1. Introduction
Today, research in the field of optical solitons has reached cruising speed. However, a lot of the attention has been concentrated on the investigatation of solitons of different suitable modeling equations describing sub-picosecond pulse propagation in optical fibers, with a plethora of mathematical integration to manipulate them. Moreover, the interesting aspect of optical solitons lies in optic fiber communication, which has revolutionized long-distance communication (i.e trans-continental and trans-oceanic). Hence, the transport of data has become fluid, and even reliable.
Over the last five years, a handful of categories of solitons like traveling-wave solutions and array breathers have been reported in the literature, such as variable sinh-Gaussian solitons, dark-combo solitons, soliton pairs, bright solitons, complexiton solitons, Gaussian-shaped solitons and singular solitons [1–12]. In addition, one of the environments that have shown themselves favorable to the propagation of these optical solitons is metamaterials.
Metamaterials are artificial materials which display unusual electromagnetic (EM) properties [13]. Because of these attractive properties, several studies on localized wave research have been successfully conducted. Furthermore, these localized solutions in nonlinear metamaterials usually take the forms of shift solitons, spatiotemporal solitons, temporal solitons and spatial solitons [13]. More recently, many works have been conducted in nonlinear metamaterials. For example, chirped soliton and periodic solutions of the generalized nonlinear Schrödinger equation, bright and dark soliton solutions for the (3+1)-dimensional coupled nonlinear Schrödinger equation with electric and magnetic fields, and the gray solitary wave of an extended nonlinear Schrödinger equation with third-order dispersion [3–5, 8, 13, 14] were investigated analytically.
The hard task today is to adopt some relevant methods to build exact traveling-wave solutions of these nonlinear Schrödinger equations which describe a short pulse propagation in nonlinear metamaterials. In this way, some mathematical tools were successfully used to investigate analytical solutions for partial differential equations (PDEs), such as the sine-Gordon expansion method [15], the auxiliary equation method [16], the first integral method [17], the new extended direct algebraic method [18, 19] and the generalized tanh method [18]. Thus, exact optical solitons in metamaterials with dispersive permeability have been reported by [13–15]. Recently, some authors have investigated optical solitons for higher-order nonlinear Schrödinger’s equations, describing a sub-picosecond pulse propagation in nonlinear metamaterials with cubic-quintic nonlinearity and fourth-order dispersion [12]. This work aims to study chirped optical solitons for the cubic nonlinear Schrödinger’s equation (CNLSE), describing a sub-picosecond pulse propagation in a nonlinear metamaterial with Kerr dispersion, low group velocity dispersion (GVD) and Kerr nonlinearity. The following governing equation will be adopted
$\begin{eqnarray}{\rm{i}}{q}_{x}+\displaystyle \frac{1}{2}{\beta }_{2}{q}_{{tt}}+{\gamma }_{1}| q{| }^{2}q-{\rm{i}}{\alpha }_{1}{\left(| {q}^{2}| q\right)}_{t}-{\rm{i}}{\alpha }_{2}q{\left(| q{| }^{2}\right)}_{t}=0.\end{eqnarray}$
Here, q(x, t) stands for a complex envelop amplitude, t represents the time (in the group velocity frame), x is the distance along the direction of propagation, while β2, α1 and α2 are, respectively, the GVD and the self-phase modulation (SPM) parameters. However, γ1 is the cubic nonlinearity term which compensates the dispersion and SPM terms to obtain the soliton.Therefore, the paper is organized as follows: section 2 is devoted to the description of the methods. In section 3 the application of the methods will be carried out, and some graphical illustrations will follow. The last section will summarize the work.
2. The sight of methods
2.1. The new extended direct algebraic methods
Step 1: adopting the PDE in the following form
$\begin{eqnarray}H(u,{u}_{t},{u}_{x},{u}_{{xx}},{u}_{{xt}},\ldots )=0,\end{eqnarray}$
where u(x, t) is an unknown function, and H is a polynomial of u. Surmise the traveling-wave hypothesis as follows and then by adopting u(x, t) = F(ξ) $\begin{eqnarray}\xi =x-{vt},\end{eqnarray}$
therefore, the PDE can turn into an ordinary differential equation $\begin{eqnarray}G(F,{F}^{{\prime} },{F}^{{\prime\prime} },{F}^{\prime\prime\prime },...)=0,\end{eqnarray}$
and prime denotes the derivative with respect to ξ.Step 2: considering that (4 ) has the solution in the following expression
$\begin{eqnarray}F(\xi )=\sum _{j=0}^{N}{h}_{i}{Q}^{{\rm{i}}}(\xi ),\quad {g}_{n}\ne 0.\end{eqnarray}$
Where gj(0 ≤ j ≤ N) are constants to be determined later, and Q(ξ) satisfies the following ordinary differential equation (ODE) $\begin{eqnarray}{Q}^{{\prime} }(\xi )=\mathrm{ln}(A)(\lambda +\mu Q(\xi )+\sigma {Q}^{2}(\xi )),\end{eqnarray}$
and $A\ne 0,1$. The solutions of the ODE (6) are:Case 1: ${\mu }^{2}-4\lambda \sigma \lt 0$ and $\sigma \ne 0$
$\begin{eqnarray}{Q}_{1}(\xi )=-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right),\end{eqnarray}$
$\begin{eqnarray}{Q}_{2}(\xi )=-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{3}(\xi ) & = & -\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\\ & & \pm \displaystyle \frac{\sqrt{-{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\sec }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{4}(\xi ) & = & -\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\\ & & \pm \displaystyle \frac{\sqrt{-{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\csc }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{5}(\xi ) & = & -\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right)\\ & & -\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right),\end{array}\end{eqnarray}$
Case 2: ${\mu }^{2}-4\lambda \sigma \gt 0$ and $\sigma \ne 0$ $\begin{eqnarray}{Q}_{6}(\xi )=-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tanh }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right),\end{eqnarray}$
$\begin{eqnarray}{Q}_{7}(\xi )=-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\coth }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{8}(\xi ) & = & -\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tanh }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\\ & & \pm {\rm{i}}\displaystyle \frac{\sqrt{{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{{\rm{sech}} }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{9}(\xi ) & = & -\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\coth }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\\ & & \pm \displaystyle \frac{\sqrt{{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cosh }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{10}(\xi ) & = & -\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{4\sigma }{\tanh }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right)\\ & & -\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{4\sigma }{\coth }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right),\end{array}\end{eqnarray}$
Case 3: $\lambda \sigma \gt 0$ and μ = 0 $\begin{eqnarray}{Q}_{11}(\xi )=\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\tan }_{A}\left(\sqrt{\lambda \sigma }\xi \right),\end{eqnarray}$
$\begin{eqnarray}{Q}_{12}(\xi )=-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\sqrt{\lambda \sigma }\xi \right),\end{eqnarray}$
$\begin{eqnarray}{Q}_{13}(\xi )=\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\tan }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\pm \sqrt{{pq}\displaystyle \frac{\lambda }{\sigma }}{\sec }_{A}\left(\sqrt{2\lambda \sigma }\xi \right),\end{eqnarray}$
$\begin{eqnarray}{Q}_{14}(\xi )=-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\pm \sqrt{{pq}\displaystyle \frac{\lambda }{\sigma }}{\csc }_{A}\left(\sqrt{2\lambda \sigma }\xi \right),\end{eqnarray}$
$\begin{eqnarray}{Q}_{15}(\xi )=\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{\lambda }{\sigma }}\left({\tan }_{A}\left(\displaystyle \frac{\sqrt{\lambda \sigma }}{2}\xi \right)-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\displaystyle \frac{\sqrt{\lambda \sigma }}{2}\xi \right)\right),\end{eqnarray}$
Case 4: λσ < 0 and μ = 0 $\begin{eqnarray}{Q}_{16}(\xi )=-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\sqrt{-\lambda \sigma }\xi \right),\end{eqnarray}$
$\begin{eqnarray}{Q}_{17}(\xi )=-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(\sqrt{-\lambda \sigma }\xi \right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{18}(\xi ) & = & -\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\\ & & \pm {\rm{i}}\sqrt{{pq}\displaystyle \frac{-{pq}\lambda }{\sigma }}{{\rm{sech}} }_{A}\left(2\sqrt{-\lambda \sigma }\xi \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{19}(\xi ) & = & -\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(2\sqrt{-\lambda \sigma }\xi \right)\\ & & \pm \sqrt{-{pq}\displaystyle \frac{\lambda }{\sigma }}{\mathrm{csch}}_{A}\left(2\sqrt{-\lambda \sigma }\xi \right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{Q}_{20}(\xi ) & = & -\displaystyle \frac{1}{2}\left(\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\displaystyle \frac{\sqrt{-\lambda \sigma }}{2}\xi \right)\right.\\ & & \left.+\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(\displaystyle \frac{\sqrt{-\lambda \sigma }}{2}\xi \right)\right),\end{array}\end{eqnarray}$
Case 5: μ = 0 and λ = σ $\begin{eqnarray}{Q}_{21}(\xi )={\tan }_{A}(\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{Q}_{22}(\xi )=-{\cot }_{A}(\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{Q}_{23}(\xi )={\tan }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\sec }_{A}(2\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{Q}_{24}(\xi )=-{\cot }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\csc }_{A}(2\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{Q}_{25}(\xi )=\displaystyle \frac{1}{2}\left({\tan }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)-{\cot }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)\right),\end{eqnarray}$
Case 6: μ = 0 and λ = −σ $\begin{eqnarray}{Q}_{26}(\xi )=-{\tanh }_{A}(\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{Q}_{27}(\xi )=-{\coth }_{A}(\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{Q}_{28}(\xi )=-{\tanh }_{A}(2\lambda \xi )\pm {\rm{i}}\sqrt{{pq}}{{\rm{sech}} }_{A}(2\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{Q}_{29}(\xi )=-{\coth }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\mathrm{csch}}_{A}(2\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{Q}_{30}(\xi )=-\displaystyle \frac{1}{2}\left({\tanh }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)+{\coth }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)\right),\end{eqnarray}$
Case 7: μ2 = 4λσ $\begin{eqnarray}{Q}_{31}(\xi )=-\displaystyle \frac{2\lambda (\mu \xi \mathrm{ln}(A)+2)}{{\mu }^{2}\xi \mathrm{ln}(A)},\end{eqnarray}$
Case 8: $\mu =k,\lambda ={mk}(m\ne 0),\ {and}\ \sigma =0,$ $\begin{eqnarray}{Q}_{32}(\xi )={A}^{\xi k}-m,\end{eqnarray}$
Case 9: μ = σ = 0 $\begin{eqnarray}{Q}_{33}(\xi )=\lambda \xi \mathrm{ln}A,\end{eqnarray}$
Case 10: μ = λ = 0 $\begin{eqnarray}{Q}_{34}(\xi )=\displaystyle \frac{-1}{\sigma \xi \mathrm{ln}A},\end{eqnarray}$
Case 11: $\mu \ne 0$ and λ = 0. $\begin{eqnarray}{Q}_{35}(\xi )=\displaystyle \frac{p\mu }{\sigma ({\cosh }_{A}(\mu \xi )-{\sinh }_{A}(\mu \xi )-p)},\end{eqnarray}$
$\begin{eqnarray}{Q}_{36}(\xi )=-\displaystyle \frac{\mu ({\sinh }_{A}(\mu \xi )+{\cosh }_{A}(\mu \xi ))}{\sigma ({\cosh }_{A}(\mu \xi )-{\sinh }_{A}(\mu \xi )+q)},\end{eqnarray}$
Case 12: μ = k and $\sigma ={mk}(m\ne 0)$ and λ = 0. $\begin{eqnarray}{Q}_{37}(\xi )=-\displaystyle \frac{{{pA}}^{k\xi }}{q-{{mpA}}^{k\xi }}.\end{eqnarray}$
Step 3: by using the homogeneous balance principle the value of N can be obtained between the highest-order derivative and high-order terms in (5).Step 4: substituting (5) and (6) into (4), then collecting all the terms of Qj(ξ) to set to zero yields a system of algebraic equations.
Step 5: with the aid of software programme, MAPLE, the results of the system of algebraic equations can be obtained, and then we use the results of (6) to construct the exact solutions of (4).
2.2. The extended auxiliary equation method
The pipe of the extended auxiliary equation method is described by the following step.
Step 1: the extended auxiliary equation method is based on the auxiliary ordinary equation method [12].
Therefore, an auxiliary ordinary equation (4 ) can be written as follows
$\begin{eqnarray}{\left(\displaystyle \frac{\partial F}{\partial \xi }\right)}^{2}={h}_{0}{F}^{2}+{h}_{1}{F}^{3}+{h}_{2}{F}^{4},\end{eqnarray}$
where h0, h1 and h2 are constants to be determined. Then, the solutions of (4) are given as follows:Case 1: ${h}_{0}\gt 0,{\rm{\Delta }}\gt 0$, where ${\rm{\Delta }}={h}_{1}^{2}-4{h}_{0}{h}_{2}$
$\begin{eqnarray}F(\xi )=\displaystyle \frac{2{h}_{0}{{\rm{sech}} }_{A}(\sqrt{{h}_{0}}\xi )}{\sqrt{{\rm{\Delta }}}-{h}_{1}{{\rm{sech}} }_{A}(\sqrt{{h}_{0}}\xi )},\end{eqnarray}$
Case 2: ${h}_{0}\gt 0,{\rm{\Delta }}\lt 0$, $\begin{eqnarray}F(\xi )=-\displaystyle \frac{2{h}_{0}{\mathrm{csch}}_{A}(\sqrt{{h}_{0}}\xi )}{\sqrt{-{\rm{\Delta }}}-{h}_{1}{{\rm{sech}} }_{A}(\sqrt{{h}_{0}}\xi )},\end{eqnarray}$
Case 3: ${h}_{0}\gt 0,{{\rm{\Delta }}}_{1}\lt 0$, where ${{\rm{\Delta }}}_{1}={h}_{1}^{2}-4{h}_{0}{h}_{2}-4{h}_{0}^{2}$, $\begin{eqnarray}F(\xi )=-\displaystyle \frac{2{h}_{0}{{\rm{sech}} }_{A}(\sqrt{{h}_{0}}\xi )}{{h}_{1}{{\rm{sech}} }_{A}(\sqrt{{h}_{0}}\xi )+\sqrt{-{{\rm{\Delta }}}_{1}}\tanh (\sqrt{{h}_{0}}\xi )-2{h}_{0}},\end{eqnarray}$
$\begin{eqnarray}F(\xi )=-\displaystyle \frac{2{h}_{0}{\mathrm{csch}}_{A}(\sqrt{{h}_{0}}\xi )}{{h}_{1}{\mathrm{csch}}_{A}(\sqrt{{h}_{1}}\xi )-\sqrt{{{\rm{\Delta }}}_{1}}\coth (\sqrt{{h}_{0}}\xi )+2{h}_{0}}.\end{eqnarray}$
3. Chirped optical solitons
To obtain the chirped optical solitons to the model, the following ansatz will be adopted like traveling-wave transformation
$\begin{eqnarray}q(x,t)={e}^{{\rm{i}}[f(\xi )]}\phi (\xi )\quad \xi =x-{vt},\end{eqnarray}$
φ(ξ) is the amplitude component of the soliton and f(ξ) is the chirp parameter, and $\delta \omega =-\tfrac{\partial }{\partial \xi }[f(\xi )]$ is the corresponding chirped optical solitons. Substituting (49) into (1) leads to $\begin{eqnarray}2{\phi }^{{\prime} }+2{\beta }_{2}{v}^{2}{\phi }^{{\prime} }{f}^{{\prime} }+{\beta }_{2}{v}^{2}\phi {f}^{{\prime\prime} }+6{\alpha }_{1}v{\phi }^{2}{\phi }^{{\prime} }+4{\alpha }_{2}v{\phi }^{2}{\phi }^{{\prime} }=0,\end{eqnarray}$
and $\begin{eqnarray}2\phi {f}^{{\prime} }+2{\alpha }_{1}v{\phi }^{3}{f}^{{\prime} }+{\beta }_{2}{v}^{2}\phi {{f}^{{\prime} }}^{2}-2{\gamma }_{1}{\phi }^{3}-{\beta }_{2}{v}^{2}{\phi }^{{\prime\prime} }=0.\end{eqnarray}$
From (51) the chirped component is recovered $\begin{eqnarray}\delta \omega =\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2}){\phi }^{2}.\end{eqnarray}$
Consequently, (50) becomes $\begin{eqnarray}\begin{array}{l}-\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}\phi -\left(\displaystyle \frac{2{\alpha }_{1}}{{\beta }_{2}v}+2{\gamma }_{1}\right){\phi }^{3}\\ \quad +\,\displaystyle \frac{1}{{\beta }_{2}}\left(\displaystyle \frac{{\alpha }_{1}}{4}(4{\alpha }_{2}-3{\alpha }_{1})+{\alpha }_{2}^{2}\right){\phi }^{5}-{\beta }_{2}{v}^{2}{\phi }^{{\prime\prime} }=0.\end{array}\end{eqnarray}$
Next, multiplying (53) by ${\phi }^{{\prime} }$, then integrating once with the zero constant of integration, we obtain $\begin{eqnarray}\begin{array}{l}-\displaystyle \frac{1}{2{\beta }_{2}{v}^{2}}{\phi }^{2}-\displaystyle \frac{1}{4}\left(\displaystyle \frac{2{\alpha }_{1}}{{\beta }_{2}v}+2{\gamma }_{1}\right){\phi }^{4}\\ +\,\displaystyle \frac{1}{6{\beta }_{2}}\left(\displaystyle \frac{{\alpha }_{1}}{4}(4{\alpha }_{2}-3{\alpha }_{1})+{\alpha }_{2}^{2}\right){\phi }^{6}-\displaystyle \frac{1}{2}{\beta }_{2}{v}^{2}{{\phi }^{{\prime} }}^{2}=0.\end{array}\end{eqnarray}$
To obtain a suitable and malleable form of (54), we surmise $\phi (\xi )=\sqrt{F(\xi )}$. Thence, $\begin{eqnarray}\begin{array}{l}-\displaystyle \frac{1}{2{\beta }_{2}{v}^{2}}{F}^{2}(\xi )-\displaystyle \frac{1}{4}\left(\displaystyle \frac{2{\alpha }_{1}}{{\beta }_{2}v}+2{\gamma }_{1}\right){F}^{3}(\xi )\\ +\,\displaystyle \frac{1}{6{\beta }_{2}}\left(\displaystyle \frac{{\alpha }_{1}}{4}(4{\alpha }_{2}-3{\alpha }_{1})+{\alpha }_{2}^{2}\right){F}^{4}(\xi )-\displaystyle \frac{1}{8}{\beta }_{2}{v}^{2}{{F}^{{\prime} }}^{2}(\xi )=0.\end{array}\end{eqnarray}$
4. Application of the new extended direct algebraic method to (55)
This section will apply the two methods described above to construct diverse exact traveling-wave solutions to the CNLSE describing a sub-picosecond pulse propagation in a nonlinear optic fiber.
Using the balance principle between F4 and ${{F}^{{\prime} }}^{2}(\xi )$, N = 1 is gained. Thereafter, (5) can be expressed as follows
$\begin{eqnarray}F(\xi )={h}_{0}+{h}_{1}Q(\xi ).\end{eqnarray}$
Next, substituting (56) and (6) into (55) a system of algebraic equations in terms of ${(Q(\xi ))}^{j}$are recovered. Then, collecting all the coefficients of the obtained algebraic equations and setting to zero gives the set of results.For ${\mu }^{2}-4\lambda \sigma \lt 0$ and $\sigma \ne 0$.
$\begin{eqnarray*}{h}_{0}={h}_{0},{h}_{1}=\displaystyle \frac{1}{12\mu \lambda }({{\ell }}_{1}-{{\ell }}_{2}-{{\ell }}_{3}+{{\ell }}_{4}-{{\ell }}_{5}+{{\ell }}_{6}),\end{eqnarray*}$
$\begin{eqnarray*}v=\displaystyle \frac{\sqrt{6\,\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}-12\,\lambda \,\sigma +6\,{\mu }^{2}}}{\left(3\,{\alpha }_{1}^{2}{\mu }^{2}-12\,{\alpha }_{1}^{2}\lambda \,\sigma +16\,{\alpha }_{1}{\alpha }_{2}\lambda \,\sigma -4\,{\alpha }_{1}{\alpha }_{2}{\mu }^{2}-4\,{\alpha }_{2}^{2}{\mu }^{2}+16\,{\alpha }_{2}^{2}\lambda \,\sigma \right){h}_{0}},\end{eqnarray*}$
$\begin{eqnarray*}{\beta }_{2}=\displaystyle \frac{\sqrt{-2\,{\mu }^{6}+16\,{\mu }^{4}\lambda \,\sigma +2\,{\mu }^{4}\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}-36{\mu }^{2}{\lambda }^{2}{\sigma }^{2}-12\,\lambda \,\sigma \,{\mu }^{2}\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}+16{\lambda }^{3}{\sigma }^{3}+16\,{\lambda }^{2}{\sigma }^{2}\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}}\left({\alpha }_{1}-2{\alpha }_{2}\right)\left(3\,{\alpha }_{1}+2{\alpha }_{2}\right){h}_{0}^{2}}{12{\text{}}\mathrm{ln}\left(A\right){\sigma }^{2}{\lambda }^{2}},\end{eqnarray*}$
$\begin{eqnarray*}{\gamma }_{1}=\displaystyle \frac{1}{3}\displaystyle \frac{({{\ell }}_{7}-{{\ell }}_{8}-{{\ell }}_{9}+{{\ell }}_{10}-{{\ell }}_{11}+{{\ell }}_{12}-{{\ell }}_{13}-{{\ell }}_{14}){{\ell }}_{15}}{{{\ell }}_{16}},{\alpha }_{1}={\alpha }_{1},{\alpha }_{2}={\alpha }_{2},\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{{\ell }}_{1}=72\lambda \,\sigma \,{\alpha }_{1}^{2}{\rm{\Gamma }},\\ {{\ell }}_{2}=18{\alpha }_{1}^{2}{\mu }^{2}{\rm{\Gamma }},\\ {{\ell }}_{3}=96{\alpha }_{2}\lambda \,\sigma \,{\alpha }_{1}{\rm{\Gamma }},\\ {{\ell }}_{4}=24{\alpha }_{1}{\alpha }_{2}{\mu }^{2}{\rm{\Gamma }},\\ {{\ell }}_{5}=96{\alpha }_{2}^{2}\lambda \,\sigma {\rm{\Gamma }},\\ {{\ell }}_{6}=24{\alpha }_{2}^{2}{\mu }^{2}{\rm{\Gamma }},\\ {{\ell }}_{7}-72\lambda \,\sigma \,{\alpha }_{1}^{2}{\rm{\Gamma }},\\ {{\ell }}_{8}=18{\alpha }_{1}^{2}{\mu }^{2}{\rm{\Gamma }},\\ {{\ell }}_{9}=96{\alpha }_{2}\lambda \,\sigma \,{\alpha }_{1}{\rm{\Gamma }},\\ {{\ell }}_{10}=24{\alpha }_{1}{\alpha }_{2}{\mu }^{2}{\rm{\Gamma }},\\ {{\ell }}_{11}=96{\alpha }_{2}^{2}\lambda \,\sigma {\rm{\Gamma }},\\ {{\ell }}_{12}=24{\alpha }_{2}^{2}{\mu }^{2}{\rm{\Gamma }},\\ {{\ell }}_{13}=\displaystyle \frac{{m}_{1}\times {m}_{2}}{3\,{\alpha }_{1}^{2}{\mu }^{2}-12\,{\alpha }_{1}^{2}\lambda \,\sigma +16\,{\alpha }_{1}{\alpha }_{2}\lambda \,\sigma -4\,{\alpha }_{1}{\alpha }_{2}{\mu }^{2}-4\,{\alpha }_{2}^{2}{\mu }^{2}+16\,{\alpha }_{2}^{2}\lambda \,\sigma },\\ {{\ell }}_{14}=\displaystyle \frac{24{\alpha }_{1}\sqrt{6}{m}_{1}{m}_{2}}{3\,{\alpha }_{1}^{2}{\mu }^{2}-12\,{\alpha }_{1}^{2}\lambda \,\sigma +16\,{\alpha }_{1}{\alpha }_{2}\lambda \,\sigma -4\,{\alpha }_{1}{\alpha }_{2}{\mu }^{2}-4\,{\alpha }_{2}^{2}{\mu }^{2}+16\,{\alpha }_{2}^{2}\lambda \,\sigma }-12\lambda \sigma ,\\ {m}_{1}=(3\,{\alpha }_{1}^{2}{\mu }^{2}-12\,{\alpha }_{1}^{2}\lambda \,\sigma +16\,{\alpha }_{1}{\alpha }_{2}\lambda \,\sigma \\ \qquad -\,4\,{\alpha }_{1}{\alpha }_{2}{\mu }^{2}-4\,{\alpha }_{2}^{2}{\mu }^{2}+16\,{\alpha }_{2}^{2}\lambda \,\sigma ),\\ {m}_{2}=(\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}-2\,\lambda \,\sigma +{\mu }^{2}),\\ {{\ell }}_{15}=\left(3\,{\alpha }_{1}^{2}{\mu }^{2}-12\,{\alpha }_{1}^{2}\lambda \,\sigma +16\,{\alpha }_{1}{\alpha }_{2}\lambda \,\sigma \right.\\ \left.-\,4\,{\alpha }_{1}{\alpha }_{2}{\mu }^{2}-4\,{\alpha }_{2}^{2}{\mu }^{2}+16\,{\alpha }_{2}^{2}\lambda \,\sigma \right){\text{}}\mathrm{ln}\left(A\right){\sigma }^{2}{\lambda }^{2},\\ {\rm{\Gamma }}=\displaystyle \frac{\left(\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}-2\lambda \,\sigma +{\mu }^{2}\right)}{3\,{\alpha }_{1}^{2}{\mu }^{2}-12\,{\alpha }_{1}^{2}\lambda \,\sigma +16\,{\alpha }_{1}{\alpha }_{2}\lambda \,\sigma -4\,{\alpha }_{1}{\alpha }_{2}{\mu }^{2}-4\,{\alpha }_{2}^{2}{\mu }^{2}+16\,{\alpha }_{2}^{2}\lambda \,\sigma },\\ {{\ell }}_{16}=\left({\mu }^{2}-4\,\lambda \,\sigma \right){h}_{0}\left(\sqrt{-4\lambda \sigma \,{\mu }^{2}+{\mu }^{4}}-2\,\lambda \,\sigma +{\mu }^{2}\right)\\ \qquad \times \,\left({\alpha }_{1}-2\,{\alpha }_{2}\right)\left(3{\alpha }_{1}+2{\alpha }_{2}\right)\sqrt{{{\ell }}_{17}-{{\ell }}_{18}},\\ {{\ell }}_{17}=-2{\mu }^{6}+16{\mu }^{4}\lambda \sigma +2\,{\mu }^{4}\sqrt{-4\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}-36{\mu }^{2}{\lambda }^{2}{\sigma }^{2},\\ {{\ell }}_{18}=4\left(3\lambda \,\sigma \,{\mu }^{2}+4\,{\lambda }^{2}{\sigma }^{2}\right)\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}+16\,{\lambda }^{3}{\sigma }^{3}.\end{array}\end{eqnarray*}$
However, we reveal new traveling waves and chirped optical solitons to (1) $\begin{eqnarray}\begin{array}{l}{q}_{1}(x,t)\,=\,\sqrt{{h}_{0}+{h}_{1}\left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right)\right]}\\ \,\times \,{{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
and the corresponding chirped $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{1}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \qquad \times \,\left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{q}_{2}(x,t)\\ =\,\sqrt{{h}_{0}+{h}_{1}\left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right)\right]}\\ \qquad \times \,{{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
and the chirped optical solitons gives $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{2}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \qquad \times \,\left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{3}(x,t) & = & \sqrt{{h}_{0}+{h}_{1}\left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\pm \displaystyle \frac{\sqrt{-{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\sec }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\right]}\\ & & \times \ {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
and the chirped $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{3} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\right.\\ & & \left.\pm \displaystyle \frac{\sqrt{-{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\sec }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{q}_{4}(x,t)=\sqrt{{h}_{0}+{h}_{1}[-\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\pm \displaystyle \frac{\sqrt{-{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\csc }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)]}\end{eqnarray}$
$\begin{eqnarray}\times \,{{\rm{e}}}^{{\rm{i}}[f(\xi )]},\,\,\,\end{eqnarray}$
and the chirped gives $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{4}(x,t)=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \left[-\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\right.\\ \left.\pm \,\displaystyle \frac{\sqrt{-{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\csc }_{A}\left(\sqrt{-({\mu }^{2}-4\lambda \sigma )}\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{5}(x,t) & = & \sqrt{{h}_{0}+{h}_{1}\left[-\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right)-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right)\right]}\\ & & \times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
and the corresponding chirped optical solitons to (1) are recovered $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{5}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \qquad \times \,\left[-\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tan }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right)\right.\\ \left.\qquad -\,\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\cot }_{A}\left(\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right)\right].\end{array}\end{eqnarray}$
For ${\mu }^{2}-4\lambda \sigma \gt 0$ and $\sigma \ne 0,{h}_{0}$, ${h}_{1}=\tfrac{1}{12\mu \lambda }({{\ell }}_{1}-{{\ell }}_{2}-{{\ell }}_{3}+{{\ell }}_{4}-{{\ell }}_{5}+{{\ell }}_{6})$,
$\begin{eqnarray*}\begin{array}{rcl}v & = & \displaystyle \frac{\sqrt{6\,\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}-12\,\lambda \,\sigma +6\,{\mu }^{2}}}{\left(3\,{\alpha }_{1}^{2}{\mu }^{2}-12\,{\alpha }_{1}^{2}\lambda \,\sigma +16\,{\alpha }_{1}{\alpha }_{2}\lambda \,\sigma -4\,{\alpha }_{1}{\alpha }_{2}{\mu }^{2}-4\,{\alpha }_{2}^{2}{\mu }^{2}+16\,{\alpha }_{2}^{2}\lambda \,\sigma \right){h}_{0}},\\ {\beta }_{2} & = & \displaystyle \frac{\sqrt{-2\,{\mu }^{6}+16\,{\mu }^{4}\lambda \,\sigma +2\,{\mu }^{4}\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}-36{\mu }^{2}{\lambda }^{2}{\sigma }^{2}-12\,\lambda \,\sigma \,{\mu }^{2}\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}+16\,{\lambda }^{3}{\sigma }^{3}+16\,{\lambda }^{2}{\sigma }^{2}\sqrt{-4\,\lambda \,\sigma \,{\mu }^{2}+{\mu }^{4}}}\left({\alpha }_{1}-2\,{\alpha }_{2}\right)\left(3\,{\alpha }_{1}+2\,{\alpha }_{2}\right){h}_{0}^{2}}{12{\text{}}\mathrm{ln}\left(A\right){\sigma }^{2}{\lambda }^{2}},\\ {\gamma }_{1} & = & \displaystyle \frac{1}{3}\displaystyle \frac{({{\ell }}_{7}-{{\ell }}_{8}-{{\ell }}_{9}+{{\ell }}_{10}-{{\ell }}_{11}+{{\ell }}_{12}-{{\ell }}_{13}-{{\ell }}_{14}){{\ell }}_{15}}{{{\rm{\Omega }}}_{1}},{\alpha }_{1}={\alpha }_{1},{\alpha }_{2}={\alpha }_{2};\end{array}\end{eqnarray*}$
Therefore, the following is recovered $\begin{eqnarray}\begin{array}{l}{q}_{6}(x,t)\\ \,=\,\sqrt{{h}_{0}+{h}_{1}\left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tanh }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right)\right]}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the corresponding chirped optical solitons give $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{6}(x,t)=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \,\times \,\left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tanh }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{7}(x,t) & = & -\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }\\ & & \times \,{\coth }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2}\xi \right)\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{8}(x,t) & = & \sqrt{{h}_{0}+{h}_{1}\left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tanh }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\pm {\rm{i}}\displaystyle \frac{\sqrt{{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{{\rm{sech}} }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\right]}\\ & & \times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
and the chirped optical solitons give $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{8} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[-\displaystyle \frac{\mu }{2\sigma }+\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\tanh }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\right.\\ & & \left.\pm {\rm{i}}\displaystyle \frac{\sqrt{{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{{\rm{sech}} }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray*}{q}_{9}(x,t)=\sqrt{{h}_{0}+{h}_{1}\left[-\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\coth }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\pm \displaystyle \frac{\sqrt{{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\mathrm{csch}}_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\right]}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray*}$
the chirp gives $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{9} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[-\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\coth }_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\right.\\ & & \left.\pm \displaystyle \frac{\sqrt{{pq}({\mu }^{2}-4\lambda \sigma )}}{2\sigma }{\mathrm{csch}}_{A}\left(\sqrt{({\mu }^{2}-4\lambda \sigma )}\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{10}(x,t) & = & {h}_{0}+{h}_{1}\left[-\displaystyle \frac{\mu }{2\sigma }-\displaystyle \frac{\sqrt{-({\mu }^{2}-4\lambda \sigma )}}{4\sigma }\right.\\ & & \times {\tanh }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right)\\ & & \left.-\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{4\sigma }{\coth }_{A}\left(\displaystyle \frac{\sqrt{({\mu }^{2}-4\lambda \sigma )}}{4}\xi \right)\right]\\ & & \times {e}^{{\rm{i}}[f(\xi )]}.\end{array}\end{eqnarray}$
For $\lambda \sigma \gt 0$ and μ = 0, we obtain ${h}_{0}=\sqrt{\tfrac{-3\lambda \sqrt{{\theta }_{1}}}{\sigma }},{h}_{1}={\left(9{\theta }_{1}\right)}^{\tfrac{1}{4}}$, $v=\tfrac{\sqrt{-\sqrt{{\theta }_{1}}\left(3{\alpha }_{1}+2{\alpha }_{2}\right)\left({\alpha }_{1}-2{\alpha }_{2}\right)}}{\mathrm{ln}(A)\sigma {\beta }_{2}}$, ${\gamma }_{1}=-\tfrac{1}{3}\mathrm{ln}(A)\sigma \left[{h}_{0}(-4{\alpha }_{2}^{2}{\alpha }_{1}^{2}-4{\alpha }_{1}{\alpha }_{2})+\tfrac{3{\alpha }_{1}}{v}\right]$, ${\alpha }_{1}={\alpha }_{1},{\alpha }_{2}={\alpha }_{2},{\beta }_{2}={\beta }_{2}$, ${\theta }_{1}=\tfrac{{\beta }_{2}^{2}{\left({\text{}}\mathrm{ln}\left(A\right)\right)}^{2}{\sigma }^{3}}{\lambda {\left(3{\alpha }_{1}+2{\alpha }_{2}\right)}^{2}{\left({\alpha }_{1}-2{\alpha }_{2}\right)}^{2}}$.
$\begin{eqnarray}{q}_{11}(x,t)=\sqrt{{h}_{0}+{h}_{1}\left[\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\tan }_{A}\left(\sqrt{\lambda \sigma }\xi \right)\right]}\times {e}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the chirped gives $\begin{eqnarray}\delta {\omega }_{11}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\left[\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\tan }_{A}\left(\sqrt{\lambda \sigma }\xi \right)\right],\end{eqnarray}$
$\begin{eqnarray}{q}_{12}(x,t)=\sqrt{{h}_{0}+{h}_{1}\left[-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\sqrt{\lambda \sigma }\xi \right)\right]}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the corresponding gives $\begin{eqnarray}\delta {\omega }_{12}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\left[-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\sqrt{\lambda \sigma }\xi \right)\right],\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{q}_{13}(x,t)\\ =\,\sqrt{{h}_{0}+{h}_{1}\left[\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\tan }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\pm \sqrt{{pq}\displaystyle \frac{\lambda }{\sigma }}{\sec }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\right]}\\ \qquad \times \,{{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the corresponding chirped gives $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{13} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\tan }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\pm \sqrt{{pq}\displaystyle \frac{\lambda }{\sigma }}{\sec }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{q}_{14}(x,t)\\ =\,\sqrt{{h}_{0}+{h}_{1}\left[-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\pm \sqrt{{pq}\displaystyle \frac{\lambda }{\sigma }}{\csc }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\right]}\\ \qquad \times \,{e}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the chirped optical solitons give $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{14}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \left[-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\pm \sqrt{{pq}\displaystyle \frac{\lambda }{\sigma }}{\csc }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{q}_{15}(\xi )\\ =\,\sqrt{{h}_{0}+{h}_{1}\left[\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{\lambda }{\sigma }}\left({\tan }_{A}\left(\displaystyle \frac{\sqrt{\lambda \sigma }}{2}\xi \right)-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\displaystyle \frac{\sqrt{\lambda \sigma }}{2}\xi \right)\right)\right]}\\ \qquad \times \,{{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the corresponding chirped gives $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{15}(\xi )=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \ \ \times \ \left[\displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{\lambda }{\sigma }}\left({\tan }_{A}\left(\displaystyle \frac{\sqrt{\lambda \sigma }}{2}\xi \right)-\sqrt{\displaystyle \frac{\lambda }{\sigma }}{\cot }_{A}\left(\displaystyle \frac{\sqrt{\lambda \sigma }}{2}\xi \right)\right)\right].\end{array}\end{eqnarray}$
For λσ < 0 and μ = 0, we gain ${h}_{0}\,=\sqrt{\tfrac{-3\lambda \sqrt{{\theta }_{1}}}{\sigma }},{h}_{1}={\left(9{\theta }_{1}\right)}^{\tfrac{1}{4}}$, $v=\tfrac{\sqrt{-\sqrt{{\theta }_{1}}\left(3{\alpha }_{1}+2{\alpha }_{2}\right)\left({\alpha }_{1}-2{\alpha }_{2}\right)}}{\mathrm{ln}(A)\sigma {\beta }_{2}}$, ${\gamma }_{1}\,=-\tfrac{1}{3}\mathrm{ln}(A)\sigma \left[{h}_{0}(-4{\alpha }_{2}^{2}{\alpha }_{1}^{2}-4{\alpha }_{1}{\alpha }_{2})+\tfrac{3{\alpha }_{1}}{v}\right]$, ${\alpha }_{1}={\alpha }_{1},{\alpha }_{2}\,={\alpha }_{2},{\beta }_{2}={\beta }_{2}$, ${\theta }_{1}=\tfrac{{\beta }_{2}^{2}{\left({\text{}}\mathrm{ln}\left(A\right)\right)}^{2}{\sigma }^{3}}{\lambda {\left(3{\alpha }_{1}+2{\alpha }_{2}\right)}^{2}{\left({\alpha }_{1}-2{\alpha }_{2}\right)}^{2}}$.
$\begin{eqnarray}{q}_{16}(x,t)=\sqrt{{h}_{0}+{h}_{1}\left[-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\sqrt{-\lambda \sigma }\xi \right)\right]}\times {e}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the chirped gives $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{16} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\sqrt{-\lambda \sigma }\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{q}_{17}(x,t)=\sqrt{{h}_{0}+{h}_{1}\left[-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(\sqrt{-\lambda \sigma }\xi \right)\right]}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the corresponding chirped gives $\begin{eqnarray}\begin{array}{rclrcl}\delta {\omega }_{17} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2}) & & & \times \left[-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(\sqrt{-\lambda \sigma }\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{q}_{18}(x,t)=\sqrt{{h}_{0}+{h}_{1}\left[-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\pm {\rm{i}}\sqrt{{pq}\displaystyle \frac{-{pq}\lambda }{\sigma }}{{\rm{sech}} }_{A}\left(2\sqrt{-\lambda \sigma }\xi \right)\right]}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the corresponding chirped is obtained $\begin{eqnarray}\delta {\omega }_{18}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\end{eqnarray}$
$\begin{eqnarray}\times \left[-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\sqrt{2\lambda \sigma }\xi \right)\pm {\rm{i}}\sqrt{{pq}\displaystyle \frac{-{pq}\lambda }{\sigma }}{{\rm{sech}} }_{A}\left(2\sqrt{-\lambda \sigma }\xi \right)\right],\end{eqnarray}$
$\begin{eqnarray}{q}_{19}(x,t)=\sqrt{{h}_{0}+{h}_{1}\left[-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(2\sqrt{-\lambda \sigma }\xi \right)\pm \sqrt{-{pq}\displaystyle \frac{\lambda }{\sigma }}{\mathrm{csch}}_{A}\left(2\sqrt{-\lambda \sigma }\xi \right)\right]}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the chirped optical solitons give $\begin{eqnarray}\begin{array}{ll}\delta {\omega }_{19}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2}) & \times \,\left[-\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(2\sqrt{-\lambda \sigma }\xi \right)\pm \sqrt{-{pq}\displaystyle \frac{\lambda }{\sigma }}{\mathrm{csch}}_{A}\left(2\sqrt{-\lambda \sigma }\xi \right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{ll}{q}_{20}(x,t) & =\,\sqrt{{h}_{0}+{h}_{1}\left[-\displaystyle \frac{1}{2}\left(\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\displaystyle \frac{\sqrt{-\lambda \sigma }}{2}\xi \right)+\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(\displaystyle \frac{\sqrt{-\lambda \sigma }}{2}\xi \right)\right)\right]}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the chirped is $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{20}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \,\times \,\left[-\displaystyle \frac{1}{2}\left(\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\tanh }_{A}\left(\displaystyle \frac{\sqrt{-\lambda \sigma }}{2}\xi \right)+\sqrt{\displaystyle \frac{-\lambda }{\sigma }}{\coth }_{A}\left(\displaystyle \frac{\sqrt{-\lambda \sigma }}{2}\xi \right)\right)\right].\end{array}\end{eqnarray}$
For μ = 0 and λ = σ, we obtain new complex optical soliton solutions to (1) ${h}_{0}={\rm{i}}{h}_{1},{h}_{1}={h}_{1}$, $v=\tfrac{{\theta }_{2}}{{h}_{1}},{\alpha }_{1}={\alpha }_{1},{\alpha }_{2}={\alpha }_{2}$, ${\beta }_{2}=-\tfrac{{h}_{1}^{2}}{3{\theta }_{2}^{2}\sigma \mathrm{ln}(A)]}$, ${\gamma }_{1}=\tfrac{{\imath }\sigma \mathrm{ln}(A){\theta }_{2}\left[-3{\imath }{\alpha }_{1}+{\alpha }_{1}^{2}{\theta }_{2}-4{\alpha }_{2}{\theta }_{2}({\alpha }_{2}-{\alpha }_{1})\right]}{-3{h}_{1}}$, ${\theta }_{2}=\sqrt{\tfrac{3}{\left(3\,{\alpha }_{1}+2\,{\alpha }_{2}\right)\left({\alpha }_{1}-2{\alpha }_{2}\right)}}$
$\begin{eqnarray}{q}_{21}(\xi )=\sqrt{{h}_{1}\left[{\rm{i}}+{\tan }_{A}(\lambda \xi )\right]}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the complex chirped soliton is obtained $\begin{eqnarray}\delta {\omega }_{21}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2}){\tan }_{A}(\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{q}_{22}(x,t)=\sqrt{{h}_{1}\left({\rm{i}}-{\cot }_{A}(\lambda \xi \right)}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the corresponding complex chirped is given $\begin{eqnarray}\delta {\omega }_{22}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2}){\cot }_{A}(\lambda \xi )),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{23}(x,t) & = & \sqrt{{h}_{1}({\rm{i}}+1)\left[{\tan }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\sec }_{A}(2\lambda \xi )\right]}\\ & & \times {e}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the complex chirped gives $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{23} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[{\tan }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\sec }_{A}(2\lambda \xi )\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{24}(x,t) & = & \sqrt{{h}_{1}({\rm{i}}+1)\left[-{\cot }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\csc }_{A}(2\lambda \xi )\right]}\\ & & \times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the complex chirped gives $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{24} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[-{\cot }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\csc }_{A}(2\lambda \xi )\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{25}(x,t) & = & \sqrt{{h}_{1}({\rm{i}}+1)\left[\displaystyle \frac{1}{2}\left({\tan }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)-{\cot }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)\right)\right]}\\ & & \times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the complex chirped optical solitons give $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{25} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[\displaystyle \frac{1}{2}\left({\tan }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)-{\cot }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)\right)\right].\end{array}\end{eqnarray}$
For μ = 0 and λ = −σ, we gain ${h}_{0}={h}_{1}\,={\left(-\tfrac{27{\beta }_{2}^{2}\mathrm{ln}{\left(A\right)}^{2}{\sigma }^{2})}{{\theta }_{2}}\right)}^{\tfrac{1}{4}}$, $v=-\tfrac{1}{\sqrt{{\beta }_{2}\sigma \mathrm{ln}(A)}}$, ${\alpha }_{1}={\alpha }_{1},{\alpha }_{2}={\alpha }_{2},{\beta }_{2}\,={\beta }_{2}$, ${\gamma }_{1}=-\tfrac{1}{{\beta }_{2}}{\beta }_{2}\sigma \mathrm{ln}(A)(3{\alpha }_{1}+{\theta }_{2}(3{\alpha }_{1}-4{\alpha }_{2}^{2}))$,
$\begin{eqnarray}{q}_{26}(x,t)=\sqrt{{h}_{1}(1-{\tanh }_{A}(\lambda \xi ))}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the chirped gives $\begin{eqnarray}\delta {\omega }_{26}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2}){\tanh }_{A}(\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{q}_{27}(x,t)={h}_{1}(1-{\coth }_{A}(\lambda \xi ))\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the chirped optical soliton gives $\begin{eqnarray}\delta {\omega }_{27}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2}){\coth }_{A}(\lambda \xi ),\end{eqnarray}$
$\begin{eqnarray}{q}_{28}(x,t)=\sqrt{{h}_{1}(1-{\tanh }_{A}(2\lambda \xi )\pm {\rm{i}}\sqrt{{pq}}{{\rm{sech}} }_{A}(2\lambda \xi ))}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the chirped gives $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{28} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[{\tanh }_{A}(2\lambda \xi )\pm {\rm{i}}\sqrt{{pq}}{{\rm{sech}} }_{A}(2\lambda \xi )\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{q}_{29}(\xi )=\sqrt{{h}_{1}(1-{\coth }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\mathrm{csch}}_{A}(2\lambda \xi ))}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the chirped is obtained $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{29}(\xi ) & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[{\coth }_{A}(2\lambda \xi )\pm \sqrt{{pq}}{\mathrm{csch}}_{A}(2\lambda \xi )\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{30}(x,t) & = & \sqrt{{h}_{1}\left(1-\displaystyle \frac{1}{2}\left({\tanh }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)+{\coth }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)\right)\right)}\\ & & \times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the chirped optical soliton is obtained $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{30} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left(\displaystyle \frac{1}{2}\left({\tanh }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)+{\coth }_{A}\left(\displaystyle \frac{\lambda }{2}\xi \right)\right)\right).\end{array}\end{eqnarray}$
$\mu =k,\lambda ={mk}(m\ne 0),{and}\sigma =0$, the following is recovered: ${h}_{0}={{mh}}_{1},{h}_{1}={h}_{1}$, $v\,={\left(-\tfrac{4}{{\beta }^{2}\mathrm{ln}{\left(A\right)}^{2}{k}^{2}}\right)}^{\tfrac{1}{4}},{\alpha }_{1}\,=-{\gamma }_{1}v{\beta }_{2}$, ${\alpha }_{2}=\tfrac{3}{2}{\gamma }_{1}v,{\beta }_{2}={\beta }_{2},{\gamma }_{1}={\gamma }_{1}$,
$\begin{eqnarray}{q}_{32}(x,t)=\sqrt{\left({h}_{1}{A}^{\xi k}-m(1-{h}_{1}\right)}\times {e}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
the chirped optical corresponds to $\begin{eqnarray}\delta {\omega }_{32}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2}){A}^{\xi k}-m.\end{eqnarray}$
For $\mu \ne 0$ and $\lambda =0$ we obtain ${h}_{0}\,={\left(12{\theta }_{2}^{2}{\beta }_{2}^{2}\mathrm{ln}{\left(A\right)}^{2}\right)}^{\tfrac{1}{4}}$, ${h}_{1}=\sigma {h}_{0}$, $v=\tfrac{\sqrt{-\tfrac{2{\theta }_{2}}{\sqrt{3}}(3{\alpha }_{1}+{\alpha }_{2})({\alpha }_{1}-2{\alpha }_{2})}}{{\beta }_{2}\mathrm{ln}(A)}$, ${\alpha }_{1}\,={\alpha }_{1},{\alpha }_{2}={\alpha }_{2},{\beta }_{2}={\beta }_{2}$, ${\gamma }_{1}=\tfrac{3\left[2{\beta }_{2}\mathrm{ln}(A)+v\mathrm{ln}(A){\left(-12{\alpha }_{1}{\theta }_{2}^{2}\mathrm{ln}{\left(A\right)}^{2}{\sigma }^{2}\right)}^{\tfrac{1}{4}}\right]}{-12{\theta }_{2}^{2}{\sigma }^{2}{\beta }_{2}^{2}\mathrm{ln}{\left(A\right)}^{2}(3{\alpha }_{1}+2{\alpha }_{2})({\alpha }_{1}-2{\alpha }_{2})}$,
$\begin{eqnarray}\begin{array}{rcl}{q}_{33}(x,t) & = & \sqrt{{h}_{0}\left(1+\sigma \displaystyle \frac{p\mu }{\sigma ({\cosh }_{A}(\mu \xi )-{\sinh }_{A}(\mu \xi )-p)}\right)}\\ & & \times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the corresponding chirped $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{33} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[\displaystyle \frac{p\mu }{\sigma ({\cosh }_{A}(\mu \xi )-{\sinh }_{A}(\mu \xi )-p)}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{q}_{34}(x,t) & = & {h}_{0}\left(1-\displaystyle \frac{\mu ({\sinh }_{A}(\mu \xi )+{\cosh }_{A}(\mu \xi ))}{({\cosh }_{A}(\mu \xi )-{\sinh }_{A}(\mu \xi )+q)}\right)\\ & & \times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{array}\end{eqnarray}$
the chirped gives $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{34} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left(\displaystyle \frac{\mu ({\sinh }_{A}(\mu \xi )+{\cosh }_{A}(\mu \xi ))}{\sigma ({\cosh }_{A}(\mu \xi )-{\sinh }_{A}(\mu \xi )+q)}\right).\end{array}\end{eqnarray}$
For μ = k and $\sigma ={mk}(m\ne 0)$ and λ = 0, we obtain ${h}_{0}={\left(12{\theta }_{2}^{2}{\beta }_{2}^{2}\mathrm{ln}{\left(A\right)}^{2}\right)}^{\tfrac{1}{4}}$, ${h}_{1}={{mh}}_{0},v=\tfrac{\sqrt{-\tfrac{2{\theta }_{2}}{\sqrt{3}}(3{\alpha }_{1}+{\alpha }_{2})({\alpha }_{1}-2{\alpha }_{2})}}{{\beta }_{2}\mathrm{ln}(A)}$, α1 = α1, α2 = α2, β2 = β2, ${\gamma }_{1}=\tfrac{3\left[2{\beta }_{2}\mathrm{ln}(A)+v\mathrm{ln}(A){\left(-12{\alpha }_{1}{\theta }_{2}^{2}\mathrm{ln}{\left(A\right)}^{2}{\sigma }^{2}\right)}^{\tfrac{1}{4}}\right]}{-12{\theta }_{2}^{2}{\sigma }^{2}{\beta }_{2}^{2}\mathrm{ln}{\left(A\right)}^{2}(3{\alpha }_{1}+2{\alpha }_{2})({\alpha }_{1}-2{\alpha }_{2})}$,
$\begin{eqnarray}{q}_{35}(\xi )=\sqrt{{h}_{0}\left(-\displaystyle \frac{{{mpA}}^{k\xi }}{q-{{mpA}}^{k\xi }}\right)}\times {{\rm{e}}}^{{\rm{i}}[f(\xi )]},\end{eqnarray}$
and lastly the chirped optical soliton to (1) is given $\begin{eqnarray}{\delta }_{35}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\displaystyle \frac{{{pA}}^{k\xi }}{q-{{mpA}}^{k\xi }}.\end{eqnarray}$
Here, in this paper, we omit the generalized hyperbolic and triangular function which are given by [18, 19].5. Application of the extended auxiliary equation method to (55)
To obtain the optical soliton solution, (55) can to be transformed as follows [12]:
$\begin{eqnarray}{\left(\displaystyle \frac{\partial F}{\partial \xi }\right)}^{2}={h}_{0}{F}^{2}(\xi )+{h}_{1}{F}^{3}(\xi )+{h}_{2}{F}^{4}(\xi ).\end{eqnarray}$
Where ${h}_{0}=-\tfrac{4}{{v}^{4}{\beta }_{2}^{2}}$, ${h}_{1}=\tfrac{4\left({\alpha }_{1}+{\gamma }_{1}{\beta }_{2}v\right)}{{\beta }_{2}^{2}{v}^{3}}$, ${h}_{2}=\tfrac{{\alpha }_{1}\left(4{\alpha }_{2}-3{\alpha }_{1}\right)+4{\alpha }_{1}^{2}}{3{v}^{2}{\beta }_{2}^{2}}$.Consequently,
S1: h0 > 0, Δ > 0, where ${\rm{\Delta }}={h}_{1}^{2}-4{h}_{0}{h}_{2}$.
$\begin{eqnarray}{q}_{35}(x,t)={e}^{{\rm{i}}[f(\xi )]}\times \sqrt{\displaystyle \frac{2{h}_{0}{\rm{sech}} (\sqrt{{{\rm{h}}}_{0}}\xi )}{\sqrt{{\rm{\Delta }}}-{h}_{1}{\rm{sech}} (\sqrt{{{\rm{h}}}_{0}}\xi )}},\end{eqnarray}$
and the chirped bright optical soliton is obtained $\begin{eqnarray}\delta {\omega }_{3}5=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}+\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\left[\displaystyle \frac{2{h}_{0}{\rm{sech}} (\sqrt{{h}_{0}}\xi )}{\sqrt{{\rm{\Delta }}}-{h}_{1}{\rm{sech}} (\sqrt{{h}_{0}}\xi )}\right].\end{eqnarray}$
S2: h0 > 0, Δ < 0,
$\begin{eqnarray}{q}_{36}(\xi )={e}^{{\rm{i}}[f(\xi )]}\times \sqrt{-\displaystyle \frac{2{h}_{0}\mathrm{csch}(\sqrt{{h}_{0}}\xi )}{\sqrt{-{\rm{\Delta }}}-{h}_{1}{\rm{sech}} (\sqrt{{h}_{0}}\xi )}},\end{eqnarray}$
the combined chirped optical soliton gives $\begin{eqnarray}\begin{array}{rcl}\delta {\omega }_{36} & = & \displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ & & \times \left[\displaystyle \frac{2{h}_{0}\mathrm{csch}(\sqrt{{h}_{0}}\xi )}{\sqrt{-{\rm{\Delta }}}-{h}_{1}{\rm{sech}} (\sqrt{{h}_{0}}\xi )}\right].\end{array}\end{eqnarray}$
Case 3: h0 > 0, Δ1 < 0, where ${{\rm{\Delta }}}_{1}={h}_{1}^{2}-4{h}_{0}{h}_{2}-4{h}_{0}^{2}$,
$\begin{eqnarray}\begin{array}{l}{q}_{37}(\xi )={e}^{{\rm{i}}[f(\xi )]}\\ \times \,\sqrt{-\displaystyle \frac{2{h}_{0}{\rm{sech}} (\sqrt{{h}_{0}}\xi )}{{h}_{1}{\rm{sech}} (\sqrt{{h}_{0}}\xi )+\sqrt{-{{\rm{\Delta }}}_{1}}\tanh (\sqrt{{h}_{0}}\xi )-2{h}_{0}}},\end{array}\end{eqnarray}$
the corresponding chirped gives $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{37}=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \ \ \times \,\left[\displaystyle \frac{2{h}_{0}{\rm{sech}} (\sqrt{{{\rm{h}}}_{0}}\xi )}{{h}_{1}{\rm{sech}} (\sqrt{{{\rm{h}}}_{0}}\xi )+\sqrt{-{{\rm{\Delta }}}_{1}}\tanh (\sqrt{{h}_{0}}\xi )-2{h}_{0}}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{q}_{38}(\xi )={e}^{{\rm{i}}[f(\xi )]}\\ \ \ \times \ \sqrt{-\displaystyle \frac{2{h}_{0}\mathrm{csch}(\sqrt{{h}_{0}}\xi )}{{h}_{1}\mathrm{csch}(\sqrt{{h}_{1}}\xi )-\sqrt{{{\rm{\Delta }}}_{1}}\coth (\sqrt{{{\rm{h}}}_{0}}\xi )+2{{\rm{h}}}_{0}}},\end{array}\end{eqnarray}$
and the chirped gives $\begin{eqnarray}\begin{array}{l}\delta {\omega }_{38}(\xi )=\displaystyle \frac{1}{{\beta }_{2}{v}^{2}}-\displaystyle \frac{1}{2{\beta }_{2}v}(3{\alpha }_{1}+2{\alpha }_{2})\\ \ \ \times \ \left[\displaystyle \frac{2{h}_{0}\mathrm{csch}(\sqrt{{h}_{0}}\xi )}{{h}_{1}\mathrm{csch}(\sqrt{{h}_{1}}\xi )-\sqrt{{{\rm{\Delta }}}_{1}}\coth (\sqrt{{{\rm{h}}}_{0}}\xi )+2{h}_{0}}\right].\end{array}\end{eqnarray}$
5.1. Graphical representation
Figures 1–2 plot the spatiotemporal evolution of the dipole bright optical solitons and the corresponding chirped of $| {q}_{35}(x,t){| }^{2}$. Moreover, by using the extended algebraic method, for certain values of the parameters β2, α1 & α2, the dipole solitary waves are obtained (see figures 1, 2, 4). However, figure 3 depicts the bright optical soliton evolution, which corresponds to the spatiotemporal shift. Figure 5 depicts the dipole optical solitons which corresponds to the spatiotemporal shift. Three dimensional plots of these chirped optical solitons may be observed in figures 1–5. The results obtained are very important in optical fibre context. Therefore, the dark optical soliton can help to carry out the communication transmission shipped to where the renewal can be collected by the dark–bright soliton conversion.
Figure 1. Spatiotemporal plots of the combined bright optical solitons $| {q}_{35}(x,t){| }^{2}$ for β2 = 1, α1 = −0.074, α2 = 0.0084, at (a) v = 25.255 and (b) v = 15.25 respectively. |
Figure 2. Spatiotemporal plots of the chirped bright soliton $| \delta {\omega }_{35}{| }^{2}$ for β2 = 1, α1 = −0.74, α2 = 0.25, γ1 = −0.025 at (a) v = 25.25, (b) v = 15.25, (c) v = 10.25, and (d) v = 10.25, respectively. |
Figure 3. Spatiotemporal plots of the combined bright optical solitons $| {q}_{36}(x,t){| }^{2}$ for β2 = 1, α1 = −0.74, α2 = 0.25, γ1 = −0.025 and (a) t = 0, (b) t = 5, (c) t = 10, (d) t = 15, (e) t = 20 at v = 5.25. |
Figure 4. Spatiotemporal plots of dipole bright optical solitons $| {q}_{37}(x,t){| }^{2}$ for β2 = 1, α1 = −0.0084, α2 = 0.0084, γ1 = −0.025 at (a) v = 10.25, (b) v = 15.25, (c) v = 20.25 and (d) v = 25.25, respectively. |
Figure 5. Spatiotemporal plots (2D) of dipole optical solitons $| {\delta }_{37}{| }^{2}$ for β2 = 1, α1 = −0.0084, α2 = 0.0084, γ1 = −0.025 at (a) v = 5.25, (b) v = 15.25, (c) v = 25.25 and (d) v = 30.25, respectively. |
6. Conclusion
In this present work, through the governing equation, we describe a propagation of sub-picosecond pulses with cubic nonlinearity and SPM in optical fibers. We investigate the behavior of the new optical soliton solutions. To target the main objective of this paper, the traveling-wave transformation was adopted to obtain the chirped component and the resulting ordinary differential equation. To solve the obtained ODE, the new extended direct algebraic scheme and the extended algebraic equation have been used. In our results, chirped dark–bright soliton solutions, trigonometric function solutions, new complex traveling-wave solutions, some rational function solutions and optical dipole solitons also emerge. Also, cases (8), (9) and (10) of the new extended direct algebraic method did not satisfy the conditions set on nonlinear ordinary differential equation (NODE) parameters, therefore, no solution was obtained. We think that the obtained one-soliton solutions should be useful in the field of solitary-wave theory. The reader should look at [21, 22] for the nonlocal Schrödinger equation.