Chirped Soliton Perturbation and Benjamin–Feir Instability of Chen–Lee–Liu Equation with Full Nonlinearity

Abstract

The objective of the present study is to detect chirped optical solitons of the perturbed Chen–Lee–Liu equation with full nonlinearity. By virtue of the traveling wave hypothesis, the discussed model is reduced to a simple form known as an elliptic equation. The latter equation, which is a second-order ordinary differential equation, is handled by the undetermined coefficient method of two forms expressed in terms of the hyperbolic secant and tangent functions. Additionally, the auxiliary equation method is applied to derive several miscellaneous solutions. Various types of chirped solitons are revealed such as W-shaped, bright, dark, gray, kink and anti-kink waves. Taking into consideration the existence conditions, the dynamical behaviors of optical solitons and their corresponding chirp are illustrated. The modulation instability of the perturbed CLL equation is examined by means of the linear stability analysis. It is found that all solutions are stable against small perturbations. These entirely new results, compared to previous works, can be employed to understand pulse propagation in optical fiber mediums and dynamic characteristics of waves in plasma.

1. Introduction

Solitons have become an attractive type of solution for nonlinear partial differential equations because of their potential role in revealing different complex aspects of nonlinear physical phenomena. Optical fibers are one of the prominent fields in science and engineering that highly benefits from the significant physical properties of solitons [1,2]. In the field of optical fibers, solitons are known as optical solitons and have been found to have excellent capacity to propagate long distances without attenuation and changing their shapes [3,4,5]. Hence, optical pulses are considered as eminent carriers of information signals in optical telecommunication systems. Due to the high intensity of propagating pulses, the self-steepening effect takes place in the medium and causes a reduction in the group velocity of the pulses [6]. The presence of self-steepening is crucial to ensure the formations of optical pulses with ultrashort width (<100 fs) in optical fiber communication systems, and thus, higher-order effects are inevitable [7,8]. To study the dynamical features of optical solitons dominated by the self-steepening influence, one can deal with the model of nonlinear Schrödinger equation and its family of equations like the Radhakrishnan–Kundu–Lakshmanan equation [9,10,11,12,13], Ginzburg–Landau equation [14,15], Schrödinger–Hirota equation [16,17,18,19], Chen–Lee–Liu equation [20,21,22,23,24] and others. Moreover, chirped solitons are closer to a realistic situation. One encounters noisy sound along fiber optic cables during such pulse propagations, which is accounted for with chirped solitons. Such sound can be heard in computer data storage rooms where all the complicated wiring of the entire firm’s software system is stored and maintained.

As stated above, the Chen–Lee–Liu (CLL) equation is one of the important models that can be discussed to investigate soliton transmission through optical fibers and nonlinear wave propagation in plasmas. This model was created by Chen et al. in 1979 [25]. The CLL equation is one of three forms of the derivative nonlinear Schrödinger equation (DNLSE) and it is known as the DNLSE-II equation. The other two forms are the Gerdjikov–Ivanov equation and the Kaup–Newell equation, which are denoted by the DNLSE-I and DNLSE-III equations. The model of the CLL equation is given by

$$i{Q}_t+a{Q}_{xx}+ib{\left|Q\right|}^2{Q}_x=0,$$

(1)

where $Q(x,t)$ is a complex-valued function representing the soliton profile. The term with parameter a stands for group velocity dispersion, while the term with b refers to the effect of the self-steepening phenomena. It is known that the standard CLL equation belongs to the family of integrable equations that can be solved by applying the inverse scattering transform. Equation (1) characterizes an optical pulse propagating in a monomode fiber. Thus, in the past, many authors [26,27,28,29,30,31,32,33,34] have attempted to derive solitonic-type solutions. Additionally, various structures of rogue wave solutions to Equation (1) have been reported by some scholars [35,36,37,38,39].

The CLL equation has been developed in distinct forms so as to be implemented in different physical media such as birefringent fibers, fiber Bragg gratings and others. The perturbed CLL equation is one of these extended forms which has a vital role in describing sub-picosecond soliton transmission through optical fibers. The dimensionless form of the perturbed CLL equation is [40]

$$i{Q}_t+a{Q}_{xx}+{ib\left|Q\right|}^2{Q}_x+i[\alpha Q_x+{\beta \left(\right|Q|}^2{Q)}_x+{\gamma \left(\right|Q|}^2{)}_{x}Q]=0.$$

(2)

This equation is a generalization of the standard CLL Equation (1) which can be recovered when the Hamiltonian perturbation terms are neglected, i.e., $\alpha =\beta =\gamma =0$. Equation (2) has been intensively studied by a lot of scholars to detect optical soliton solutions [41,42,43,44,45]. Miscellaneous forms of soliton solutions are procured such as bright, dark and singular optical solitons. Further to this, different periodic wave structures in terms of trigonometric and Jacobi elliptic functions are obtained.

Our intention in this work is to investigate the chirped optical solitons of the perturbed CLLE with full nonlinearity, which is a generalized form to Equation (2). The details of this article are described in the following sections. Section 2 elucidates the strategy of the undetermined coefficient method, which has two functional forms [46]. In Section 3, the discussed model is analyzed and reduced to an elliptic-type equation by using the traveling wave theory. Section 4 displays the derivation of wave solutions by the auxiliary equation scheme and the undetermined coefficient method with two functional structures of the hyperbolic secant and tangent functions. In Section 5, the technique of linear stability analysis is employed to diagnose the modulation instability of the discussed model. The interpretations of obtained results are presented in Section 6. Finally, our conclusion is summarized in Section 7.

2. Description of Method

Herein, we present the process of applying the undetermined coefficient method as follows. Consider a nonlinear partial differential equation (NLPDE) in the form

$$P(u,u_x,u_{xx},u_{tt},u_{tx},u_{xxx},\dots )=0,$$

(3)

where $u=u(x,t)$ is an unknown function and P is a polynomial in u and its various partial derivatives. Based on the traveling wave transformation given by

$$u(x,t)=\mathsf{\Lambda }\left(\xi \right),\xi =x-\nu t,$$

(4)

the NLPDE (3) reduces to a nonlinear ordinary differential equation (NLODE) of the form

$$H(\mathsf{\Lambda },{\mathsf{\Lambda }}^{\prime },{\mathsf{\Lambda }}^{″},{\mathsf{\Lambda }}^{‴},\dots )=0,$$

(5)

where prime denotes the derivative with respect to $\xi$. We assume that Equation (5) has a solution in the form of a finite series as

$$\mathsf{\Lambda }\left(\xi \right)=\sum _{j=1}^m{\eta }_j\mathsf{\Gamma }\left(\xi \right),$$

(6)

where $\mathsf{\Gamma }\left(\xi \right)$ is expressed in two different functional structures. The first expression of $\mathsf{\Gamma }\left(\xi \right)$ has a hyperbolic tangent function, introduced as

$$\mathsf{\Gamma }\left(\xi \right)={tanh}^j\left(ϵ\xi \right),$$

(7)

while the second form has a hyperbolic secant function addressed as

$$\mathsf{\Gamma }\left(\xi \right)={\displaystyle \frac{{sech}^j\left(ϵ\xi \right)}{1+{sech}^j\left(ϵ\xi \right)}},$$

(8)

where $ϵ$ and ${\eta }_j,(j=0,1,\dots ,m)$ are constants to be identified. The parameter m is a positive integer which can be identified by balancing the highest-order derivative term with the nonlinear term in Equation (5). The purpose of expressing the function $\mathsf{\Gamma }\left(\xi \right)$ in terms of the hyperbolic tangent function is that its derivatives are expressed in the form of the hyperbolic secant function. On the other hand, it provides us with targeted soliton solutions.

To achieve the goal of implementing this method, we substitute (6) along with (7) into Equation (5) to arrive at a polynomial in ${tanh}^j\left(ϵ\xi \right)$ with different powers. Collecting all coefficients with the same powers of ${tanh}^j\left(ϵ\xi \right)$ and equating them to zero leads to a set of algebraic equations for $ϵ$ and ${\eta }_j$. Finally, solving this set of equations simultaneously, various exact solutions of Equation (3) can be derived through inserting the obtained values of $ϵ$ and ${\eta }_j$ into (6) together with (7). Similarly, performing the same process using (8) instead of (7), this gives us a system of algebraic equations that yields distinct values for $ϵ$ and ${\eta }_j$ which are plugged into (6) alongside (8) to create several cases of solutions to Equation (3).

3. Governing Model and Mathematical Analysis

The perturbed CLLE with full nonlinearity is described by [47]

$$i{Q}_t+a{Q}_{xx}+{ib\left(\right|Q|}^{2n}{Q}_x)+i[\alpha Q_x+{\beta \left(\right|Q|}^{2n}{Q)}_x+{\gamma \left(\right|Q|}^{2n}{)}_{x}Q]=0.$$

(9)

One can see that Equation (9) is a generalization of the perturbed CLL Equation (2) when $n=1$. The unknown function $Q(x,t)$ represents the soliton profile which depends on the spatial variable x and the temporal coordinate t. In Equation (9), the first term stands for the evolution term, the second term denotes the chromatic dispersion (CD) and the third term accounts for the nonlinear dispersion (ND). The terms with the coefficients $\alpha ,\beta$ and $\gamma$ define, respectively, the inter-modal dispersion (IM), the self-steepening (SS) phenomena and the nonlinear dispersion terms.

The model (9) is discussed by Kudryashov [48] using the traveling wave hypothesis to investigate various exact solutions. In particular, periodic and solitary waves are detected with the effect of full nonlinearity. In addition, the exact chirped solutions of Equation (9) are investigated in [49] using the trial equation method and complete discrimination system for polynomials. Different forms of solutions are obtained such as rational solutions, solitary wave solutions, triangular function solutions and doubly periodic elliptic function solutions. Further to this, the qualitative analysis and explicit solutions of Equation (9) are studied in [50]. By means of bifurcation analysis, several explicit solutions including Jacobian function solutions and hyperbolic function solutions are created.

To handle Equation (9) analytically, let us consider the traveling wave transformation

$$Q(x,t)=q\left(\xi \right)e^{i\left(\phi \right(\xi )-\omega t)},$$

(10)

where $\xi =x-\nu t$ represents the traveling coordinate of a soliton wave while $q\left(\xi \right)$ and $\phi \left(\xi \right)$ are real functions identifying the amplitude component of the soliton and the nonlinear phase shift, respectively. The parameters $\omega$ and $\nu$ are real constants denoting the wave number and the soliton velocity.

Substituting (10) into Equation (9) leads to its decomposition into real and imaginary components. The equation of the real part is given by

$$a{q}^{″}+(b+\beta )(\kappa -{\phi }^{\prime })q^{1+2n}-[\omega -\alpha \kappa +a{\kappa }^2+a{\phi }^{\prime 2}-(\nu -\alpha +2a\kappa ){\phi }^{\prime }]q=0,$$

(11)

while the equation of the imaginary part has the form

$$-(\nu -\alpha +2a\kappa )q^{\prime }+2a{q}^{\prime }{\phi }^{\prime }+aq{\phi }^{″}+(2n(\beta +\gamma )+b+\beta )q^{2n}{q}^{\prime }=0,$$

(12)

where $q=q\left(\xi \right),\phi =\phi \left(\xi \right)$ and the prime is the derivative with respect to $\xi$. Equation (12) can be integrated, after multiplying by q, to arrive at

$${\phi }^{\prime }={\displaystyle \frac{\nu -\alpha +2a\kappa }{2a}}-{\displaystyle \frac{2n(\beta +\gamma )+b+\beta }{2a(n+1)}}{q}^{2n},$$

(13)

where the integration constant is set to zero. As is known, the general formula for a chirp is introduced as

$$\delta \omega (x,t)=-{\displaystyle \frac{\partial }{\partial x}}[\phi \left(\xi \right)-\omega t]=-{\phi }^{\prime }\left(\xi \right).$$

(14)

Hence, from Equation (13), the chirping expression can be created as

$$\delta \omega =-{\displaystyle \frac{\nu -\alpha +2a\kappa }{2a}}+{\displaystyle \frac{2n(\beta +\gamma )+b+\beta }{2a(n+1)}}{q}^{2n}.$$

(15)

Upon substituting Equation (13) into Equation (11), one can come to an equation with the form

$$\begin{array}{cc}\hfill & 4a^2{(n+1)}^2{q}^{″}+{(n+1)}^2({\nu }^2+{\alpha }^2-4a\omega -\nu (2\alpha -4a\kappa ))q-2{(n+1)}^2(\nu -\alpha )(b+\beta )q^{2n+1}\hfill \\ \hfill & +\left[(n+1)(b+\beta )(4n(\beta +\gamma )+2(b+\beta ))-{(2n(\beta +\gamma )+(b+\beta ))}^2\right]q^{4n+1}=0.\hfill \end{array}$$

(16)

Multiplying by $q^{\prime }$ and integrating with respect to $\xi$, this gives rise to

$$\begin{array}{cc}\hfill & 4a^2{(n+1)}^2{q}^{\prime 2}+{(n+1)}^2({\nu }^2+{\alpha }^2-4a\omega -\nu (2\alpha -4a\kappa ))q^2-2(n+1)(\nu -\alpha )(b+\beta )q^{2n+2}\hfill \\ \hfill & +\left[(n+1)(b+\beta )(4n(\beta +\gamma )+2(b+\beta ))-{(2n(\beta +\gamma )+(b+\beta ))}^2\right]{\displaystyle \frac{q^{4n+2}}{(2n+1)}}=0,\hfill \end{array}$$

(17)

where the integration constant is taken to be zero. In order to extract an analytic closed-form solution for the proposed model, the complicated structure of Equation (17) has to be reformulated in a simple form. Thus, multiplying Equation (17) by $q^{2n-2}$ reduces it to the form

$$4c_1{\left[{\left(q^n\right)}^{\prime }\right]}^2+c_2{q}^{2n}-2c_3{q}^{4n}+c_4{q}^{6n}=0,$$

(18)

where the constants $c_1,c_2,c_3,c_4$ are defined as

$$\left.\begin{array}{cc}\hfill & c_1={\displaystyle \frac{a^2{(n+1)}^2}{n^2}},\hfill \\ \hfill & c_2={(n+1)}^2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega )),\hfill \\ \hfill & c_3=(n+1)(\nu -\alpha )(b+\beta ),\hfill \\ \hfill & c_4={\displaystyle \frac{\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}{(2n+1)}}.\hfill \end{array}\right\}$$

(19)

Let us introduce the variable transformation of the form

$$V=q^{2n},$$

(20)

which converts Equation (18) into

$$c_1{V}^{\prime 2}+c_2{V}^2-2c_3{V}^3+c_4{V}^4=0.$$

(21)

For convenience, Equation (21) is converted to a second-order differential equation by differentiating it once to induce

$$c_1{V}^{″}+c_{2}V-3c_3{V}^2+2c_4{V}^3=0.$$

(22)

This type of equation is found to appear in many physical and natural applications. It has been found by many authors to reveal distinct forms of solutions via implementing several mathematical tools such as the Sinh–Gordon method [51], new modified sub-ODE method [52], Jacobi elliptic method [53], $(G^{\prime }/G^2)$-expansion method [54], and soliton ansatz [55]. The various solutions to the elliptic Equation (22) combined with the relations (10) and (20) give rise to the general form of exact solutions for Equation (9) as

$$Q(x,t)=V{\left(\xi \right)}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},$$

(23)

where the nonlinear phase shift represented by the function $\phi \left(\xi \right)$ can be acquired through integrating Equation (13) as

$$\phi \left(\xi \right)={\displaystyle \frac{\nu -\alpha +2a\kappa }{2a}}\xi -{\displaystyle \frac{2n(\beta +\gamma )+b+\beta }{2a(n+1)}}\int q{\left(\xi \right)}^{2n}\mathrm{d}\xi +\theta ,$$

(24)

where $\theta$ is a constant phase. Our aim now is to deal with Equation (22) by using the undetermined coefficient method so as to extract the soliton-type solutions for the model (9).

4. Chirped Soliton Solutions

The chirped soliton solutions of the perturbed CLLE are retrieved here by means of the auxiliary equation scheme and the undetermined coefficient method [46], which is proposed in two forms with the hyperbolic secant and tangent functions as follows.

4.1. Ansatz with Hyperbolic Tangent Function

We assume that Equation (22) has a solution expressed as

$$V\left(\xi \right)=f+gtanh\left(ϵ\xi \right)+h{tanh}^2\left(ϵ\xi \right),$$

(25)

where $f,g,h$ and $ϵ$ are constants to be identified. Making use of ansatz (25), Equation (22) generates a polynomial in $tanh\left(ϵ\xi \right)$ with different powers. Equating all coefficients of all powers of $tanh\left(ϵ\xi \right)$ to zero, this yields a system of algebraic equations that identifies the values of constants $f,g,h$ and $ϵ$. Solving this system leads to the following cases of solutions.

Case 1.

$$\left.\begin{array}{cc}\hfill & f=0,\hfill \\ \hfill & g=\sqrt{-{\displaystyle \frac{2(2n+1){(n+1)}^2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}}},\hfill \\ \hfill & h=0,\hfill \\ \hfill & ϵ={\displaystyle \frac{n}{a}}\sqrt{2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},\hfill \end{array}\right\}$$

(26)

under the constraint conditions

$$\nu =\alpha ,\mathrm{or}b=-\beta .$$

(27)

Using the outcome (26) with (23), one can reach an exact solution in the form of a chirped dark soliton for Equation (9) as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[\sqrt{-{\displaystyle \frac{(2n+1){(n+1)}^2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}}}\right.\hfill \\ \hfill & {\left.\times tanh\left({\displaystyle \frac{n}{2a}}\sqrt{2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(28)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))>0$ and $\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )<0$. (Figure 1).

Case 2.

$$\left.\begin{array}{cc}\hfill & f={\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & g=0,\hfill \\ \hfill & h=-{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & ϵ={\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},\hfill \end{array}\right\}$$

(29)

under the constraint conditions

$$\gamma ={\displaystyle \frac{\beta +b(2n+1)}{2n}},\mathrm{or}\gamma =-{\displaystyle \frac{b+\beta (2n+1)}{2n}}.$$

(30)

Applying these findings to (23), an exact solution in the form of a chirped bright soliton for Equation (9) is obtained as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times {sech}^2\left({\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(31)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))<0$ and $(\nu -\alpha )(b+\beta )<0$. (Figure 2).

Case 3.

$$\left.\begin{array}{cc}\hfill & f=-{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{6(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & g=0,\hfill \\ \hfill & h={\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & ϵ={\displaystyle \frac{n}{2a}}\sqrt{({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},\hfill \end{array}\right\}$$

(32)

under the constraint conditions

$$\gamma ={\displaystyle \frac{\beta +b(2n+1)}{2n}},\mathrm{or}\gamma =-{\displaystyle \frac{b+\beta (2n+1)}{2n}}.$$

(33)

Making use of (32) with (23), an exact solution in the form of a chirped soliton for Equation (9) is secured as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[-{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{6(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times \left\{1-3{tanh}^2\left({\displaystyle \frac{n}{2a}}\sqrt{({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)\right\}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(34)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))>0$ and $(\nu -\alpha )(b+\beta )<0$. (Figure 3).

Case 4.

$$\left.\begin{array}{cc}\hfill & f={\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & g={\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & h=0,\hfill \\ \hfill & ϵ={\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},\hfill \end{array}\right\}$$

(35)

under the constraint conditions

$${\displaystyle \frac{\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}{(2n+1)}}={\displaystyle \frac{{(\nu -\alpha )}^2{(b+\beta )}^2}{{(\nu -\alpha )}^2+4a(\kappa \nu -\omega )}}.$$

(36)

Utilizing the result (35) with (23), an exact solution in the form of a chirped kink-type soliton for Equation (9) is acquired as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times \left\{1+tanh\left({\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)\right\}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(37)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))<0$ and $(\nu -\alpha )(b+\beta )<0$. (Figure 4).

Case 5.

$$\left.\begin{array}{cc}\hfill & f={\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & g=-{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & h=0,\hfill \\ \hfill & ϵ={\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},\hfill \end{array}\right\}$$

(38)

under the constraint conditions

$${\displaystyle \frac{\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}{(2n+1)}}={\displaystyle \frac{{(\nu -\alpha )}^2{(b+\beta )}^2}{{(\nu -\alpha )}^2+4a(\kappa \nu -\omega )}}.$$

(39)

Employing these findings in (23), an exact solution in the form of a chirped anti-kink soliton for Equation (9) is extracted as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times \left\{1-tanh\left({\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)\right\}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(40)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))<0$ and $(\nu -\alpha )(b+\beta )<0$. (Figure 5).

4.2. Ansatz with Hyperbolic Secant Function

We suppose that Equation (22) has an exact soliton solution presented as

$$V\left(\xi \right)=p+{\displaystyle \frac{rsech\left(\sigma \xi \right)}{1+sech\left(\sigma \xi \right)}}+{\displaystyle \frac{s{sech}^2\left(\sigma \xi \right)}{1+{sech}^2\left(\sigma \xi \right)}},$$

(41)

where $p,r,s$ and $\sigma$ are constants to be determined. Substituting (41) into Equation (22) yields a polynomial in $sech\left(\sigma \xi \right)$ of various powers. Equating each coefficient in this polynomial to zero induces a set of algebraic equations. Solving them simultaneously gives rise to the following cases of solutions.

Case 1.

$$\left.\begin{array}{cc}\hfill & p=0,\hfill \\ \hfill & r=0,\hfill \\ \hfill & s={\displaystyle \frac{3(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & \sigma ={\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},\hfill \end{array}\right\}$$

(42)

under the constraint conditions

$${\displaystyle \frac{\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}{(2n+1)}}={\displaystyle \frac{8{(\nu -\alpha )}^2{(b+\beta )}^2}{9({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}}.$$

(43)

By virtue of (42) with (23), one can arrive at an exact solution in the form of a chirped bright soliton for Equation (9) as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{3(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times {\displaystyle \frac{{sech}^2\left(\frac{n}{2a}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}{1+{sech}^2\left(\frac{n}{2a}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(44)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))<0$ and $(\nu -\alpha )(b+\beta )<0$. (Figure 6).

Case 2.

$$\left.\begin{array}{cc}\hfill & p=0,\hfill \\ \hfill & r={\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & s=0,\hfill \\ \hfill & \sigma ={\displaystyle \frac{n}{a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},\hfill \end{array}\right\}$$

(45)

under the constraint conditions

$$\gamma ={\displaystyle \frac{\beta +b(2n+1)}{2n}},\mathrm{or}\gamma =-{\displaystyle \frac{b+\beta (2n+1)}{2n}}.$$

(46)

Based on these findings, an exact solution in the form of a chirped bright soliton for Equation (9) is derived as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times {\displaystyle \frac{sech\left(\frac{n}{2a}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}{1+sech\left(\frac{n}{2a}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(47)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))<0$ and $(\nu -\alpha )(b+\beta )<0$. (Figure 7).

Case 3.

$$\left.\begin{array}{cc}\hfill & p={\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{3(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & r=-{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & s=0,\hfill \\ \hfill & \sigma ={\displaystyle \frac{n}{a}}\sqrt{({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},\hfill \end{array}\right\}$$

(48)

under the constraint conditions

$$\gamma ={\displaystyle \frac{\beta +b(2n+1)}{2n}},\mathrm{or}\gamma =-{\displaystyle \frac{b+\beta (2n+1)}{2n}}.$$

(49)

As a consequence of using (48) in company with (23), an exact solution in the form of a chirped soliton structure for Equation (9) is deduced as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{3(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times \left\{1-{\displaystyle \frac{3sech\left(\frac{n}{2a}\sqrt{({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}{1+sech\left(\frac{n}{2a}\sqrt{({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}}\right\}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(50)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))>0$ and $(\nu -\alpha )(b+\beta )>0$. (Figure 8).

Case 4.

$$\left.\begin{array}{cc}\hfill & p={\displaystyle \frac{(3-\sqrt{17})(9-\sqrt{17})(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(17-9\sqrt{17})(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & r=0,\hfill \\ \hfill & s=-{\displaystyle \frac{8(3-\sqrt{17})(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(17-9\sqrt{17})(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & \sigma ={\displaystyle \frac{2n}{a}}\sqrt{{\displaystyle \frac{2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(-17+9\sqrt{17})}}},\hfill \end{array}\right\}$$

(51)

under the constraint conditions

$${\displaystyle \frac{\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}{(2n+1)}}={\displaystyle \frac{(17-9\sqrt{17}){(\nu -\alpha )}^2{(b+\beta )}^2}{{(3-\sqrt{17})}^2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}}.$$

(52)

Exploiting these findings along with (23) leads to retrieving an exact solution in the form of a chirped soliton wave for Equation (9) as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(3-\sqrt{17})(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(17-9\sqrt{17})(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times \left\{\left(9-\sqrt{17}\right)-{\displaystyle \frac{16{sech}^2\left(\frac{2n}{a}\sqrt{\frac{2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(-17+9\sqrt{17})}}\xi \right)}{1+{sech}^2\left(\frac{2n}{a}\sqrt{\frac{2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(-17+9\sqrt{17})}}\xi \right)}}\right\}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(53)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))>0$ and $(\nu -\alpha )(b+\beta )>0$. (Figure 9).

Case 5.

$$\left.\begin{array}{cc}\hfill & p={\displaystyle \frac{(3+\sqrt{17})(9+\sqrt{17})(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(17+9\sqrt{17})(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & r=0,\hfill \\ \hfill & s=-{\displaystyle \frac{8(3+\sqrt{17})(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(17+9\sqrt{17})(\nu -\alpha )(b+\beta )}},\hfill \\ \hfill & \sigma ={\displaystyle \frac{2n}{a}}\sqrt{-{\displaystyle \frac{2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(17+9\sqrt{17})}}},\hfill \end{array}\right\}$$

(54)

under the constraint conditions

$${\displaystyle \frac{\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}{(2n+1)}}={\displaystyle \frac{(17+9\sqrt{17}){(\nu -\alpha )}^2{(b+\beta )}^2}{{(3+\sqrt{17})}^2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}}.$$

(55)

With the help of (54) together with (23), an exact solution in the form of a chirped gray soliton for Equation (9) is created as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(3+\sqrt{17})(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{2(17+9\sqrt{17})(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times \left\{\left(9+\sqrt{17}\right)-{\displaystyle \frac{16{sech}^2\left(\frac{2n}{a}\sqrt{-\frac{2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(17+9\sqrt{17})}}\xi \right)}{1+{sech}^2\left(\frac{2n}{a}\sqrt{-\frac{2({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(17+9\sqrt{17})}}\xi \right)}}\right\}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(56)

where $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))<0$ and $(\nu -\alpha )(b+\beta )<0$. (Figure 10).

4.3. Auxiliary Equation Method

Our purpose here is to apply the auxiliary equation method to obtain the solution of Equation (21). We consider the transformation of the form

$$V\left(\xi \right)=W\left(\zeta \right),\zeta =\tau \xi ,$$

(57)

which converts Equation (21) to

$$W^{\prime 2}=L_2{W}^2+L_3{W}^3+L_4{W}^4,$$

(58)

where $W^{\prime }$ is the derivative of W with respect to $\zeta$, $\tau$ is a constant to be determined, and the constants $L_2,L_3,L_4$ are given by

$$L_2=-{\displaystyle \frac{c_2}{{\tau }^2{c}_1}},L_3={\displaystyle \frac{2c_3}{{\tau }^2{c}_1}},L_4=-{\displaystyle \frac{c_4}{{\tau }^2{c}_1}}.$$

(59)

It is found that Equation (58) admits several hyperbolic function solutions [56]. As a result, a variety of optical soliton solutions for Equation (9) are retrieved below.

Family 1. The bright optical soliton is secured as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times {\displaystyle \frac{sech\left(\frac{n}{a}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}{\sqrt{1-\frac{c_2{c}_4}{c_3^2}}+sech\left(\frac{n}{a}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(60)

which demands

$$\tau ={\displaystyle \frac{n}{a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},$$

(61)

and $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))<0$, $c_3^2>c_2{c}_4$. (Figure 11).

Family 2. The singular optical soliton is obtained as

$$\begin{array}{cc}\hfill Q(x,t)=& \left[{\displaystyle \frac{(n+1)({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}{(\nu -\alpha )(b+\beta )}}\right.\hfill \\ \hfill & {\left.\times {\displaystyle \frac{csch\left(\frac{n}{a}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}{\sqrt{\frac{c_2{c}_4}{c_3^2}-1}+csch\left(\frac{n}{a}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))}\xi \right)}}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(62)

which similarly demands

$$\tau ={\displaystyle \frac{n}{a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},$$

(63)

and $({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))<0$, $c_3^2<c_2{c}_4$.

Family 3. The bright optical soliton in another form is acquired as

$$Q(x,t)={\left[{\displaystyle \frac{{sech}^2\left(\tau \xi \right)}{2\sqrt{\frac{\eta +c_4}{c_2}}tanh\left(\tau \xi \right)-{tanh}^2\left(\tau \xi \right)-\frac{\eta }{c_2}}}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},$$

(64)

which implies

$$\tau ={\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},$$

(65)

and $\eta =c_2-2c_3$, $\eta +c_4<0$, $c_2<0$. (Figure 12).

Family 4. One can retrieve the soliton structure presented as

$$Q(x,t)={\left[{\displaystyle \frac{{csch}^2\left(\tau \xi \right)}{2\sqrt{\frac{\eta +c_4}{c_2}}coth\left(\tau \xi \right)+{coth}^2\left(\tau \xi \right)+\frac{\eta }{c_2}}}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},$$

(66)

which similarly requests

$$\tau ={\displaystyle \frac{n}{2a}}\sqrt{-({(\nu -\alpha )}^2+4a(\kappa \nu -\omega ))},$$

(67)

and $\eta =c_2-2c_3$, $\eta +c_4<0$, $c_2<0$. (Figure 13).

Family 5. The new structure of the optical soliton is given as

$$\begin{array}{cc}\hfill Q(x,t)& =\left[{\displaystyle \frac{(2n+1)(n+1)(\nu -\alpha )(b+\beta )}{2\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )}}\right.\hfill \\ \hfill & {\left.\times \left\{1\pm {tanh}_{fg}\left(\sqrt{{\displaystyle \frac{-n^2(2n+1){(\nu -\alpha )}^2{(b+\beta )}^2}{a^2(2n(b-\gamma )+b+\beta )(2n(\beta +\gamma )+b+\beta )}}}\xi \right)\right\}\right]}^{{\displaystyle \frac{1}{2n}}}{e}^{i\left(\phi \right(\xi )-\omega t)},\hfill \end{array}$$

(68)

which necessitates $c_2{c}_4=c_3^2$, $\left(2n\right(b-\gamma )+b+\beta )\left(2n\right(\beta +\gamma )+b+\beta )<0,(\nu -\alpha )(b+\beta )<0$, where

$${tanh}_{fg}\left(\chi \right)={\displaystyle \frac{f{e}^{\chi }-g{e}^{-\chi }}{f{e}^{\chi }+g{e}^{-\chi }}},$$

(69)

and $f,g$ are arbitrary constants. (Figure 14).

5. Modulation Instability Analysis

The linear stability analysis technique is utilized here to diagnose the modulation instability of the perturbed CLLE (9). To use this strategy, assume that Equation (9) has the perturbed steady-state solution given by

$$Q(x,t)=\left[\sqrt{\rho }+\mathrm{Y}(x,t)\right]e^{i\delta \rho t},$$

(70)

where $\rho$ is the normalized optical power while $\mathrm{Y}(x,t)$ is a small perturbation and $\mathrm{Y}\ll \rho$. The perturbation $\mathrm{Y}(x,t)$ is examined by utilizing linear stability analysis. Substituting Equation (70) into Equation (9) and collecting the linearized terms, the following equation is obtained:

$$\begin{array}{cc}\hfill & i{\displaystyle \frac{\partial \mathrm{Y}}{\partial t}}-\delta \rho \mathrm{Y}+a{\displaystyle \frac{{\partial }^2\mathrm{Y}}{\partial x^2}}+i\left[\alpha +b{\rho }^n+(n+1)\beta {\rho }^n+n\gamma {\rho }^n\right]{\displaystyle \frac{\partial \mathrm{Y}}{\partial x}}+i(\beta +\gamma )n{\rho }^n{\displaystyle \frac{\partial {\mathrm{Y}}^{*}}{\partial x}}=0,\hfill \end{array}$$

(71)

where * denotes the conjugate of the complex function $\mathrm{Y}$. Considering that the solution of Equation (71) has the form

$$\mathrm{Y}(x,t)=\lambda e^{i(\widehat{\kappa }x-\widehat{\omega }t)}+\mu e^{-i(\widehat{\kappa }x-\widehat{\omega }t)},$$

(72)

where $\widehat{\kappa }$ and $\widehat{\omega }$ are the normalized wave number and frequency of perturbation, respectively. Substituting ansatz (72) into Equation (71), we find a couple of equations in $\lambda$ and $\mu$ by splitting the coefficients of $exp\left\{i(\widehat{\kappa }x-\widehat{\omega }t)\right\}$ and $exp\{-i(\widehat{\kappa }x-\widehat{\omega }t)\}$ presented as

$$\begin{array}{cc}\hfill & \left[n(\lambda +\mu )(\beta +\gamma )+\lambda (\beta +b)\right]\widehat{\kappa }{\rho }^n+\left[a{\widehat{\kappa }}^2+\rho \delta +\alpha \widehat{\kappa }-\widehat{\omega }\right]\lambda =0,\hfill \\ \hfill & \left[n(\lambda +\mu )(\beta +\gamma )+\mu (\beta +b)\right]\widehat{\kappa }{\rho }^n-\left[a{\widehat{\kappa }}^2+\rho \delta -\alpha \widehat{\kappa }+\widehat{\omega }\right]\mu =0.\hfill \end{array}$$

(73)

The coupled equations (73) can be formulated in matrix form for the coefficients of $\lambda$ and $\mu$. The existence of a non-trivial solution to this matrix implies that the determinant must be zero. Hence, it brings about the dispersion relation of the form

$$\left(\widehat{\omega }-\alpha \widehat{\kappa }-(\beta +b)\widehat{\kappa }{\rho }^n\right)\left(\widehat{\omega }-\alpha \widehat{\kappa }-(2n(\beta +\gamma )+\beta +b)\widehat{\kappa }{\rho }^n\right)-{(a{\widehat{\kappa }}^2+\rho \delta )}^2=0.$$

(74)

One can reach the solution to the dispersion relation (74) for $\widehat{\omega }$ given by

$$\widehat{\omega }=[\alpha +(n(\beta +\gamma )+\beta +b){\rho }^n]\widehat{\kappa }\pm \sqrt{{(a{\widehat{\kappa }}^2+\rho \delta )}^2+{(\beta +\gamma )}^2{n}^2{\widehat{\kappa }}^2{\rho }^{2n}}.$$

(75)

This relation between the frequency and wave number reveals the situation of the steady-state stability which is dependent upon the existence of a complex solution for $\widehat{\omega }$. Obviously, it can be noted that ${(a{\widehat{\kappa }}^2+\rho \delta )}^2+{(\beta +\gamma )}^2{n}^2{\widehat{\kappa }}^2{\rho }^{2n}$ is always $\ge 0$. Accordingly, this implies that $\widehat{\omega }$ is real for all values of $\widehat{\kappa }$ and therefore, the steady state is stable against small perturbations. The dependence of perturbation frequency on the normalized wave number is described in Figure 15.

6. Results and Discussion

As it has been seen above, the algorithm employed to investigate the chirped optical solitons appears to be very efficient in extracting distinct forms of solutions. Each expression of the two forms of undetermined coefficient schemes generated five different solitonic solutions. The obtained solutions are illustrated graphically to shed light on their dynamic behaviors. It is found that these solutions describe miscellaneous structures of chirped solitons such as W-shaped, bright, dark, gray, singular, kink and anti-kink waves. The optical solitons are depicted in 2D and 3D plots along with their corresponding chirp by selecting suitable parameter values when $n=1$ and $n=2$.

Firstly, we discuss the behaviors of solutions obtained by the undetermined coefficient approach in the form of a hyperbolic tangent function. Figure 1 shows the intensity profile of the dark optical soliton for solution (28) which is plotted with parameter values $a=\beta =\omega =0.5,b=0.25,\nu =\alpha =0.65,\gamma =0.75,\kappa =n=1$. Figure 2 characterizes the bright soliton wave for solution (31) with parameter values $a=\beta =0.5,b=0.25,\nu =0.35,\alpha =\omega =0.65,\gamma =0.625,\kappa =n=1$. Further to this, we observed that solution (34) has the profile of a W-shaped wave as demonstrated in Figure 3 with thesame parameter values as in Figure 2 except $\nu =0.75$. One can see that Figure 4 exhibits a kink wave profile for solution (37) while Figure 5 describes an anti-kink wave profile for solution (40), where both graphs are plotted with the same parameter values as in Figure 2 except $\gamma =0.75$. The behaviors of solutions retrieved by the undetermined coefficient approach that is expressed in the form of the hyperbolic secant function are elucidated as follows. Figure 6 displays the intensity profile of the bright optical soliton for solution (44) with the same parameter values as in Figure 4 except $\omega =0.478$. Similarly, Figure 7 presents the structure of a bright soliton wave for solution (47) with the same parameter values as in Figure 2. Moreover, the graphs in Figure 8 and Figure 9 depict the profile of a W-shaped wave for solutions (50) and (53), respectively, with the same parameter values as in Figure 3. The behavior of solution (56) represents a gray soliton structure as delineated in Figure 10 with the same parameter values as in Figure 4. One can observe that solutions (60), (64), and (66) obtained by the auxiliary equation technique represent a bright soliton wave as shown in Figure 11, Figure 12 and Figure 13. Finally, the illustration of solution (68) describes a kink wave profile as presented in Figure 14.

The influence of physical parameters such as chromatic dispersion (CD), nonlinear dispersion (ND) and self-steepening (SS) on the evolution of optical solitons is reported. In particular, the domination of these parameters has been examined on the amplitude variation of the dark soliton wave given by solution (28). It can be seen in Figure 16a that the uniform increase in CD results in an approximately regular rise in the soliton amplitude. In comparison to CD, the large values of ND and SS cause the amplitude to soar up rapidly as exhibited in Figure 16b,c.

Similarly, the effects of CD, ND and SS on the dispersion relation are detected as shown in Figure 17. One can see from Figure 17a that CD causes remarkable variation in perturbation frequency, $\widehat{\omega }$, while ND has a negligible influence on $\widehat{\omega }$ as presented in Figure 17b. The impact of SS on $\widehat{\omega }$ is more or less moderate as plotted in Figure 17c.

Comparing the results shown here with those obtained in previous studies, Kudryashov [41] discussed the optical soliton solutions of the perturbed CLL equation by exploiting the Weierstrass and Jacobi elliptic functions. Several solitary waves are extracted including bright, dark and singular solutions as well as periodic wave solutions. Furthermore, Zhang [49] dealt with Equation (9) using the trial equation method and a complete discrimination system to acquire exact chirped solutions. The obtained solutions are Jacobi elliptic functions, singular periodic functions and singular solitons. Additionally, Li [50] applied bifurcation analysis to Equation (9) so as to perform qualitative analysis and study explicit solutions. The constructed explicit solutions consist of Jacobi function solutions and hyperbolic function solutions. Some soliton solutions are created such as bright and kink waves. All of the soliton wave solutions obtained in those previous studies are entirely different from the ones procured in this work.

7. Conclusions

The current work studied the chirped optical solitons of the perturbed Chen–Lee–Liu equation with full nonlinearity. Having two forms in terms of hyperbolic secant and tangent functions, the method of undetermined coefficients is exploited to generate the solutions. Each form of the implemented scheme results in five different soliton solutions. Further to this, utilizing the auxiliary equation technique provided five distinct solutions as well. The structures of the derived chirped solitons include W-shaped, bright, dark, gray, singular, kink and anti-kink waves. The validity conditions for the existence of these solitons are given. The intensity profiles of the retrieved solitons together with their associated chirping are illustrated. The repercussions of chromatic dispersion, nonlinear dispersion and self-steepening on the evolution of soliton amplitude are inspected. A high amount of them is found to dramatically enhance the amplitude of dark solitons. In addition to this, the modulation instability of the perturbed CLL model is examined with the aid of linear stability analysis. It has been noticed that chromatic dispersion and self-steepening exert a significant effect on the dispersion relation compared to the nonlinearity impact, which has a negligible effect in varying the perturbation frequency. Based on the obtained solutions, it can be concluded that the undetermined coefficient approach is a powerful technique to derive several forms of optical soliton solutions. Additionally, the extracted results can be exploited for applications related to optical fiber systems as well as plasma physics.