-24{A}_{0}^{2}{A}_{1}^{2}b+12{A}_{0}{A}_{1}^{2}{ck}\\ \quad +\,4\alpha {A}_{1}^{2}k+12{A}_{0}{A}_{1}^{2}\beta k+4{A}_{1}^{2}\omega =0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}4{{aA}}_{0}{A}_{1}{\eta }^{2}r-16{A}_{0}{A}_{1}^{3}b+4{A}_{1}^{3}{ck}\\ \quad +\,4{A}_{1}^{3}\beta k=0,3{{aA}}_{1}^{2}{\eta }^{2}r-4{A}_{1}^{4}b=0,\end{array}\end{eqnarray*}$
upon which when solved results in the acquisition of the following special solutions sets:Set-I
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & {A}_{0},\quad {A}_{1}=-{A}_{0}\sqrt{r},\\ \omega & = & \displaystyle \frac{1}{4}\left(-5a{\eta }^{2}-4{{ak}}^{2}-4\alpha k\right),\\ b & = & \displaystyle \frac{3a{\eta }^{2}}{4{A}_{0}^{2}},\quad c=\displaystyle \frac{2a{\eta }^{2}-{A}_{0}\beta k}{{A}_{0}k},\quad \end{array}\end{eqnarray}$
that leads to the following exact solution $\begin{eqnarray}\begin{array}{rcl}{q}_{}(x,t) & = & \sqrt{{A}_{0}-\displaystyle \frac{4{A}_{0}d\sqrt{r}}{4{d}^{2}{{\rm{e}}}^{\eta (x-\upsilon t)}+{{r}{\rm{e}}}^{-\eta (x-\upsilon t)}}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(-5a{\eta }^{2}-4{{ak}}^{2}-4\alpha k\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A0, d and r are any non-zero arbitrary constants.Set-II
$\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & {A}_{0},\quad {A}_{1}={A}_{0}\sqrt{r},\quad \omega =\displaystyle \frac{1}{4}\left(-5a{\eta }^{2}-4{{ak}}^{2}-4\alpha k\right),\\ b & = & \displaystyle \frac{3a{\eta }^{2}}{4{A}_{0}^{2}},\quad c=\displaystyle \frac{2a{\eta }^{2}-{A}_{0}\beta k}{{A}_{0}k},\quad \end{array}\end{eqnarray}$
that leads to the following exact solution $\begin{eqnarray}\begin{array}{rcl}{q}_{}(x,t) & = & \sqrt{{A}_{0}+\displaystyle \frac{4{A}_{0}d\sqrt{r}}{4{d}^{2}{{\rm{e}}}^{\eta (x-\upsilon t)}+{{r}{\rm{e}}}^{-\eta (x-\upsilon t)}}}\\ & & \times \,\exp \left(\iota \left(-{kx}+\displaystyle \frac{1}{4}t\left(-5a{\eta }^{2}-4{{ak}}^{2}-4\alpha k\right)+\vartheta \right)\right),\end{array}\end{eqnarray}$
where A0, d and r are any non-zero arbitrary constants.6. Graphical illustrations and discussion of results
This section gives the graphical illustrations of some of the constructed exact solitonic solutions for the governing two models and further discusses the results concerning the application of three Kudryashov-based methods [8, 9], [28, 29]. In fact, we have shown in figures 1, 2 and 3 the two-dimensional illustrations of Set-I solutions for the standard Gerdjikov–Ivanov equation using the classical Kudryashov's method, the modified Kudryashov's method, and the enhanced Kudryashov's method, respectively; while figures 4, 5 and 6 gives the same graphical illustrations for perturbed Gerdjikov–Ivanov equation, correspondingly. In addition, these figures further examine the influence of the variational nature of Kudryashov's index d on the propagation of nonlinear waves in the governing media. Besides, all the related parameters are set to be unity while simulating the beseeched structure graphically; see figures 1–6; while r is assumed to be 10; that is, r = 10. In addition, it is worth noting here that despite the existing studies on the governing models—see [4–7], the present study stands unique, looking at how three Kudryashov-based methods are assessed with regard to the governing Gerdjikov–Ivanov equations. Furthermore, the acquired solutions in the form of exponential functions are indeed generalized ones, being able to recast them to the popularly known singular and bright solitonic solutions as addressed and clearly implemented in Elsherbeny et al [29]. Besides, we deeply examine the acquired solutions concerning the respective models in what follows.
Figure 1. Influence of Kudryashov's index d on the Set-I solution in ( |
Figure 2. Influence of Kudryashov's index d on the Set-I solution in ( |
Figure 3. Influence of Kudryashov's index d on the Set-I solution in ( |
Figure 4. Influence of Kudryashov's index d on the Set-I solution in ( |
Figure 5. Influence of Kudryashov's index d on the Set-I solution in ( |
Figure 6. Influence of Kudryashov's index d on the Set-I solution in ( |
6.1. Standard Gerdjikov–Ivanov equation
Here, the constructed Set-I solutions for the standard Gerdjikov–Ivanov equation are shown in figures 1, 2 and 3 via the applications of the classical Kudryashov's method, the modified Kudryashov's method, and the enhanced Kudryashov's method, respectively. Indeed, from figure 1, where the Set-I solution was shown via the use of the classical Kudryashov's method, figure 1(a) shows the influence of the Kudryashov's index d on the real part of the solution, figure 1(b) shows the influence of the Kudryashov's index d on the imaginary part of the solution, while figure 1(c) shows the influence of the Kudryashov's index d on the intensity of the solution (that is, ∣q(x, t)∣2). Evidently, one may note from both figure 1(a) and (b) that the wave dies out as the spatial variable x increases; this is true from the assumed solution form in equation (4 ) since the soliton's frequency is assumed positive, which is preceded by a negative sign. Moreover, from both figure 1(a) and (b), an increase in d quickens the dying out of the wave; certainly, this affirms the bounded condition at a large distance. In addition, from figure 1(c), one observes an increase in d lowers the wave intensity. Further, figure 2 has the same interpretation as that of figure 1; however, the presence of additional parameter r in the modified Kudryashov's method serves as a controlling parameter that speeds up the convergence of the solution to zero as in the cases of figure 2(a) and (b); while in figure 2(c) the influence of d reduces significantly. Furthermore, we have shown in figure 3 the acquired solution for the examining model using the enhanced Kudryashov's method, where such influence of the Kudryashov's index d is seen at the crest and trough as in the cases of figures 3(a) and (b), while the intensity plot in figure 3(c) is significantly influenced by the variation of d that draws the wave back.
6.2. Perturbed Gerdjikov–Ivanov equation
Equally, the acquired Set-I solutions for the perturbed Gerdjikov–Ivanov equation are shown in figures 4, 5, and 6 via the applications of the classical Kudryashov's method, the modified Kudryashov's method, and the enhanced Kudryashov's method, respectively. Thus, without further delay, the interpretation of these figures follows that of the above subsection for the standard Gerdjikov–Ivanov equation. However, the presence of the perturbations terms in the examining perturbed model plays a vital role in the wave propagation in the nonlinear media. Indeed, looking at the introduced parameters due to the perturbations, which are α (the inter-modal dispersion), β (the self-steepening effect), and γ (higher-order dispersion effect), the wave tread remains the same as in the case of the unperturbed model, only that a delay has been noted in the wave range.
7. Conclusion
As a concluding note, the current study made use of three Kudryashov-based analytical methods, including the classical Kudryashov method, the modified Kudryashov method, and the improved Kudryashov method to inspect the class of Gerdjikov–Ivanov equations, comprising the standard Gerdjikov–Ivanov equation and the perturbed Gerdjikov–Ivanov equation. Indeed, various optical solitonic solutions have been constructed for the considered nonlinear Schrödinger equations. Certainly, as the reported solitonic structures happened to be exponential functions, diverse true solitonic solutions can easily be resorted to upon suitably fixing the involving parameters, including mainly the bright and singular optical solitons. Moreover, the acquired solutions can be used as benchmark results for numerical and experimental analysis of various processes described by Gerdjikov–Ivanov equations. In the end, the study graphically examined some of the constructed solitonic structures, which portrayed some interesting known shapes in the theory of nonlinear Schrödinger equations. Besides, this study can be applied to optical fiber designs; while the application of the adopted methodology can be extended to higher-order nonlinear complex-valued evolution equations.
Author's contributions
FAA, AAA, BOB, RIN: Conceptualization, Methodology, Writing- Original draft preparation, Methodology, Data curation, Supervision. All authors read and approved the final manuscript.
Consent to participate
All authors approved the final manuscript.
Consent for publication
All authors approved the publication of the manuscript.