Exact bright and dark solitary wave solutions of the generalized higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber

During the last few decades, nonlinear partial differential equations (NPDEs) have seriously draw the attention of the world scientific research. These NPDEs permit to model phenomena in several fields of sciences, notably in physics, such as fluid mechanics, electronics, plasma physics, condensate mater and optical fiber communication [1–7].

The numerical or analytical studies of NPDEs are one of the very important and essential tasks in nonlinear sciences, but the construction of exact solutions plays a very important role, because they can provide much information for a best comprehension of a physical problem. That is why some mathematical methods have been developed to solve NPDEs. We can quote the tanh method, the extended tanh method, the Jacobi's elliptic function algorithm, the Riccati's equation and G'/G method [8–13]. The use of these different methods leads to the achievement of travelling wave solutions, periodic solutions and solitary wave solutions. Solitary waves are particular type of solutions of NPDEs with special property. They propagate without changing their shape.

The importance of soliton theory in nonlinear science is no more to prove. In area of optical fiber, soliton can be observed, due to the equilibrium between the dispersion and nonlinear effects in the media [14, 15]. In communication sciences, solitary wave solutions act as information carrier bits through optical fibers [16, 17]. The long distance wired communication across transoceanic and transcontinental distances is now done using optical solitons, and the quantity of information and their transmission speed are unbeatable [18].

The propagation of the light in a media is suspected to the dispersion effects and nonlinear effects, which can be of higher-order or not [19, 20]. For nonlinear effects, we can note here: Kerr effect or self phase modulation, stimulated brillouin scattering, self-steepening and stimulated Raman scattering [20, 21].

Our aim in this paper is to construct exact bright and dark solitary wave solutions of a higher-order nonlinear Schrodinger equation (HONLSE), which is the ordinary cubic nonlinear equation accompanied by effects of third order dispersion and nonlinearity such as self-steepening and stimulated Raman scattering, via the mathematical method calls Bogning-Djeumen Tchaho-Kofané method (BDKm) [22–29].

This paper is organized as follow: in section 2, we define the governing equation; in section 3 we present the BDKm; in section 4, we investigate on the HONLSE and discuss about the obtained results, and section 5 is the conclusion of our work.

The nonlinear and partial differential equation modeling the propagation of electromagnetic wave with higher order nonlinear effect such as self-steepening and stimulated Raman scattering and high order dispersion is given by the following equation [21]

$$\begin{eqnarray}&&{E}_{z}-i{\alpha }_{{\rm{1}}}{E}_{tt}-i{\alpha }_{{\rm{2}}}{| E| }^{2}E-{\alpha }_{{\rm{3}}}{E}_{ttt}-{\alpha }_{{\rm{4}}}{({| E| }^{2}E)}_{t}-{\alpha }_{{\rm{5}}}{({| E| }^{2})}_{t}E=0,\end{eqnarray} \tag{ 1 }$$

were α₁ and α₃ are the coefficients of dispersion, α₂, α₄ and α₅ the nonlinear coefficients.

The generalized equation of equation (1) can be written as

$$\begin{eqnarray}&&{{\rm{n}}}_{{\rm{0}}}{E}_{z}+i{{\rm{n}}}_{{\rm{1}}}{E}_{tt}+i{{\rm{n}}}_{{\rm{2}}}{| E| }^{2}E+{{\rm{n}}}_{{\rm{3}}}{E}_{ttt}+{{\rm{n}}}_{{\rm{4}}}{({| E| }^{2}E)}_{t}+{{\rm{n}}}_{{\rm{5}}}{({| E| }^{2})}_{t}E=0,\end{eqnarray} \tag{ 2 }$$

were E is the slow varying envelope of the electric field, n₀ the coefficients related to the variation of the electric field which respect to z, n₁ and n₃ the group velocity dispersion coefficient and the third order dispersion coefficient, n₂, n₄ and n₅ respectively the coefficient related to the self-phase modulation, the self-steepening and stimulated Raman scattering.

To simplify the complexity of the study, we set

$$\begin{eqnarray}&&E(z,t)=\bar{E}(z,t){e}^{-i(kz-\omega t)}=\bar{E}(z,t){e}^{-i\phi },\end{eqnarray} \tag{ 3 }$$

Where $\overline{E}$ is a real or complex function envelope, and ϕ linear phase shift.

Inserting equation (3) into equation (2) and removing the exponential term yields to the new equation

$$\begin{eqnarray}&&-i(k{{\rm{n}}}_{{\rm{0}}}+{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}+{{\rm{n}}}_{{\rm{3}}}{\omega }^{3})\bar{E}+i({{\rm{n}}}_{{\rm{1}}}+3{{\rm{n}}}_{{\rm{3}}}\omega ){\bar{E}}_{tt}+i({{\rm{n}}}_{{\rm{2}}}+{{\rm{n}}}_{{\rm{4}}}\omega ){\bar{E}}^{3}\\ &&\,+\,(-2{{\rm{n}}}_{{\rm{1}}}\omega -3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2}){\bar{E}}_{{\rm{t}}}+{{\rm{n}}}_{{\rm{0}}}{\bar{E}}_{{\rm{z}}}+{{\rm{n}}}_{{\rm{3}}}{\bar{E}}_{ttt}+({{\rm{3n}}}_{{\rm{4}}}+{{\rm{2n}}}_{{\rm{5}}}){| \bar{E}| }^{2}{\bar{E}}_{t}=0.\end{eqnarray} \tag{ 4 }$$

The BDKm is a method based on the construction of exact solutions of certain types of nonlinear equations of the form

$$\begin{eqnarray}&&P(W,{W}_{t},{W}_{z},{W}_{tt},{W}_{zz},\mathrm{...},\,{W}_{ttt,}{W}_{zzz},\mathrm{...},\,{| W| }^{2},{(W{| W| }^{2})}_{t},\mathrm{...},\,W{({| W| }^{2})}_{t},\mathrm{...})=0,\end{eqnarray} \tag{ 5 }$$

where W(z, t) is an unknown function to determine. P is a polynomial function of W(z, t) and its partial derivatives where the highest order derivatives and nonlinear terms are involved.

The main steps of the BDKm are as follow

Step 1: We try to find traveling wave solutions of equation (5) by taking

$$\begin{eqnarray}&&W(z,t)=W(\xi ),\xi =\alpha (z-\beta t),\end{eqnarray} \tag{ 6 }$$

where β and α real constant number. We then transform equation (5) into and ordinary differential equation (ODE)

$$\begin{eqnarray}&&{P}_{ode}(W,W^{\prime} ,W^{\prime\prime} ,\mathrm{...},\,{| W| }^{2}W^{\prime} ,W({| W| }^{2})^{\prime} \mathrm{...})=0,\end{eqnarray} \tag{ 7 }$$

where $W^{\prime} =\displaystyle \frac{\partial W}{\partial \xi }.$

Step 4: We assume that the solution of equation (7) can be expressed under the shape:

$$\begin{eqnarray}&&W(\xi )={a}_{1}{\sinh }^{m}(\xi )\text{sec}{h}^{n}(\xi )+\mathrm{...}\,+\,{a}_{l}{\sinh }^{q}(\xi )\text{sec}{h}^{r}(\xi ),\end{eqnarray} \tag{ 8 }$$

where m, n, q and r are numbers to determine.

Inserting equation (8) into equation (7), and taking into account the transformations related to the BDKm [27], we get the equation of the form indicated below

$$\begin{eqnarray}&&\displaystyle \sum _{i,j}F({a}_{i},\alpha ,\beta ,\omega ,k){\cosh }^{j}(\xi )+\displaystyle \sum _{i,j}H({a}_{i},\alpha ,\beta ,\omega ,k){\text{sech}}^{k}(\xi )\\ &&\,+\displaystyle \sum _{i,s}G({a}_{i},\alpha ,\beta ,\omega ,k)\sinh (\xi )\text{sec}{h}^{s}(\xi )\\ &&\,+\displaystyle \sum _{i,s}T({a}_{i},\alpha ,\beta ,\omega ,k)\sinh (\xi )\cos \,{h}^{s}(\xi )+\displaystyle \sum _{i}V({a}_{i},\alpha ,\beta ,\omega ,k)=0.\end{eqnarray} \tag{ 9 }$$

Step 5: If the functions F(a_i, α, β, ω, k), H(a_i, α, β, ω, k), G(a_i, α, β, ω, k), T(a_i, α, β, ω, k), V(a_i, α, β, ω, k) can be null for some values of the coefficients a_i, α, β, ω or k (nontrivial values), the ansatz given in equation (9) can be supported by equation (5) for any values of m, n, ..., q, r and can be considered as solution of equation.

Step 6: We look for the values of m, n, q and r which can make some terms of equation (9) merge, and set the coefficients of hyperbolic function equal to zero to determine the constants a_i, α, β, ω and k of equation (8).

Setting the change of variable

$$\begin{eqnarray}&&\bar{E}(z,t)=\bar{E}(\xi ),\xi =\alpha (t-\beta z),\end{eqnarray} \tag{ 10 }$$

Equation (2) becomes

$$\begin{eqnarray}&&-\alpha (\beta {{\rm{n}}}_{{\rm{0}}}+2{{\rm{n}}}_{{\rm{1}}}\omega +3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2}){\bar{E}}_{\xi }+{{\rm{n}}}_{{\rm{3}}}{\alpha }^{3}{\bar{E}}_{\xi \xi \xi }+\alpha ({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}}){({| \bar{E}| }^{2})}_{\xi }\bar{E}+\alpha {{\rm{n}}}_{{\rm{4}}}{| \bar{E}| }^{2}{\bar{E}}_{\xi }\\ &&\,+\,i[-(k{{\rm{n}}}_{{\rm{0}}}+{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}+{{\rm{n}}}_{{\rm{3}}}{\omega }^{3})\bar{E}+{\alpha }^{2}({{\rm{n}}}_{{\rm{1}}}+3{{\rm{n}}}_{{\rm{3}}}\omega ){\bar{E}}_{\xi \xi }+({{\rm{n}}}_{{\rm{2}}}+{{\rm{n}}}_{{\rm{4}}}\omega ){\bar{E}}^{2}E]=0.\end{eqnarray} \tag{ 11 }$$

We look for the solution of equation (11) in the form

$$\begin{eqnarray}&&\overline{E}(t)=a{\sinh }^{m}\alpha (t-\beta z)\text{sec}{h}^{n}\alpha (t-\beta z)+(b+ic){\sinh }^{q}\alpha (t-\beta z)\text{sec}{h}^{r}\\ &&\,\,\,\times \alpha (t-\beta z),\end{eqnarray} \tag{ 12 }$$

where a, b and c are real constants. α is the pulse width and β the shift of inverse group velocity.

In the following lines, we are going to precede our analysis in two steps.

Case 1: b ≠ 0 with c = 0. $\bar{E}$ becomes in this case a real physical size and permits to rewrite equation (11) in the form

$$\begin{eqnarray}&&[\alpha (\beta {{\rm{n}}}_{{\rm{0}}}-2{{\rm{n}}}_{{\rm{1}}}\omega -3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2}){\bar{E}}_{\xi }{+n}_{{\rm{3}}}{\alpha }^{3}{\bar{E}}_{\xi \xi \xi }+\alpha ({{\rm{3n}}}_{{\rm{4}}}{+\mathrm{2n}}_{{\rm{5}}}){\bar{E}}^{2}{\bar{E}}_{\xi }]\\ &&\,+\,i[-(k{{\rm{n}}}_{{\rm{0}}}+{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}+{{\rm{n}}}_{{\rm{3}}}{\omega }^{3})\bar{E}+{\alpha }^{2}({{\rm{n}}}_{{\rm{1}}}+3{{\rm{n}}}_{{\rm{3}}}\omega ){\bar{E}}_{\xi \xi }+({{\rm{n}}}_{{\rm{2}}}+{{\rm{n}}}_{{\rm{4}}}\omega ){\bar{E}}^{2}E]=0.\end{eqnarray} \tag{ 13 }$$

Equation (13) can also write

$$\begin{eqnarray}&&-\alpha {\kappa }_{1}{\bar{E}}_{\xi }+\alpha {\kappa }_{2}{\bar{E}}^{2}{\bar{E}}_{\xi }+{\alpha }^{3}{\kappa }_{3}{\bar{E}}_{\xi \xi \xi }-i{\kappa }_{4}\bar{E}+i{\kappa }_{5}{\bar{E}}^{3}+i{\alpha }^{2}{\kappa }_{6}{\bar{E}}_{\xi \xi }=0,\end{eqnarray} \tag{ 14 }$$

were,

$$\begin{eqnarray}&&{\kappa }_{1}=\beta {{\rm{n}}}_{{\rm{0}}}+2{{\rm{n}}}_{{\rm{1}}}\omega +3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2},{\kappa }_{2}={{\rm{3n}}}_{{\rm{4}}}{+\mathrm{2n}}_{{\rm{5}}},{\kappa }_{3}={{\rm{n}}}_{{\rm{3}}},\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&{\kappa }_{4}=k{{\rm{n}}}_{{\rm{0}}}+{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}+{{\rm{n}}}_{{\rm{3}}}{\omega }^{3},{\kappa }_{5}={{\rm{n}}}_{{\rm{2}}}+{{\rm{n}}}_{{\rm{4}}}\omega ,{\kappa }_{6}={{\rm{n}}}_{{\rm{1}}}+3{{\rm{n}}}_{{\rm{3}}}\omega .\end{eqnarray} \tag{ 16 }$$

Inserting equation (12) into equation (14), we obtain a main equation permitting to make all analyses,

$$\begin{eqnarray}&&\alpha {\kappa }_{{\rm{2}}}{a}^{{\rm{3}}}(m{\sinh }^{{\rm{3m}}-{\rm{1}}}\xi \text{sec}{h}^{3n-1}\xi -n{\sinh }^{\mathrm{3m}+1}\xi \text{sec}{h}^{3n+1}\xi )\\ &&\,+\,\alpha {\kappa }_{{\rm{2}}}q{b}^{{\rm{3}}}(q{\sinh }^{{\rm{3q}}-{\rm{1}}}\xi \text{sec}{h}^{3r-1}\xi -r{\sinh }^{\mathrm{3q}+1}\xi \text{sec}{h}^{3r+1}\xi )\\ &&\,+\,a{b}^{{\rm{2}}}\alpha {\kappa }_{{\rm{2}}}[(m+2q){\sinh }^{m+2q-{\rm{1}}}\xi \text{sec}{h}^{n+2r-1}\xi -(n+2r){\sinh }^{m+2q+{\rm{1}}}\xi \text{sec}{h}^{n+2r+1}\xi ]\\ &&\,+\,{a}^{{\rm{2}}}b\alpha {\kappa }_{{\rm{2}}}[(q+2m){\sinh }^{2m+q-{\rm{1}}}\xi \text{sec}{h}^{2n+r-1}\xi -(r+2n){\sinh }^{2m+q+{\rm{1}}}\xi \text{sec}{h}^{2n+r+1}\xi ]\\ &&\,-\,a\alpha {m}({\rm{2}}{\kappa }_{{\rm{3}}}{\alpha }^{{\rm{2}}}+3{\alpha }^{{\rm{2}}}{\rm{mn}}{\kappa }_{{\rm{3}}}-{\rm{3}}{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{3}}}m+{\kappa }_{{\rm{1}}})\sinh \,{\xi }^{m-1}\text{sec}h{\xi }^{n-1}\\ &&\,+\,a\alpha n({\rm{3}}{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{3}}}n+3{\alpha }^{{\rm{2}}}{\rm{mn}}{\kappa }_{{\rm{3}}}+2{\kappa }_{{\rm{3}}}{\alpha }^{{\rm{2}}}+{\kappa }_{{\rm{1}}})\sinh \,{\xi }^{m+1}\text{sec}h{\xi }^{n+1}\\ &&\,-\,\alpha bq({\rm{2}}{\kappa }_{{\rm{3}}}{\alpha }^{{\rm{2}}}+3{\alpha }^{{\rm{2}}}qr{\kappa }_{{\rm{3}}}-{\rm{3}}{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{3}}}q+{\kappa }_{{\rm{1}}})\sinh \,{\xi }^{q-1}\text{sec}h{\xi }^{r-1}\\ &&\,+\,\alpha r({\rm{2}}{\kappa }_{{\rm{3}}}{\alpha }^{{\rm{2}}}+3{\alpha }^{{\rm{2}}}qr{\kappa }_{{\rm{3}}}+3{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{3}}}r+{\kappa }_{{\rm{1}}})\sinh \,{\xi }^{q+1}\text{sec}h{\xi }^{r+1}\\ &&\,+\,{\alpha }^{3}{\kappa }_{{\rm{3}}}a[m(m-1)(m-2){\sinh }^{{\rm{m}}-{\rm{3}}}\xi \text{sec}{h}^{n-3}\xi -n(n-1)(n-2){\sinh }^{m+3}\xi \text{sec}{h}^{n+3}\xi ]\\ &&\,+\,{\alpha }^{3}{\kappa }_{{\rm{3}}}b[q(q-1)(q-2){\sinh }^{q-{\rm{3}}}\xi \text{sec}{h}^{r-3}\xi -r(r-1)(r-2){\sinh }^{q+{\rm{3}}}\xi \text{sec}{h}^{r+3}\xi ]\\ &&\,+\,i\{{\rm{3}}{\kappa }_{{\rm{5}}}{a}^{2}b{\sinh }^{2m+q}\xi \text{sec}{h}^{2n+r}\xi +{\rm{3}}{\kappa }_{{\rm{5}}}a{b}^{2}{\sinh }^{2q+m}\xi \text{sec}{h}^{2r+m}\xi \\ &&\,+\,{\kappa }_{{\rm{5}}}{a}^{3}{\sinh }^{3m}\xi \text{sec}{h}^{3n}\xi +{\kappa }_{{\rm{5}}}{b}^{3}{\sinh }^{3q}\xi \text{sec}{h}^{3r}\xi \\ &&\,-\,a({\kappa }_{{\rm{4}}}+{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{6}}}n+2{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{6}}}{\rm{mn}}-{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{6}}}m){\sinh }^{m}\xi \text{sec}{h}^{n}\xi \\ &&\,-\,b({\kappa }_{{\rm{4}}}+{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{6}}}r+2{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{6}}}qr-{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{6}}}q){\sinh }^{q}\xi \text{sec}{h}^{r}\xi \\ &&\,+\,{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{6}}}a[m(m-1){\sinh }^{m-2}\xi \text{sec}{h}^{n-2}\xi +n(n+1){\sinh }^{m+2}\xi \text{sec}{h}^{n+2}\xi ]\\ &&\,+\,{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{6}}}b[q(q-1){\sinh }^{q-2}\xi \text{sec}{h}^{r-2}\xi +r(r+1){\sinh }^{q+2}\xi \text{sec}{h}^{r+2}\xi ]\}.\end{eqnarray} \tag{ 17 }$$

Let analyze the main equation (17) in order to obtain exact solutions of HONLSE as above, for different values of m, n, q, and r.

When m = n = 0 and q = r, some terms of equation (17) merge for q = r = {0, ±1, ±2, ±3, ±4}.

• For m = n = q = r = 0, equation (17) becomes
  $$\begin{eqnarray}&&(a+b)[{\kappa }_{{\rm{5}}}{(a+b)}^{{\rm{2}}}-{\kappa }_{{\rm{4}}}]=0,\end{eqnarray} \tag{ 18 }$$
  and the constant solution of equation (2) is
  $$\begin{eqnarray}&&E(x,t)=\pm \sqrt{\displaystyle \frac{{\kappa }_{{\rm{4}}}}{{\kappa }_{{\rm{5}}}}\,=}\,\pm \sqrt{\displaystyle \frac{{{\rm{kn}}}_{{\rm{0}}}+{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}+{{\rm{n}}}_{{\rm{3}}}{\omega }^{3}}{{{\rm{n}}}_{{\rm{2}}}+{{\rm{n}}}_{{\rm{4}}}\omega }}=cst.\end{eqnarray} \tag{ 19 }$$
• For m = n = 0 and q = r = ±1 equation (17) becomes
  $$\begin{eqnarray}&&\alpha b({\rm{4}}{\alpha }^{{\rm{2}}}{\kappa }_{3}+{\kappa }_{2}{b}^{2}+{\kappa }_{2}{a}^{2}-{\kappa }_{1})\text{sec}{h}^{2}\xi -\alpha b({\kappa }_{2}{b}^{2}+6{\alpha }^{{\rm{2}}}{\kappa }_{3})\text{sec}{h}^{4}\xi \\ &&+\,2\alpha {\kappa }_{2}a{b}^{2}\,\sinh \,\xi \text{sec}{h}^{3}\xi i\{a({\kappa }_{5}{a}^{2}+3{\kappa }_{5}{b}^{2}-{\kappa }_{4})-{\rm{3}}{\kappa }_{5}a{b}^{2}\text{sec}{h}^{2}\xi \\ &&+\,b(-{\kappa }_{4}+3{\kappa }_{5}{a}^{2}+{\kappa }_{5}{b}^{2}-{\kappa }_{4})\sinh \,\xi \text{sec}h\xi -b({\kappa }_{5}{b}^{2}+2{\alpha }^{{\rm{2}}}{\kappa }_{6})\sinh \,\xi \text{sec}{h}^{3}\xi \}=0.\end{eqnarray} \tag{ 20 }$$
  By setting to zero the coefficients of hyperbolic functions of the real and the imaginary we obtain the algebraic system
  $$\begin{eqnarray}&&{\rm{4}}{\alpha }^{{\rm{2}}}{\kappa }_{3}+{\kappa }_{2}{b}^{2}+{\kappa }_{2}{a}^{2}-{\kappa }_{1}=0,\,{\kappa }_{2}{b}^{2}+6{\alpha }^{{\rm{2}}}{\kappa }_{3}=0,\end{eqnarray} \tag{ 21 }$$
  $$\begin{eqnarray}&&{\rm{2}}\alpha {\kappa }_{2}a{b}^{2}=0,a({\kappa }_{5}{a}^{2}+3{\kappa }_{5}{b}^{2}-{\kappa }_{4})=0,\,{\kappa }_{5}a{b}^{2}=0,\end{eqnarray} \tag{ 22 }$$
  $$\begin{eqnarray}&&b(-{\kappa }_{4}+3{\kappa }_{5}{a}^{2}+{\kappa }_{5}{b}^{2}-{\kappa }_{4})=0,\,b({\kappa }_{5}{b}^{2}+2{\alpha }^{{\rm{2}}}{\kappa }_{6})=0.\end{eqnarray} \tag{ 23 }$$
  The resolution of this algebraic system taking into account equations (15) and (16) permits to obtain the values of the parameters
  $$\begin{eqnarray}&&a=0,\,b=\pm \sqrt{\displaystyle \frac{-6{{\rm{n}}}_{{\rm{3}}}}{{{\rm{3n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\alpha ,\,\omega =\displaystyle \frac{{{\rm{3n}}}_{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-{{\rm{n}}}_{{\rm{1}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})}{{{\rm{6n}}}_{{\rm{3}}}({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})},\end{eqnarray} \tag{ 24 }$$
  $$\begin{eqnarray}&&k=(-2{\alpha }^{2}{{\rm{n}}}_{{\rm{1}}}-6{\alpha }^{2}{{\rm{n}}}_{{\rm{3}}}\omega -{\omega }^{2}{{\rm{n}}}_{{\rm{1}}}-{{\rm{n}}}_{{\rm{3}}}{\omega }^{3})/{{\rm{n}}}_{{\rm{0}}},\,\beta =(-{\rm{2}}{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-2{{\rm{n}}}_{{\rm{1}}}\omega -3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2})/{{\rm{n}}}_{{\rm{0}}}.\end{eqnarray} \tag{ 25 }$$
  The exact dark soliton solitary wave solution and the travelling wave solution of the HONLSE when n₃(3n₄ + 2n₅) < 0 equation read
  $$\begin{eqnarray}&&E(z,t)=\pm \sqrt{\displaystyle \frac{-6{{\rm{n}}}_{{\rm{3}}}}{{{\rm{3n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\alpha \,\tanh \,\alpha \left(t-\displaystyle \frac{-{\rm{2}}{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-2{{\rm{n}}}_{{\rm{1}}}\omega -3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2}}{{{\rm{n}}}_{{\rm{0}}}}z\right)\\ &&\,\,\,\,\times \,{\exp }^{-i\left(\begin{array}{l}\displaystyle \frac{-2{\alpha }^{2}{{\rm{n}}}_{{\rm{1}}}-6{\alpha }^{2}{{\rm{n}}}_{{\rm{3}}}\omega -{\omega }^{2}{{\rm{n}}}_{{\rm{1}}}-{{\rm{n}}}_{{\rm{3}}}{\omega }^{3}}{{{\rm{n}}}_{{\rm{0}}}}z\\ -\displaystyle \frac{{{\rm{3n}}}_{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-{{\rm{n}}}_{{\rm{1}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})}{{{\rm{6n}}}_{{\rm{3}}}({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})}t\end{array}\right)},\end{eqnarray} \tag{ 26 }$$
  and
  $$\begin{eqnarray}&&E(z,t)=\pm \sqrt{\displaystyle \frac{-6{{\rm{n}}}_{{\rm{3}}}}{{{\rm{3n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\alpha \,\cot \,anh\alpha \left(t-\displaystyle \frac{-{\rm{2}}{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-2{{\rm{n}}}_{{\rm{1}}}\omega -3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2}}{{{\rm{n}}}_{{\rm{0}}}}z\right)\\ &&\,\,\,\,\times {\exp }^{-i\left(\begin{array}{l}\displaystyle \frac{-2{\alpha }^{2}{{\rm{n}}}_{{\rm{1}}}-6{\alpha }^{2}{{\rm{n}}}_{{\rm{3}}}\omega -{\omega }^{2}{{\rm{n}}}_{{\rm{1}}}-{{\rm{n}}}_{{\rm{3}}}{\omega }^{3}}{{{\rm{n}}}_{{\rm{0}}}}z\\ -\displaystyle \frac{{{\rm{3n}}}_{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-{{\rm{n}}}_{{\rm{1}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})}{{{\rm{6n}}}_{{\rm{3}}}({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})}t\end{array}\right)}.\end{eqnarray} \tag{ 27 }$$
  No solution is found for m = n = q = r and ∣r∣ ≥ 2.
• For m = 0, n = 0, q = 0 and r = 1, equation (17) becomes
  $$\begin{eqnarray}&&\alpha b({\kappa }_{{\rm{1}}}-{\kappa }_{{\rm{2}}}{a}^{2}-{\alpha }^{2}{\kappa }_{{\rm{3}}})\sinh \,\xi {{\rm{sech}} }^{{\rm{2}}}\xi -{\rm{2}}\alpha {\kappa }_{{\rm{2}}}a{b}^{2}\,\sinh \,\xi {{\rm{sech}} }^{{\rm{3}}}\xi \\ &&\,+\,\alpha b(-{\kappa }_{{\rm{2}}}{b}^{2}+6{\alpha }^{2}{\kappa }_{{\rm{3}}})\sinh \,\xi {{\rm{sech}} }^{{\rm{4}}}\xi \\ &&\,+\,i[a(-{\kappa }_{{\rm{4}}}+{\kappa }_{{\rm{5}}}{a}^{2})+(-{\kappa }_{{\rm{4}}}b+3{\kappa }_{{\rm{5}}}{a}^{2}b+{\alpha }^{2}{\kappa }_{{\rm{6}}}b){\rm{sech}} \,\xi +3{\kappa }_{{\rm{5}}}a{b}^{2}{{\rm{sech}} }^{{\rm{2}}}\xi \\ &&\,+\,({\kappa }_{{\rm{5}}}{b}^{3}-{\rm{2}}{\alpha }^{2}{\kappa }_{{\rm{6}}}b){{\rm{sech}} }^{{\rm{3}}}\xi ]=0.\end{eqnarray} \tag{ 28 }$$
  The resolution of the algebraic system resulting from this equation gives the following results:
  $$\begin{eqnarray}&&a=0,\,b=\pm \sqrt{\displaystyle \frac{{{\rm{6n}}}_{{\rm{3}}}}{{{\rm{3n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\alpha ,\end{eqnarray} \tag{ 29 }$$
  $$\begin{eqnarray}&&\omega =\displaystyle \frac{{{\rm{3n}}}_{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-3{{\rm{n}}}_{{\rm{1}}}{{\rm{n}}}_{{\rm{4}}}-{{\rm{2n}}}_{{\rm{1}}}{{\rm{n}}}_{{\rm{5}}}}{{{\rm{6n}}}_{{\rm{3}}}({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})},k=({\alpha }^{2}{{\rm{n}}}_{{\rm{1}}}+3{\alpha }^{2}{{\rm{n}}}_{{\rm{3}}}\omega -{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}-{{\rm{n}}}_{{\rm{3}}}{\omega }^{3})/{{\rm{n}}}_{{\rm{0}}},\end{eqnarray} \tag{ 30 }$$
  and
  $$\begin{eqnarray}&&\beta =\displaystyle \frac{{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-2{{\rm{n}}}_{{\rm{1}}}\omega -3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2}}{{{\rm{n}}}_{{\rm{0}}}}\\ &&\,=\displaystyle \frac{12{\alpha }^{{\rm{2}}}{{{\rm{n}}}_{{\rm{3}}}}^{2}{({{\rm{n}}}_{{\rm{4}}}{+n}_{{\rm{5}}})}^{2}-{{\rm{4n}}}_{{\rm{1}}}({{\rm{n}}}_{{\rm{4}}}{+n}_{{\rm{5}}})(3{{\rm{n}}}_{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-3{{\rm{n}}}_{{\rm{1}}}{{\rm{n}}}_{{\rm{4}}}-2{{\rm{n}}}_{{\rm{1}}}{{\rm{n}}}_{{\rm{4}}})-{(3{{\rm{n}}}_{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-3{{\rm{n}}}_{{\rm{1}}}{{\rm{n}}}_{{\rm{4}}}-2{{\rm{n}}}_{{\rm{1}}}{{\rm{n}}}_{{\rm{5}}})}^{2}}{12{{\rm{n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}{({{\rm{n}}}_{{\rm{4}}}{+n}_{{\rm{5}}})}^{2}}.\end{eqnarray} \tag{ 31 }$$
  The bright solitary wave solution obtained here is written
  $$\begin{eqnarray}&&E(z,t)=\pm \sqrt{\displaystyle \frac{{{\rm{6n}}}_{{\rm{3}}}}{{{\rm{3n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\alpha \text{sech}\alpha \left(t-\displaystyle \frac{{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-2{{\rm{n}}}_{{\rm{1}}}\omega -3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2}}{{{\rm{n}}}_{{\rm{0}}}}z\right)\\ &&\,\,\,\,\times {e}^{-i\left(\displaystyle \frac{{\alpha }^{2}{{\rm{n}}}_{{\rm{1}}}+3{\alpha }^{2}{{\rm{n}}}_{{\rm{3}}}\omega -{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}-{{\rm{n}}}_{{\rm{3}}}{\omega }^{3}}{{{\rm{n}}}_{{\rm{0}}}}z-\displaystyle \frac{{{\rm{3n}}}_{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}-3{{\rm{n}}}_{{\rm{1}}}{{\rm{n}}}_{{\rm{4}}}-{{\rm{2n}}}_{{\rm{1}}}{{\rm{n}}}_{{\rm{5}}}}{{{\rm{6n}}}_{{\rm{3}}}({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})}t\right)},\end{eqnarray} \tag{ 32 }$$
  with n₃(3n₄ + 2n₅) > 0.
• No solution is found for m = 0, n = 1, q = 1 and r = 1.From equations (26) and (32) when 3n₄ + 2n₅ > 0 (3n₄ + 2n₅ < 0), dark and bright solitary wave can propagate in optical fiber for n₃ < 0 (n₃ > 0) and n₃ > 0 (n₃ < 0) respectively. Thus, the parameter of dispersion of third order n₃, can be the selector of solitary wave –type propagating in the fiber, when n₄ and n₅ are fixed.Case 2: b = 0 with c ≠ 0. Setting
  $$\begin{eqnarray}&&A=a{\sinh }^{m}(\xi )\text{sec}{h}^{n}(\xi ),B=c{\sinh }^{q}(\xi )\text{sec}{h}^{r}(\xi ),\end{eqnarray} \tag{ 33 }$$
  and rewrite equation (12) as $\bar{E}(z,t)=A+iB,$ equation (11) becomes
  $$\begin{eqnarray}&&{\sigma }_{{\rm{2}}}{\rm{B}}-\alpha {\sigma }_{{\rm{1}}}{{\rm{A}}}_{\xi }-{\alpha }^{2}{\sigma }_{{\rm{3}}}{{\rm{B}}}_{\xi \xi }+{\alpha }^{3}{{\rm{n}}}_{{\rm{3}}}{{\rm{A}}}_{\xi \xi \xi }+\alpha {\sigma }_{{\rm{5}}}{{\rm{A}}}^{{\rm{2}}}{{\rm{A}}}_{\xi }+2\alpha {\sigma }_{{\rm{6}}}{{\rm{ABB}}}_{\xi }\\ &&\,+\,\alpha {{\rm{n}}}_{{\rm{4}}}{{\rm{B}}}^{{\rm{2}}}{{\rm{A}}}_{\xi }-{\sigma }_{{\rm{4}}}({{\rm{B}}}^{{\rm{3}}}{+A}^{{\rm{2}}}{\rm{B}})i[-{\sigma }_{{\rm{2}}}{\rm{A}}-\alpha {\sigma }_{{\rm{1}}}{{\rm{B}}}_{\xi }+{\alpha }^{2}{\sigma }_{{\rm{3}}}{{\rm{A}}}_{\xi \xi }\\ &&\,+\,{\alpha }^{3}{{\rm{n}}}_{{\rm{3}}}{{\rm{B}}}_{\xi \xi \xi }+\alpha {\sigma }_{{\rm{5}}}{{\rm{B}}}^{{\rm{2}}}{{\rm{B}}}_{\xi }+2\alpha {\sigma }_{{\rm{6}}}{{\rm{ABA}}}_{\xi }+\alpha {{\rm{n}}}_{{\rm{4}}}{{\rm{A}}}^{{\rm{2}}}{{\rm{B}}}_{\xi }+{\sigma }_{{\rm{4}}}({{\rm{A}}}^{{\rm{3}}}{+B}^{{\rm{2}}}{\rm{A}})]\\ &&\,=\,0,\end{eqnarray} \tag{ 34 }$$
  where
  $$\begin{eqnarray}&&{\sigma }_{1}=\beta {{\rm{n}}}_{{\rm{0}}}+2{{\rm{n}}}_{{\rm{1}}}\omega +3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2},\,{\sigma }_{2}=k{{\rm{n}}}_{{\rm{0}}}+{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}+{{\rm{n}}}_{{\rm{3}}}{\omega }^{3},\,{\sigma }_{3}={{\rm{n}}}_{{\rm{1}}}+3{{\rm{n}}}_{{\rm{3}}}\omega ,\end{eqnarray} \tag{ 35 }$$
  and
  $$\begin{eqnarray}&&{\sigma }_{4}={{\rm{n}}}_{{\rm{2}}}+{{\rm{n}}}_{{\rm{4}}}\omega ,{\sigma }_{5}=3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}},{\sigma }_{6}={{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}}.\end{eqnarray} \tag{ 36 }$$
  Substituting equation (33) into (34), and separating the imaginary and real parts leads to the following new main equations
  $$\begin{eqnarray}&&b({\sigma }_{{\rm{2}}}+{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}r-{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}q+2{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}qr){\sinh }^{q}\xi \text{sec}{h}^{r}\xi -{\sigma }_{{\rm{4}}}{b}^{3}{\sinh }^{3q}\xi \text{sec}{h}^{3r}\xi \\ &&\,-{\sigma }_{{\rm{4}}}b{a}^{2}{\sinh }^{2m+q}\xi \text{sec}{h}^{2n+r}\xi +{\alpha }^{{\rm{3}}}{{\rm{n}}}_{{\rm{3}}}{\rm{am}}(m-1)(m-2){\sinh }^{m-3}\xi \text{sec}{h}^{n-3}\xi \\ &&\,-\alpha {\rm{am}}({\rm{3}}{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}{\rm{mn}}-{\rm{3}}{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}m+2{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}+{\sigma }_{{\rm{1}}}){\sinh }^{m-1}\xi \text{sec}{h}^{n-1}\xi \\ &&\,\,+\alpha {\sigma }_{{\rm{5}}}{a}^{3}m{\sinh }^{3m-1}\xi \text{sec}{h}^{3n-1}\xi \\ &&\,+\alpha a{b}^{2}({{\rm{n}}}_{{\rm{4}}}m+2{\sigma }_{{\rm{6}}}q){\sinh }^{m+2q-1}\xi \text{sec}{h}^{n+2r-1}\xi -\alpha {\sigma }_{{\rm{5}}}{a}^{3}n{\sinh }^{3m+1}\xi \text{sec}{h}^{3n+1}\xi \\ &&\,-a{b}^{{\rm{2}}}\alpha ({{\rm{n}}}_{{\rm{4}}}n+2{\sigma }_{{\rm{6}}}r){\sinh }^{m+2q+1}\xi \text{sec}{h}^{n+2r+1}\xi \\ &&\,+\alpha an({\sigma }_{{\rm{1}}}+3{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}\mathrm{mn}+2{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}+3{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}n){\sinh }^{m+1}\xi \text{sec}{h}^{n+1}\xi \\ &&\,\,-{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}bq(q-{\rm{1}}){\sinh }^{q-2}\xi \text{sec}{h}^{r-2}\xi \\ &&\,-{\alpha }^{{\rm{3}}}{{\rm{n}}}_{{\rm{3}}}an(n+1)(n+2){\sinh }^{m+3}\xi \text{sec}{h}^{n+3}\xi -{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}br(r+1){\sinh }^{q+2}\xi \text{sec}{h}^{r+2}\xi =0,\end{eqnarray} \tag{ 37 }$$
  and
  $$\begin{eqnarray}&&{\sigma }_{{\rm{4}}}a{b}^{2}{\sinh }^{m+2q}\xi \text{sec}{h}^{n+2r}\xi -a({\sigma }_{{\rm{2}}}+{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}n-{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}m+2{\alpha }^{{\rm{2}}}{\kappa }_{{\rm{3}}}{\rm{mn}}){\sinh }^{m}\xi \text{sec}{h}^{n}\xi \\ &&\,+{\sigma }_{{\rm{4}}}{a}^{3}{\sinh }^{3m}\xi \text{sec}{h}^{3n}\xi +{\alpha }^{{\rm{3}}}{{\rm{n}}}_{{\rm{3}}}bq(q-1)(q-2){\sinh }^{q-3}\xi \text{sec}{h}^{r-3}\xi \\ &&\,-\alpha bq({\rm{3}}{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}qr-{\rm{3}}{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}q+2{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}+{\sigma }_{{\rm{2}}}){\sinh }^{q-1}\xi \text{sec}{h}^{r-1}\xi +\alpha {\sigma }_{{\rm{5}}}{b}^{3}q{\sinh }^{3q-1}\xi \text{sec}{h}^{3r-1}\xi \\ &&\,+\alpha b{a}^{2}({{\rm{n}}}_{{\rm{4}}}q+2{\sigma }_{{\rm{6}}}m){\sinh }^{q+2m-1}\xi \text{sec}{h}^{r+2n-1}\xi -\alpha {\sigma }_{{\rm{5}}}{b}^{3}r{\sinh }^{3q+1}\xi \text{sec}{h}^{3r+1}\xi \\ &&\,+\alpha br({\rm{3}}{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}r+2{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}+3{\alpha }^{{\rm{2}}}{{\rm{n}}}_{{\rm{3}}}qr+{\sigma }_{{\rm{1}}}){\sinh }^{q+1}\xi \text{sec}{h}^{r+1}\xi \\ &&\,-\alpha {a}^{2}b({{\rm{n}}}_{{\rm{4}}}r+2{\sigma }_{{\rm{6}}}n){\sinh }^{2m+q+1}\xi \text{sec}{h}^{2n+r+1}\xi +{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}{\rm{am}}(m-1){\sinh }^{m+2}\xi \text{sec}{h}^{n+2}\xi \\ &&\,-{\alpha }^{{\rm{3}}}{{\rm{n}}}_{{\rm{3}}}br(r+1)(r+2){\sinh }^{q+3}\xi \text{sec}{h}^{r+3}\xi +{\alpha }^{{\rm{2}}}{\sigma }_{{\rm{3}}}an(n+1){\sinh }^{m+2}\xi \text{sec}{h}^{n+2}\xi =0.\end{eqnarray} \tag{ 38 }$$
• For m = 0, n = 0 and q = r = ±1 we obtain

$$\begin{eqnarray}&&a=\displaystyle \frac{\alpha \gamma }{b},\,b=\pm \alpha \sqrt{\displaystyle \frac{-6{{\rm{n}}}_{{\rm{3}}}}{3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\,{\rm{with}}\,\gamma =\left[\displaystyle \frac{3({{\rm{n}}}_{{\rm{2}}}-2{{\rm{n}}}_{{\rm{4}}}\omega -2{{\rm{n}}}_{{\rm{5}}}\omega )-{{\rm{n}}}_{{\rm{1}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})}{(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})}\right],\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&\beta =\displaystyle \frac{\left\{\begin{array}{l}6{{\rm{n}}}_{{\rm{3}}}({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})(2{{\rm{n}}}_{{\rm{3}}}{\alpha }^{2}+2{{\rm{n}}}_{{\rm{1}}}\omega +3{{\rm{n}}}_{{\rm{3}}}{\omega }^{2})+{{\rm{n}}}_{{\rm{2}}}({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}}){(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})}^{2}{\gamma }^{2}\\ -6{{\rm{n}}}_{{\rm{3}}}({{\rm{n}}}_{{\rm{2}}}+{{\rm{n}}}_{{\rm{4}}}\omega )[{{\rm{n}}}_{{\rm{1}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})+3{{\rm{n}}}_{{\rm{3}}}(-{{\rm{n}}}_{{\rm{2}}}+2{{\rm{n}}}_{{\rm{4}}}\omega +2{{\rm{n}}}_{{\rm{5}}}\omega )]\end{array}\right\}}{-6{{\rm{n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})},\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&k=\displaystyle \frac{({a}^{2}+{b}^{2})({{\rm{n}}}_{{\rm{2}}}{+n}_{{\rm{4}}}\omega )-{{\rm{n}}}_{{\rm{1}}}{\omega }^{2}-{{\rm{n}}}_{{\rm{3}}}{\omega }^{3}}{{{\rm{n}}}_{{\rm{0}}}}\\ &&\,=\,\displaystyle \frac{6{{\rm{n}}}_{{\rm{3}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})({{\rm{n}}}_{{\rm{1}}}{\omega }^{2}+{{\rm{n}}}_{{\rm{3}}}{\omega }^{3})+({{\rm{n}}}_{{\rm{2}}}+{{\rm{n}}}_{{\rm{4}}}\omega )[{(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})}^{2}{\gamma }^{2}+36{\alpha }^{2}{{{\rm{n}}}_{{\rm{3}}}}^{2}]}{-6{{\rm{n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})}.\end{eqnarray} \tag{ 41 }$$

Other solutions of the HONLSE found are:

$$\begin{eqnarray}&&E(z,t)=\pm \left\{\sqrt{\displaystyle \frac{3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}{-6{{\rm{n}}}_{{\rm{3}}}}}\left[\displaystyle \frac{\begin{array}{l}3({{\rm{n}}}_{{\rm{2}}}-2{{\rm{n}}}_{{\rm{4}}}\omega -2{{\rm{n}}}_{{\rm{5}}}\omega )\\ -{{\rm{n}}}_{{\rm{1}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})\end{array}}{(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})}\right]+i\alpha \sqrt{\displaystyle \frac{-6{{\rm{n}}}_{{\rm{3}}}}{3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\,\tanh \,\alpha (t-\beta z)\right\}\\ &&\,\,\,\times \,{e}^{-i(kz-\omega t)},\end{eqnarray} \tag{ 42 }$$

and

$$\begin{eqnarray}&&E(z,t)=\pm \left\{\sqrt{\displaystyle \frac{3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}{-6{{\rm{n}}}_{{\rm{3}}}}}\left[\displaystyle \frac{3({{\rm{n}}}_{{\rm{2}}}-2{{\rm{n}}}_{{\rm{4}}}\omega -2{{\rm{n}}}_{{\rm{5}}}\omega )-{{\rm{n}}}_{{\rm{1}}}(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})=0}{(3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}})({{\rm{n}}}_{{\rm{4}}}+{{\rm{n}}}_{{\rm{5}}})}\right]\right.\\ &&\left.\,\,\,\,+\,i\alpha \sqrt{\displaystyle \frac{-6{{\rm{n}}}_{{\rm{3}}}}{3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\,\cot \,anh\alpha (t-\beta z)\right\}{e}^{-i(kz-\omega t)}.\end{eqnarray} \tag{ 43 }$$

• m = n = 0, q = r ≥ 2, no solution is found
• m = n = 0, q = 0, r = 1, under the constraint equations n₄ = −n₅ and 3n₂n₃ = n₁n₄ the constants take the values
  $$\begin{eqnarray}&&b=\pm \alpha \sqrt{\displaystyle \frac{6{{\rm{n}}}_{{\rm{3}}}}{3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}},\,\omega =-\displaystyle \frac{{{\rm{n}}}_{{\rm{2}}}}{{{\rm{n}}}_{{\rm{4}}}},\,\beta =\displaystyle \frac{{{{\rm{n}}}_{{\rm{1}}}}^{2}+3{\alpha }^{2}{{{\rm{n}}}_{{\rm{3}}}}^{2}}{{{\rm{3n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}},\,k=-\displaystyle \frac{2{{{\rm{n}}}_{{\rm{1}}}}^{3}}{{{\rm{n}}}_{{\rm{0}}}{{{\rm{n}}}_{{\rm{3}}}}^{2}},\end{eqnarray} \tag{ 44 }$$
  and another bright solitary wave solution of equation (2) obtained is
  $$\begin{eqnarray}&&E(z,t)=\left\{a\pm i\alpha \sqrt{\displaystyle \frac{6{{\rm{n}}}_{{\rm{3}}}}{3{{\rm{n}}}_{{\rm{4}}}+2{{\rm{n}}}_{{\rm{5}}}}}\text{sec}h\alpha \left(t-\displaystyle \frac{{{{\rm{n}}}_{{\rm{1}}}}^{2}+3{\alpha }^{2}{{{\rm{n}}}_{{\rm{3}}}}^{2}}{{{\rm{3n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}}z\right)\right\}{e}^{-i\left(-\displaystyle \frac{2{{{\rm{n}}}_{{\rm{1}}}}^{3}}{{{\rm{n}}}_{{\rm{0}}}{{{\rm{n}}}_{{\rm{3}}}}^{2}}z+\displaystyle \frac{{{\rm{n}}}_{{\rm{2}}}}{{{\rm{n}}}_{{\rm{4}}}}t\right)}.\end{eqnarray} \tag{ 45 }$$
  When 3(n₂ − 2n₄ω − 2n₅ω) − n₁(3n₄ + 2n₅) = 0 or ω = ω_d = [3n₂ − n₁(3n₄ + 2n₅)]/6(n₄ + n₅), in equation (42) and a = 0 in equation (45), darklike and brightlike solitary wave solutions given by those equations can be reduced to single dark and bright solitary wave solutions. In the case a ≠ 0, we can consider those solutions as solitary wave solution with their asymptotic values which approach a as time variable approach infinity. They appear in the same condition previously mentioned like dark and bright solitary wave respectively.
• For m = 0, n = 1, q = 1 and r = 1, under the constraint relation n₄ = −n₅, 3n₂n₃ = n₁n₄, constants take the values

$$\begin{eqnarray}&&a=\pm \sqrt{{b}^{2}+\displaystyle \frac{6{\alpha }^{2}{{\rm{n}}}_{{\rm{3}}}}{{{\rm{n}}}_{{\rm{4}}}}},\,\omega =-\displaystyle \frac{{{\rm{n}}}_{{\rm{2}}}}{{{\rm{n}}}_{{\rm{4}}}},\,\beta =\displaystyle \frac{{b}^{2}{{\rm{n}}}_{{\rm{4}}}+{\alpha }^{2}{{{\rm{n}}}_{{\rm{3}}}}^{2}-21{{{\rm{n}}}_{{\rm{1}}}}^{2}}{{{\rm{n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}},\,k=\displaystyle \frac{{{{\rm{18n}}}_{{\rm{1}}}}^{3}}{{{\rm{n}}}_{{\rm{0}}}{{{\rm{n}}}_{{\rm{3}}}}^{2}}.\end{eqnarray} \tag{ 46 }$$

The new solitary wave solution read

$$\begin{eqnarray}&&E(z,t)=\left\{\begin{array}{l}\pm \sqrt{{b}^{2}+\displaystyle \frac{6{\alpha }^{2}{{\rm{n}}}_{{\rm{3}}}}{{{\rm{n}}}_{{\rm{4}}}}}\text{sec}h\alpha \left(t-\displaystyle \frac{{b}^{2}{{\rm{n}}}_{{\rm{4}}}+{\alpha }^{2}{{{\rm{n}}}_{{\rm{3}}}}^{2}-21{{{\rm{n}}}_{{\rm{1}}}}^{2}}{{{\rm{n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}}z\right)\\ +ib\,\tanh \,\alpha \left(t-\displaystyle \frac{{b}^{2}{{\rm{n}}}_{{\rm{4}}}+{\alpha }^{2}{{{\rm{n}}}_{{\rm{3}}}}^{2}-21{{{\rm{n}}}_{{\rm{1}}}}^{2}}{{{\rm{n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}}z\right)\end{array}\right\}{e}^{-i\left(\displaystyle \frac{{{{\rm{18n}}}_{{\rm{1}}}}^{3}}{{{\rm{n}}}_{{\rm{0}}}{{{\rm{n}}}_{{\rm{3}}}}^{2}}z+\displaystyle \frac{{{\rm{n}}}_{{\rm{2}}}}{{{\rm{n}}}_{{\rm{4}}}}t\right)}.\end{eqnarray} \tag{ 47 }$$

Solution found in equation (47) is a complex mixing of bright and dark solitary wave. If n₄ > 0, the soliton amplitude ${| E| }^{2}={b}^{2}+\tfrac{6{\alpha }^{2}{{\rm{n}}}_{{\rm{3}}}}{{{\rm{n}}}_{{\rm{4}}}}\text{sec}{h}^{2}\alpha \left(t-\tfrac{{b}^{2}{{\rm{n}}}_{{\rm{4}}}+{\alpha }^{2}{{{\rm{n}}}_{{\rm{3}}}}^{2}-21{{{\rm{n}}}_{{\rm{1}}}}^{2}}{{{\rm{n}}}_{{\rm{0}}}{{\rm{n}}}_{{\rm{3}}}}z\right)$ shows that it is a brightlike (darklike) solitary wave solution when n₃ > 0 (n₃ < 0), and can be reduced to the single bright or dark solitary wave solution when b = 0.

Figures 1(a) and (b) respectively show the numerical propagation of an initial bright soliton solution, and the propagation of this perturbated bright soliton solution with an amplitude 10% smaller than that of the exact solution.

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Figures 2(a) and (b) respectively illustrate the numerical propagation of an initial dark soliton solution, and the propagation of the perturbated soliton in amplitude (less than 10% of the initial amplitude).

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These results clearly indicate that dark and bright solitary waves given by the complex mixing of bright and dark solitary waves of equation (47) remain unchanged along a significant propagation distance. Our numerical studied also reveal that these solitary waves remain stable again small perturbations in amplitude.

Using the BDK method to construct exact bright and dark solutions of the generalized higher-order nonlinear Schrödinger equation, we have obtained new exact dark and bright solitary wave solutions. When n₀ = 1, the solutions obtained in equations (26) and (32) correspond to those found by Kumar H and Chand F in their studies [30], but for other values of n₀ and for the other solutions construct in this letters, they appear like new analytical solutions of the generalized higher-order nonlinear Schrödinger equation. It is well known that the higher-order nonlinear Schrodinger equation governing the propagation of ultra-short pulse is integrable when the triplet (n₃, n₄, n₄ + n₅)∈{(0, 1, 1), (0, 1, 0), (1, 6, 0), (1, 6, 3)} for n₀ = −n₂ = −2n₁ = 1. Those four triplets respectively correspond to the integrable derivative nonlinear Schrödinger equation-type I, integrable derivative nonlinear Schrödinger equation-type II, Hirota equation and Sasa –Satsuma equations [31–34]. In the case of nonzero coefficients n_i ≠ 0(i = 0, 1, 2, 3, 4, 5). We have succeeded in this work to construct some exact bright and dark solitary wave solutions propagating in the optical fiber under certain parametric conditions, different from conditions previously mentioned. In addition, we know that for picoseconds pulse propagation, the term of second order dispersion is the selector of solitary wave type propagating in the optical fiber [35], otherwise, for high bit rate transmission in optical fiber communication, ultra-short pulse transmission is uncountable. But thin width pulses will induce higher-order effects like third order dispersion which can be the new selector of the solitary wave-type (bright or dark) propagating in the optical fiber. Those results can be useful for femtosecond laser sources such like fiber laser doped with rare earth ions (ytterbium or erbium) which are well suited for their development. Their spectral band of amplification which of the order of 6 THz is large enough to produce pulse impulsions of less than 100 fs [36]. In view of the constraint relations obtained on the existence and the type of solutions that can propagate in the nonlinear optical fiber, we can also conclude that new optical fibers could emerge, with their fabrication answering to the corresponding constraint equations. Moreover, another application of these results could be in the design of optical pulse compressors and solitary wave based communication links [30]. The numerical investigations show that dark and bright solitons can propagate in those new optical fibers, and they still stable again small perturbations. All those results are obtained utilizing the BDKm which is an intelligent method to solve NLPDEs.