Grey-Black Optical Solitons, Homoclinic Breather, Combined Solitons via Chupin Liu’s Theorem for Improved Perturbed NLSE with Dual-Power Law Nonlinearity

Abstract

In this article, we consider the improved perturbed nonlinear Schrödinger Equation (IP-NLSE) with dual power law nonlinearity, which arises in optical fibers and photovoltaic-photo-refractive materials. We found grey and black optical solitons of the governing equation by means of a suitable complex envelope ansatz solution. By using the Chupin Liu’s theorem (CLT) for the grey and black solitons, we evaluated new categories of combined optical soliton (COS) solutions to the IP-NLSE. The propagation behaviors for homoclinic breathers (HB), multiwaves and M-shape solitons will be analytically examined. All new analytical solutions will be found by an ansatz function scheme and suitable transformations. Multiwave solitons have been reported by using a three-waves technique. Furthermore, two kinds of interactions for M-shape soliton through exponential functions will be examined.

1. Introduction

Optical solitons have been extensively applied in high-speed and high-capacity communication systems as an information carrier. The nonlinear Schrödinger Equation (NLSE) is one of the rarely used models in this context and describes the evolution in optics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Recently, the study of exact rational solutions, their interactions, homoclinic breathers and multiwaves has been receiving attention from many scientists for their applications in biophysics, nonlinear fibers, oceanography and plasma. There are some adequate techniques that are applied to achieve the retrieval of rational solutions to the nonlinear systems [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34].

The IP-NLSE in its dimensionless form is given as [35],

$$\begin{array}{c}\hfill i{\mathsf{\Omega }}_t+a{\mathsf{\Omega }}_{tx}+b{\mathsf{\Omega }}_{xx}+cF\left({\left|\mathsf{\Omega }\right|}^2\right)\mathsf{\Omega }=i\left(c_3{\mathsf{\Omega }}_x+c_4{\left({\left|\mathsf{\Omega }\right|}^{2m}\mathsf{\Omega }\right)}_x+c_5{\left({\left|\mathsf{\Omega }\right|}^{2m}\right)}_x\mathsf{\Omega }\right)\end{array}$$

(1)

where x and t stand for the spatial and temporal variables, respectively, while $\mathsf{\Omega }(x,t)$ is the soliton pulse profile. The 1st term interprets the linear evolution. The m stands for a nonlinearity factor that is applied to put the model into a generalized setting. $c_4$ represents the self-steepening perturbation term, while the coefficient of $c_3$ is the inter-modal dispersion and $c_5$ is the nonlinear dispersion coefficient. The coefficient of b accounts for the group velocity dispersion and the coefficient of a accounts for spatio-temporal dispersion and hence Equation (1) stays integrable. The coefficient c interprets the non-Kerr-law nonlinear term. Now, in order to use the functional F, it is important to consider the smoothness of $F\left({\left|\mathsf{\Omega }\right|}^2\mathsf{\Omega }\right):C\to C$. Treating the complex plane C as a two-dimensional linear space $R^2$ the function $F\left({\left|\mathsf{\Omega }\right|}^2\mathsf{\Omega }\right)$ is continuously differentiable, so that

$$\begin{array}{c}\hfill F\left({\left|\mathsf{\Omega }\right|}^2\right)\mathsf{\Omega }\in \bigcup _{n,m=1}^{\infty }{C}^k\left(\left(-m,m\right)\times \left(-n,n\right):R^2\right).\end{array}$$

(2)

2. Dual-Power Nonlinearity

This nonlinearity is used in photovoltaic photorefractive materials such as $LiNb{O}_3$ [35],

$$\begin{array}{c}\hfill F\left(s\right)=c_1{s}^n+c_2{s}^{2n}\end{array}$$

(3)

where $c_1$ and $c_2$ are constants. Therefore, Equation (1) reduces to [35],

$$\begin{array}{c}\hfill i{\mathsf{\Omega }}_t+a{\mathsf{\Omega }}_{tx}+b{\mathsf{\Omega }}_{xx}+\left(c_1{\left|\mathsf{\Omega }\right|}^{2n}+c_2{\left|\mathsf{\Omega }\right|}^{4n}\right)\mathsf{\Omega }=i\left(c_3{\mathsf{\Omega }}_x+c_4{\left({\left|\mathsf{\Omega }\right|}^{2m}\mathsf{\Omega }\right)}_x+c_5{\left({\left|\mathsf{\Omega }\right|}^{2m}\right)}_x\mathsf{\Omega }\right),\end{array}$$

(4)

since for dual power law nonlinearity, $m=n$. Furthermore, as expected, for $n=1$, the dual power law changes to the parabolic law and Equation (4) will be entertained in the upcoming sections of the paper. Some researchers worked on the stated model. For instance, Biswas et al. investigated optical soliton perturbation via a trial function method [35] and Ekici et al. studied the analysis of solitons in nonlinear materials [36], but the motivation of this document is to find the grey and black optical solitons for the governing equation by means of a suitable complex envelope ansatz solution. With the application of CLT to the grey and black solitons, we evaluate new sets of COS for the model. Furthermore, we will discuss the propagation of homoclinic breathers, multiwaves and rational solitons [37,38,39].

The rest of the article is as follows: In Section 3, we will give the evaluation of grey and black optical solitons by using a complex envelope ansatz solution. Our Section 4 contains the CLT extended to optical solitons with its applications to the stated model. In Section 5, we will give the evaluation of COS of type-I and in Section 6 the evaluation of COS of type-II. In Section 7, we learn the multiwave solution for the governing problem via a three-waves scheme. In Section 8, the HB scheme is presented and we compute new solitons. In Section 9, we will form the rational solitons and their interactions in Section 10 and Section 11 respectively. Our Section 12, contains the discussions and finally our Section 13 contains the conclusion.

3. Grey and Black Optical Solitons (GBOS)

For these solitons, we apply the following transformation in Equation (4) [40],

$$\mathsf{\Omega }(x,t)=B(x,t)e^{i\Xi }with\Xi (x,t)=k_{2}t-k_{1}x+v$$

(5)

where the $B(x,t)$ is a complex function, $\mathsf{\Omega }(x,t)$ stands for phase shift, v denotes the phase constant, $k_2$ represents frequency and $k_1$ represents wave number. To find GBOS for Equation (4), we choose [40]:

$$B(x,t)=tanh\left[\eta \right(x-vt\left)\right]\lambda +i\beta$$

(6)

where v is pulse width and $\eta$ denotes inverse group velocity shift. The amplitude of $B(x,t)$ is

$$|B\left(x,t\right)|={\left[{\lambda }^2+{\beta }^2-{\lambda }^{2}sec{h}^2\left(\eta \left(x-vt\right)\right)\right]}^{\frac{1}{2}}$$

(7)

and, furthermore, the nonlinear phase is

$$G_{NL}={tan}^{-1}\left[\frac{\beta }{\lambda tanh\left(\eta \left(x-vt\right)\right)}\right]$$

(8)

For $\beta =0$ in Equation (6), a dark soliton is known as a black soliton. However, for $\beta \ne 0$ in Equation (6), a GOS is attained.

Substituting Equation (6) into Equation (3):

$$\begin{array}{c}\hfill i{c}_3{k}_1\beta +ib{k}_1^2\beta -i{k}_2\beta +ia{k}_1{k}_2\beta -i{c}_1{\beta }^3+i{c}_4{k}_1{\beta }^4-i{c}_2{\beta }^5-i{c}_2{tanh}^4\left(\eta \right)\beta {\lambda }^4\\ \hfill +\left(-2i{c}_4{k}_1{\beta }^3-2i{c}_5{k}_1{\beta }^3\right)\sqrt{{\beta }^2+{tanh}^2\left(\eta \right){\lambda }^2}+\left(-i{c}_1\beta {\lambda }^2+i{c}_4{k}_1\beta {\lambda }^2-2i{c}_2{\beta }^3{\lambda }^2\right){tanh}^2\left(\eta \right)+\\ \hfill \left(-c_3{k}_1\lambda -b{k}_1^2\lambda +k_2\lambda -a{k}_1{k}_2\lambda +c_1{\beta }^2\lambda -c_4{\beta }^4\lambda +c_1\lambda \right)+\\ \hfill \left(6c_4{k}_1{\beta }^2\lambda +6c_5{k}_1{\beta }^3\lambda \right)\sqrt{{\beta }^2+{tanh}^2\left(\eta \right){\lambda }^2}tanh\left(\eta \right)+\\ \hfill c_2{\lambda }^5{tanh}^4\left(\eta \right)+\left(-c_4{k}_1{\lambda }^3+2c_2{\beta }^2{\lambda }^3\right)-c_1{\lambda }^{3}sec{h}^2\left(\eta \right)tanh\left(\eta \right)\\ \hfill +\left(6i{c}_4{k}_1\beta {\lambda }^2+6i{c}_5\beta {\lambda }^2\right)tan{h}^2\left(\eta \right)\sqrt{{\beta }^2+tan{h}^2\left(\eta \right){\lambda }^2}+\\ \hfill \left(-2c_4{k}_1{\lambda }^3-2c_5{k}_1{\lambda }^3\right)tan{h}^3\left(\eta \right)\sqrt{{\beta }^2+tan{h}^2\left(\eta \right){\lambda }^2}=0\end{array}$$

(9)

Solving distinct equations from the coefficients of tanh, $tanh{tanh}^2$ and ${tanh}^2$ we find the following:

• Constants:

  $$i{c}_3{k}_1\beta +ib{k}_1^2\beta -i{k}_2\beta +ia{k}_1{k}_2\beta -i{c}_1{\beta }^3+i{c}_4{k}_1{\beta }^4-i{c}_2{\beta }^5=0,$$

  (10)
• ${tanh}^2$:

  $$-i{c}_1\beta {\lambda }^2+i{c}_4{k}_1\beta {\lambda }^2-2i{c}_2{\beta }^3{\lambda }^2=0,$$

  (11)
• $\sqrt{{\beta }^2+{tanh}^2\left(\eta \right){\lambda }^2}$:

  $$-2i{c}_4{k}_1{\beta }^3-2i{c}_5{k}_1{\beta }^3=0,$$

  (12)
  $\sqrt{{\beta }^2+{tanh}^2\left(\eta \right){\lambda }^2}tanh\left(\eta \right)$:

  $$6c_4{k}_1{\beta }^2\lambda +6c_5{k}_1{\beta }^3\lambda =0,$$

  (13)
• $tanh\left(\eta \right)$:

  $$-c_3{k}_1\lambda -b{k}_1^2\lambda +k_2\lambda -a{k}_1{k}_2\lambda +c_1{\beta }^2\lambda -c_4{\beta }^4\lambda +c_1\lambda =0,$$

  (14)
  $\sqrt{{\beta }^2+{tanh}^2\left(\eta \right){\lambda }^2}tanh{\left(\eta \right)}^2$:

  $$6i{c}_4{k}_1\beta {\lambda }^2+6i{c}_5\beta {\lambda }^2=0,$$

  (15)
• $tanh{\left(\eta \right)}^3$:

  $$-2c_4{k}_1{\lambda }^3-2c_5{k}_1{\lambda }^3=0$$

  (16)

3.1. Type 1: Grey Optical Soliton (GOS) ($\beta \ne 0$)

By evaluation of equations from Equation (10) to Equation (15) for $\beta \ne =0$, we get [40]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_1=\frac{2c_2{\beta }^2}{c_4},\hfill \\ k_2=-\frac{c_2{\beta }^2\left(4b{\beta }^2{c}_2+{\beta }^2{c}_4^2+2c_3{c}_4\right)}{c_4\left(2a{\beta }^2{c}_2-c_4\right)},\hfill \\ c_5=-c_4,\hfill \\ \lambda =\lambda ,\hfill \\ \beta =\beta .\hfill \end{array}\right.\end{array}$$

(17)

Hence, the GOS of Equation (3) is

$${\mathsf{\Omega }}_1\left(x,t\right)=exp\left[i\left(v-\frac{2c_{2}x{\beta }^2}{c_4}+-\frac{c_2{\beta }^2\left(4b{\beta }^2{c}_2+{\beta }^2{c}_4^2+2c_3{c}_4\right)t}{c_4\left(2a{\beta }^2{c}_2-c_4\right)}\right)\right]\left(-i\beta +\lambda tanh\left(\eta \right)\right).$$

(18)

The pulse intensity is

$$\left|{\mathsf{\Omega }}_1\left(x,t\right)\right|={\left\{{\beta }^2+{\lambda }^2-{\lambda }^{2}sec{h}^2\left[\eta \left(x-vt\right)\right]\right\}}^{\frac{1}{2}}.$$

(19)

The nonlinear phase shift is

$${\mathsf{\Omega }}_{1NL}=arctan{\left[\frac{\beta }{\lambda tanh\left[\eta \left(x-vt\right)\right]}\right]}^{}.$$

(20)

The soliton will exist provided $\eta >0$ and $v>0$.

3.2. Type 2: Black Optical Soliton (BOS) ($\beta =0$)

By evaluation of equations from Equation (10) to Equation (15) for $\beta =0$, we get [40]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_2=-\frac{b{k}_1^2-c_1{\lambda }^2+c_3{k}_1}{a{k}_1-1},\hfill \\ c_5=-c_4,\hfill \\ v=v,\hfill \\ \beta =\beta ,\hfill \\ \lambda =\lambda .\hfill \end{array}\right.\end{array}$$

(21)

Hence, the black optical soliton of Equation (3) is

$${\mathsf{\Omega }}_2\left(x,t\right)=\lambda exp\left[i\left(v-k_{1}x+\frac{\left(b{k}_1^2-c_1{\lambda }^2+c_3{k}_1\right)t}{a{k}_1-1}\right)\right]tanh\left(\eta \right).$$

(22)

Its intensity is

$$\left|{\mathsf{\Omega }}_2\left(x,t\right)\right|={\left\{{\lambda }^2-{\lambda }^{2}sec{h}^2\left[\eta \left(x-vt\right)\right]\right\}}^{\frac{1}{2}}.$$

(23)

4. Optical Solitons Extension via Chupin Liu’s Theorem (CLT)

The CLT shows a relationship among kink–bell and kink-type solutions [40]. This theorem is also used for nonlinear fibre optics. This theorem can be used to compute the COS results of a nonlinear model. For this reason, we will utilize the CLT to extract the COS of governing model via the GOS and BOS Equations (18) and (22). Hence, the modified Chupin Liu’s Theorem on the relationship among dark–bright and dark soliton solutions is given by the subsequent definition. Suppose we are given a nonlinear model,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\Delta }_1\left(\mathsf{\Omega },\mathsf{\Lambda },{\mathsf{\Omega }}_x,{\mathsf{\Lambda }}_x,{\mathsf{\Omega }}_t,{\mathsf{\Lambda }}_t,{\mathsf{\Omega }}_{xx},{\mathsf{\Lambda }}_{xx},\dots \right)=0,\hfill \\ {\Delta }_2\left(\mathsf{\Omega },\mathsf{\Lambda },{\mathsf{\Omega }}_x,{\mathsf{\Lambda }}_x,{\mathsf{\Omega }}_t,{\mathsf{\Lambda }}_t,{\mathsf{\Omega }}_{xx},{\mathsf{\Lambda }}_{xx},\dots \right)=0.\hfill \end{array}\right.\end{array}$$

(24)

Here, $\mathsf{\Omega }=\mathsf{\Omega }(x,t)$, $\mathsf{\Lambda }=\mathsf{\Lambda }(x,t)$, ${\Delta }_1$ and ${\Delta }_2$ are polynomials about $\mathsf{\Omega }$, $\mathsf{\Lambda }$ and their derivatives. If Equation (24) has dark soliton solutions

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathsf{\Omega }=G_k\left\{tanh\left[B\left(x-vt+{\varsigma }_0\right)\right]\right\},\hfill \\ \mathsf{\Lambda }=H_m\left\{tanh\left[B\left(x-vt+{\varsigma }_0\right)\right]\right\}.\hfill \end{array}\right.\end{array}$$

(25)

then Equation (24) has COS

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathsf{\Omega }=G_k\left\{tanh\left[2B\left(x-vt+{\varsigma }_0\right)\right]+i\mathrm{sech}\left[2B\left(x-vt+{\varsigma }_0\right)\right]\right\},\hfill \\ \mathsf{\Lambda }=H_m\left\{tanh\left[2B\left(x-vt+{\varsigma }_0\right)\right]+i\mathrm{sech}\left[2B\left(x-vt+{\varsigma }_0\right)\right]\right\}.\hfill \end{array}\right.\end{array}$$

(26)

where $G_k$ and $H_m$ are polynomials.

5. COS Category-I

By using the CLT to the GOS Equation (18), we derive the following COS of category-I,

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_3\left(x,t\right)=exp\left[i\left(v-\frac{2c_{2}x{\beta }^2}{c_4}+-\frac{c_2{\beta }^2\left(4b{\beta }^2{c}_2+{\beta }^2{c}_4^2+2c_3{c}_4\right)t}{c_4\left(2a{\beta }^2{c}_2-c_4\right)}\right)\right]\left(-i\beta +\lambda tanh\left(\eta \right)+isech\left(\eta \right)\right)\end{array}$$

(27)

with intensity

$$\begin{array}{c}\hfill \left|{\mathsf{\Omega }}_1\left(x,t\right)\right|={\left\{{\lambda }^2+{\beta }^2+2\lambda \beta sech\left[2\eta \left(x-vt\right)\right]\right\}}^{\frac{1}{2}}\end{array}$$

(28)

and nonlinear phase shift

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{2NL}=arctan{\left[\frac{\beta +\lambda sech\left[2\eta \left(x-vt\right)\right]}{\lambda tanh\left[2\eta \left(x-vt\right)\right]}\right]}^{}\end{array}$$

(29)

6. COS Category-II

With the application of CLT to the BOS in Equation (22), we compute the COS of category-II:

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_4\left(x,t\right)=\lambda \left(tanh\left(\eta \right)+i\mathrm{sech}\left(\eta \right)\right)exp\left[i\left(v-k_{1}x+\frac{\left(b{k}_1^2-c_1{\lambda }^2+c_3{k}_1\right)t}{a{k}_1-1}\right)\right]\end{array}$$

(30)

with intensity

$$\begin{array}{c}\hfill {\left|{\mathsf{\Omega }}_4\left(x,t\right)\right|}^2=\lambda \end{array}$$

(31)

and nonlinear phase shift

$$\begin{array}{c}\hfill {\mathsf{\Omega }}_{3NL}=arctan{\left[\frac{sech\left[2\eta \left(x-vt\right)\right]}{tanh\left[2\eta \left(x-vt\right)\right]}\right]}^{}\end{array}$$

(32)

7. Multiwaves Solitons

The ansatz is [41]:

$$\begin{array}{c}\hfill \mathsf{\Omega }(x,t)=\psi \left(\zeta \right)e^{i\theta },\zeta =k_{1}x-vt,\theta =k_{2}x-vt.\end{array}$$

(33)

By substituting Equation (33) into Equation (3) and extracting real and imaginary parts respectively:

$$\begin{array}{c}\hfill c_3{k}_2\psi -b{k}_2^2\psi +v\psi +a{k}_{2}v\psi +c_1{\psi }^3+3c_4{k}_2{\psi }^3+2c_5{k}_2{\psi }^4+c_2{\psi }^5+b{k}_1^2{\psi }^{{}^{″}}-a{k}_{1}v{\psi }^{{}^{″}}=0,\end{array}$$

(34)

and

$$\begin{array}{c}\hfill c_3{k}_1{\psi }^{{}^{\prime }}-2b{k}_2{k}_1{\psi }^{{}^{\prime }}+v{\psi }^{{}^{\prime }}+a{k}_{1}v{\psi }^{{}^{\prime }}+a{k}_{2}v{\psi }^{{}^{\prime }}+c_4{k}_1{\psi }^2{\psi }^{{}^{\prime }}+2c_4{k}_1{\psi }^3{\psi }^{{}^{\prime }}+2c_5{k}_1{\psi }^3{\psi }^{{}^{\prime }}=0.\end{array}$$

(35)

We apply the following logarithmic transformation to Equations (34) and (35):

$$\psi =2{\left(lnf\right)}_{\zeta },$$

(36)

and obtain,

$$\begin{array}{c}\hfill c_3{k}_2{f}^4{f}^{{}^{\prime }}-b{k}_2^2{f}^4{f}^{{}^{\prime }}+v{f}^4{f}^{{}^{\prime }}+a{k}_{2}v{f}^4{f}^{{}^{\prime }}+4c_1{f}^2{f}^{{}^{\prime }3}+2b{k}_1^2{f}^2{f}^{{}^{\prime }3}+12c_4{k}_2{f}^2{f}^{{}^{\prime }3}-2a{k}_{1}v{f}^2{f}^{{}^{\prime }3}\\ \hfill +16c_5{k}_{2}f{f}^{{}^{\prime }4}+16c_2{f}^{{}^{\prime }5}-3b{k}_1^2{f}^3{f}^{{}^{\prime }}{f}^{{}^{″}}+3a{k}_{1}v{f}^3{f}^{{}^{\prime }}{f}^{{}^{″}}+b{k}_1^2{f}^4{f}^{{}^{‴}}-a{k}_{1}v{f}^4{f}^{{}^{‴}}=0.\end{array}$$

(37)

Moreover, the three-waves ansatz [41] is:

$$f=b_{0}cosh(a_1\zeta +a_2)+b_{1}cos(a_3\zeta +a_4)+b_{2}cosh(a_5\zeta +a_6),$$

(38)

where $a_i$ are any constants. Substituting Equation (38) into Equation (37) and solving the equations from of cos, sinh and cosh coefficients:

• Set I.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_2=\frac{av+c_3+\sqrt{a^2{v}^2+2a{c}_{3}v+4bv{c}_3^2}}{2b},\hfill \\ k_1=\frac{av}{b},\hfill \\ a_5=a_5.\hfill \end{array}\right.\end{array}$$

(39)

We have

$${\mathsf{\Omega }}_5(x,t)=\frac{2e^{i\left(-vt+\frac{\left(av+c_3+\sqrt{a^2{v}^2+2a{c}_{3}v+4bv{c}_3^2}\right)x}{2b}\right)}\left(-a_3{b}_{1}cos\left[a_4+a_3\left(-vt+\frac{avx}{b}\right)\right]sin\left[a_4+a_3\left(-vt+\frac{avx}{b}\right)\right]+{\mathsf{\Pi }}_1\right)}{b_{1}cos\left[a_4+a_3\left(-vt+\frac{avx}{b}\right)\right]+b_{0}cos\left[a_2+a_1\left(-vt+\frac{avx}{b}\right)\right]+b_{2}cosh\left[a_6+a_5\left(-vt+\frac{avx}{b}\right)\right]},$$

(40)

where ${\mathsf{\Pi }}_1=a_1{b}_{0}cos\left[a_2+a_1\left(-vt+\frac{avx}{b}\right)\right]sinh\left[a_2+a_1\left(-vt+\frac{avx}{b}\right)\right]+{\mathsf{\Pi }}_2$, and ${\mathsf{\Pi }}_2=a_5{b}_{2}cosh\left[a_6+a_5\left(-vt+\frac{avx}{b}\right)\right]sinh\left[a_6+a_5\left(-vt+\frac{avx}{b}\right)\right]$.

• Set II. When

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_2=\frac{av+c_3+\sqrt{a^2{v}^2+2a{c}_{3}v+4bv{c}_3^2}}{2b},\hfill \\ k_1=0,\hfill \\ b_0=b_0,\hfill \\ a_5=a_5,\hfill \end{array}\right.\end{array}$$

  (41)

by usage of the above values,

$${\mathsf{\Omega }}_6(x,t)=\frac{2exp\left[i\left(-vt+\frac{\left(av+c_3+\sqrt{a^2{v}^2+2a{c}_{3}v+4bv{c}_3^2}\right)x}{2b}\right)\right]\left(-a_3{b}_{1}cos\left(a_4-a_{3}vt\right)sin\left(a_4-a_{3}vt\right)+{\mathsf{\Pi }}_3\right)}{b_{1}cos\left(a_4-a_{3}vt\right)+b_{0}cosh\left(a_2-a_{1}vt\right)+b_{2}cosh\left(a_6-a_{5}vt\right)},$$

(42)

where ${\mathsf{\Pi }}_3=a_1{b}_{0}cosh\left(a_2-a_{1}vt\right)sin\left(a_4-a_{3}vt\right)+a_5{b}_{2}cosh\left(a_2-a_{1}vt\right)sinh\left(a_2-a_{1}vt\right)$.

8. HB Approach

We consider f as follows [41]:

$$f=e^{-p\left(a_2+a_1\zeta \right)}+b_1{e}^{p\left(a_4+a_3\zeta \right)}+b_{0}cos\left(p_1\left(a_6+a_5\zeta \right)\right),$$

(43)

where $a_1$, $a_2$, $a_3$, $a_4$, $b_0$, $b_1$, $a_5$ and $a_6$ are constants. Inserting Equation (43) into Equation (37) and selecting coefficients of exp, cos and sin and evaluating them:

• Set I. When

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{c}_1=-\frac{-32a_1^4{c}_2{p}^2+3a{a}_1^2{k}_{1}v+a{a}_1{a}_3{k}_{1}v-2a{a}_3^2{k}_{1}v}{4a_1^2},\hfill \\ k_2=-\frac{-128a_1^4{c}_2{p}^2+3a{a}_1^2{k}_{1}v-3a{a}_3^2{k}_{1}v}{48a_1^3{c}_{5}p},a_5=0,\hfill \\ c_3=-\frac{256a_1^7{c}_2{c}_5{p}^5-48a{a}_1^4{a}_3{c}_5{k}_1{p}^{3}v+48a{a}_1^3{a}_3^2{c}_5{k}_1{p}^{3}v-128a{a}_1^4{c}_2{p}^{2}v+3a_2^2{a}_1^2{k}_1{v}^2-3a^2{a}_3^2{k}_1{v}^2+48a_1^3{c}_{5}pv}{-128a_1^4{c}_2{p}^2+3a{a}_1^2{k}_{1}v-3a{a}_3^2{k}_{1}v},\hfill \end{array}\right.\end{array}$$

  (44)

via the above values and Equation (6), we obtain

$${\mathsf{\Omega }}_7\left(x,t\right)=\frac{2\left(-a_1{e}^{-p\left(a_2+a_1\left(-vt+k_{1}x\right)\right)}p+a_3{b}_{1}p{e}^{-p\left(a_2+a_1\left(-vt+k_{1}x\right)\right)}\right)e^{i\left(-vt+-\frac{\left(-128a_1^4{c}_2{p}^2+3a{a}_1^2{k}_{1}v-3a{a}_3^2{k}_{1}v\right)x}{48a_1^3{c}_{5}p}\right)}}{e^{-p\left(a_2+a_1\left(-vt+k_{1}x\right)\right)}p+b_{1}p{e}^{-p\left(a_2+a_1\left(-vt+k_{1}x\right)\right)}+b_{0}cos\left(a_6{p}_1\right)}.$$

(45)

• Set II. When

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{c}_1=-\frac{8\left(9a_1-5a_3\right)a_1^2{c}_2{p}^2}{3a_1-3a_3},\hfill \\ k_1=-\frac{8}{p^2\left(a_1-a_3\right)\left(a_1-7a_3\right)a},\hfill \\ v=-\frac{16a_1^4{c}_2{p}^4\left(a_1-7a_3\right)}{3a_1-3a_3},a_5=0,\hfill \\ k_2=0.\hfill \end{array}\right.\end{array}$$

  (46)

via the above values and Equation (6), we get

$${\mathsf{\Omega }}_8\left(x,t\right)=\frac{2\left(-a_{1}p{e}^{-p\left(a_2+a_1\left(\frac{16a_1^4{c}_2{p}^4\left(a_1-7a_3\right)t}{3a_1-3a_3}--\frac{8x}{p^2\left(a_1-a_3\right)\left(a_1-7a_3\right)a}\right)\right)}+{\mathsf{\Pi }}_4\right)e^{\frac{16a_1^4{c}_2{p}^4\left(a_1-7a_3\right)t}{3a_1-3a_3}}}{e^{-p\left(a_2+a_1\left(\frac{16a_1^4{c}_2{p}^4\left(a_1-7a_3\right)t}{3a_1-3a_3}--\frac{8x}{p^2\left(a_1-a_3\right)\left(a_1-7a_3\right)a}\right)\right)}+b_{0}cos\left(a_6{p}_1\right)+{\mathsf{\Pi }}_5},$$

(47)

where ${\mathsf{\Pi }}_4=a_1{b}_{1}p{e}^{-p\left(a_2+a_1\left(\frac{16a_1^4{c}_2{p}^4\left(a_1-7a_3\right)t}{3a_1-3a_3}--\frac{8x}{p^2\left(a_1-a_3\right)\left(a_1-7a_3\right)a}\right)\right)}$ and ${\mathsf{\Pi }}_5=b_1{e}^{p\left(a_2+a_1\left(\frac{16a_1^4{c}_2{p}^4\left(a_1-7a_3\right)t}{3a_1-3a_3}--\frac{8x}{p^2\left(a_1-a_3\right)\left(a_1-7a_3\right)a}\right)\right)}$.

9. Evaluation of M-Shape Solitons

For this type of solitons, we choose f as [41,42]:

$$f={(d_1\zeta +d_2)}^2+{(d_3\zeta +d_4)}^2+d_5,$$

(48)

where $d_i(1\le i\le 5)$ are some constants. Using Equation (20) in Equation (7) and extracting equations from coefficients of $\zeta$ and their evaluation gives:

• Set I. When $d_2=d_4=0$,

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{c}_1=-\frac{8c_2\left(d_1^2+d_3^2\right)}{d_5},v=\frac{c_2{d}_5\left(d_1^2+d_3^2\right)\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)}{6c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)},\hfill \\ k_1=0,\hfill \\ k_2=\frac{c_2{d}_5\left(d_1^2+d_3^2\right)}{6c_4}.\hfill \end{array}\right.\end{array}$$

  (49)

Using the above values we get,

$${\mathsf{\Omega }}_9\left(x,t\right)=\frac{2\left(-\frac{c_2{d}_1^2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{6c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}-\frac{c_2{d}_3^2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{3c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}\right)N_1}{d_5+d_1^2{\left(\frac{c_2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{6c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}\right)}^2+d_3^2{\left(\frac{c_2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{3c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}\right)}^2},$$

(50)

where $N_1=exp\left[i\left(-\frac{c_2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{3c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}+\frac{c_2{d}_5\left(d_1^2+d_3^2\right)x}{6c_4}\right)\right]$.

• Set II. When $d_4=d_2=0$,

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{c}_1=-\frac{8c_2\left(d_1^2+d_3^2\right)}{d_5},\hfill \\ v=\frac{c_2{d}_5\left(d_1^2+d_3^2\right)\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)}{6c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)},\hfill \\ k_1=\frac{c_2{d}_5\left(d_1^2+d_3^2\right)a\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)}{6c_{4}b\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)},\hfill \\ k_2=\frac{c_2{d}_5\left(d_1^2+d_3^2\right)}{6c_4}.\hfill \end{array}\right.\end{array}$$

  (51)

Using the above values and Equation (6) we get,

$${\mathsf{\Omega }}_{10}\left(x,t\right)=\frac{\left(2\left(2d_1^2\left(-\frac{c_2{d}_1^2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{6c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}-\frac{c_2{d}_3^2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{3c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}\right)\right)+2\right)N_1}{d_5+d_1^2{\left(\frac{c_2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{6c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}+\frac{c_2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)x}{3c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}\right)}^2+N_2^2},$$

(52)

where $N_1=exp\left[i\left(-\frac{c_2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{3c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}+\frac{c_2{d}_5\left(d_1^2+d_3^2\right)x}{6c_4}\right)\right]$ and,

$N_2=d_3^2\left(-\frac{c_2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)t}{6c_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}+\frac{a{c}_2\left(d_1^2+d_3^2\right)d_5\left(b{c}_2{d}_1^2{d}_5+b{d}_3^2{d}_5-6c_3{c}_4\right)x}{6b{c}_4\left(a{c}_2{d}_1^2{d}_5+ac{d}_3^2{d}_5+6c_4\right)}\right)$.

10. One-Kink Interaction for M-Shape Soliton

We choose f [41,42]:

$$f={\mathsf{\Theta }}_1^2+{\mathsf{\Theta }}_2^2+d_5+e^{{\mathsf{\Theta }}_3},$$

(53)

the ${\mathsf{\Theta }}_1=d_1\zeta +d_2,{\mathsf{\Theta }}_2=d_3\zeta +d_4,{\mathsf{\Theta }}_3=d_6\zeta +d_7$$d_i(1\le i\le 7),$ are constants. Applying Equations (53) and (37) and setting equations of $\zeta$ and $exp$ coefficients:

• Set I. If $d_4=d_2=0$,

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{c}_1=-\frac{48c_4{k}_2}{d_5^2},\hfill \\ c_2=\frac{k_2\left(-c_5^2{d}_1^2{d}_5^3-c_5^2{d}_3^2{d}_5^2+324c_4^2\right)}{90c_4{d}_5\left(d_1^2+d_3^2\right)},\hfill \\ v=\frac{a{d}_3^2{d}_5^3{v}^2+192c_4{d}_1^4+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4}{d_5^{3}a{d}_3^2},\hfill \\ d_6=\frac{18c_4}{6c_5{d}_5^2},k_1=0,\hfill \\ c_3=\frac{a_2^2{d}_3^2{d}_5^3{k}_2{v}_2-ab{d}_3^2{d}_5^2{k}_2^2+192a{c}_4{d}_1^4{k}_2+384a{c}_4{d}_1^2{d}_3^2{k}_2+192a{c}_4{d}_3^4{k}_2+F_1}{d_5^3{k}_2{d}_3^{2}a},\hfill \\ F_1=a{d}_3^2{d}_5^2{v}_2+192c_4{d}_1^2+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4.\hfill \end{array}\right.\end{array}$$

  (54)

Using the above values we get,

$${\mathsf{\Omega }}_{11}\left(x,t\right)=\frac{2\left(\frac{18c_4{e}^{d_7-\frac{18c_4\left(a{d}_3^2{d}_5^3{v}^2+192c_4{d}_1^4+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4\right)t}{6c_5{d}_5^2{d}_5^{3}a{d}_3^2}}-}{6c_5{d}_5^2}\frac{2t\left(a{d}_3^2{d}_5^3{v}^2+192c_4{d}_1^4+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4\right)}{d_5^{3}a}-H_1\right)H_2}{d_5+e^{d_7-\frac{18c_4\left(a{d}_3^2{d}_5^3{v}^2+192c_4{d}_1^4+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4\right)t}{6c_5{d}_5^2{d}_5^{3}a{d}_3^2}}+H_3},$$

(55)

where $H_1=\frac{2d_1^{2}t\left(a{d}_3^2{d}_5^3{v}^2+192c_4{d}_1^4+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4\right)}{d_3^2{d}_5^{3}a}$,

$H_2=exp\left[i\left(-\frac{t\left(a{d}_3^2{d}_5^3{v}^2+192c_4{d}_1^4+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4\right)}{d_3^2{d}_5^{3}a}+k_{2}x\right)\right]$ and

$H_3=\frac{d_1^2{t}^2{\left(a{d}_3^2{d}_5^3{v}^2+192c_4{d}_1^4+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4\right)}^2}{d_3^4{d}_5^6{a}^2}+\frac{t^2{\left(a{d}_3^2{d}_5^3{v}^2+192c_4{d}_1^4+384c_4{d}_1^2{d}_3^2+192c_4{d}_3^4\right)}^2}{d_3^4{d}_5^6{a}^2}$.

• Set II. When $d_4=d_2=0$,

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{c}_1=\frac{-16c_5}{4}{\left(\frac{-c_5{d}_1^4-2c_5{d}_1^2{d}_3^2-c_5{d}_3^4}{81c_4}\right)}^{\frac{1}{3}}{k}_2,\hfill \\ c_2=\frac{c_5{k}_2}{30{\left(\frac{-c_5{d}_1^4-2c_5{d}_1^2{d}_3^2-c_5{d}_3^4}{81c_4}\right)}^{\frac{1}{3}}},\hfill \\ d_5=\frac{d_1^2+d_3^2}{3{\left(\frac{-c_5{d}_1^4-2c_5{d}_1^2{d}_3^2-c_5{d}_3^4}{81c_4}\right)}^{\frac{1}{3}}},\hfill \\ d_6=2{\left(\frac{-c_5{d}_1^4-2c_5{d}_1^2{d}_3^2-c_5{d}_3^4}{81c_4}\right)}^{\frac{1}{3}}.\hfill \end{array}\right.\end{array}$$

  (56)

Using the above values we get,

$${\mathsf{\Omega }}_{12}\left(x,t\right)=\frac{2\left(2{\left(\frac{-c_5{d}_1^4-2c_5{d}_1^2{d}_3^2-c_5{d}_3^4}{81c_4}\right)}^{\frac{1}{3}}{e}^{d_7+\frac{2{\left(\frac{-c_5{d}_1^4-2c_5{d}_1^2{d}_3^2-c_5{d}_3^4}{81c_4}\right)}^{\frac{1}{3}}\left(-vt+k_{1}x\right)}{{33}^{\frac{1}{3}}}}+Q_1\right)e^{i\left(-vt+k_{2}x\right)}}{\frac{{33}^{\frac{1}{3}}\left(d_1^2+d_3^2\right)}{3{\left(\frac{-c_5{d}_1^4-2c_5{d}_1^2{d}_3^2-c_5{d}_3^4}{81c_4}\right)}^{\frac{1}{3}}}+e^{d_7+\frac{2{\left(\frac{-c_5{d}_1^4-2c_5{d}_1^2{d}_3^2-c_5{d}_3^4}{81c_4}\right)}^{\frac{1}{3}}\left(-vt+k_{1}x\right)}{{33}^{\frac{1}{3}}}}+Q_2},$$

(57)

where $Q_1=2d_1^2\left(-vt+k_{1}x\right)+2d_3^2\left(-vt+k_{1}x\right)$,

$Q_2=d_3^2\left(-vt+k_{1}x\right)+d_3^2\left(-vt+k_{1}x\right)$.

11. Two-Kink Interaction for M-Shape Soliton

We consider [41,42]:

$$f=b_1{e}^{-a_1\zeta +a_2}+b_2{e}^{a_3\zeta +a_4},$$

(58)

where $a_1$, $a_2$, $a_3$ and $a_4$ are some constants. By usage of Equations (37) and (58) and solving the equations from coefficients of the exp functions:

• Set I.

  $$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_1=\frac{a{a}_3^2{c}_3+b}{ab\left(a_3-1\right)\left(a_3+1\right)a_3^2},\hfill \\ k_2=\frac{a{a}_3^2{c}_3+b}{ab\left(a_3-1\right)\left(a_3+1\right)},\hfill \\ v=\frac{a{a}_3^2{c}_3+b}{a^2\left(a_3-1\right)\left(a_3+1\right)a_3^2},\hfill \\ c_1=0,a_5=a_5,a_1=0.\hfill \end{array}\right.\end{array}$$

  (59)

Using the above values we have,

$${\mathsf{\Omega }}_{13}\left(x,t\right)=\frac{2a_3{b}_2{e}^{a_4+a_3\left(-\frac{\left(a{a}_3^2{c}_3+b\right)t}{ab\left(a_3-1\right)\left(a_3+1\right)a_3^2}+\frac{\left(a{a}_3^2{c}_3+b\right)x}{a\left(a_3-1\right)\left(a_3+1\right)a_3^2}\right)}exp\left[i\left(-\frac{\left(a{a}_3^2{c}_3+b\right)t}{ab\left(a_3-1\right)\left(a_3+1\right)a_3^2}+\frac{\left(a{a}_3^2{c}_3+b\right)x}{a\left(a_3-1\right)\left(a_3+1\right)a_3^2}\right)\right]}{b_1{e}^{a_2}+b_2{e}^{a_4+a_3\left(-\frac{\left(a{a}_3^2{c}_3+b\right)t}{ab\left(a_3-1\right)\left(a_3+1\right)a_3^2}+\frac{\left(a{a}_3^2{c}_3+b\right)x}{a\left(a_3-1\right)\left(a_3+1\right)a_3^2}\right)}}.$$

(60)

12. Results and Discussion

We have successfully obtained the soliton solutions by assigning appropriate values to parameters and they show a discrepancy of waves. First, by applying the complex envelope ansatz approach, we have evaluated two solution sets for the governing model and their profiles are plotted. Notice that the 3D, 2D and contour grey optical soliton graphs of ${\mathsf{\Omega }}_1(x,t)$ in Equation (18) are presented with distinct parameters $c_2=4, \beta =10, c_3=30, c_4=5, a=20, b=1.4 \lambda =10, v=5, \eta =1$ in Figure 1. Moreover, the black optical soliton graphs of ${\mathsf{\Omega }}_2(x,t)$ in Equation (22) are constructed with distinct parameters $c_2=4, c_3=30, c_4=5, a=20, b=1.4, \lambda =10, v=5, \eta =1$ in the 3D, 2D and contour expressions (see Figure 2). Secondly, by applying the complex envelope ansatz approach and Chupin Liu’s theorem the combined optical solitons of type-I and type-II are generated. The combined optical soliton type-I graphs of ${\mathsf{\Omega }}_3(x,t)$ in Equation (27) are presented with distinct parameters $c_2=4, \beta =10, c_3=30, c_4=5, a=20, b=1.4,\lambda =10, v=5, \eta =1$ in Figure 3. Similarly, the combined optical soliton type-II profiles for solution ${\mathsf{\Omega }}_4(x,t)$ in Equation (30) are presented with various parameters $c_2=4, \beta =10, c_3=30, c_4=5, a=20, b=1.4, \lambda =10, v=5, \eta =1$ in Figure 4. Furthermore by applying the three-waves assumption on f in Equation (37) we have generated the multiwave solutions respectively. The multiwave graphs of ${\mathsf{\Omega }}_6(x,t)$ in Equation (42) are plotted with distinct parameters $a_3=0.25, a_4=3, a_1=10, a_2=1.5, a_5=10, a_6=3, b_0=0.5, c_3=3.5, b_1=3, b_2=-2.5, a=3, b=1, v=0.1$ in Figure 5. Moreover, the multiwave graphs of ${\mathsf{\Omega }}_6(x,t)$ in Equation (42) are plotted with distinct parameters $a_3=2, a_4=1, a_1=5, a_2=5, a_5=4, a_6=3, b_0=0.5, c_3=-2, b_1=-5, b_2=-2.5, a=1, b=1, v=5$ in Figure 6. Similarly, the multiwave profiles for the solution ${\mathsf{\Omega }}_6(x,t)$ in Equation (42) are plotted via distinct parameters $a_3=0.5, a_4=7, a_1=2, a_2=1, a_5=4, a_6=3, b_0=0.5, c_3=-2, b_1=-5, b_2=-2.5, a=3, b=1, v=5$ in Figure 7 respectively. In addition, with the usage of the homoclinic breather (HB) scheme we have computed the 3D, 2D and contour HB graphs of ${\mathsf{\Omega }}_6(x,t)$ in Equation (42). These are plotted with distinct parameters $a_1=-1, a_3=0.25, a_4=5, a_6=3, b_0=0.5, p=3.5, b_1=3, b_2=-2.5, k_1=3, c_2=1, c_5=3, p_1=4, v=5$ in Figure 8. The 3D, 2D and contour HB profiles of ${\mathsf{\Omega }}_6(x,t)$ in Equation (42) are plotted with distinct parameters $a_1=-1, a_3=3, a_4=10, a_6=3, b_0=0.5, p=3.5, b_1=3, b_2=-2.5, k_1=3, c_2=1, c_5=3, p_1=4, v=5$ in Figure 9. Furthermore, the HB shapes of ${\mathsf{\Omega }}_6(x,t)$ in Equation (42) are plotted with distinct parameters $a_1=0.25, a_3=3, a_4=10, a_6=3, b_0=0.5, p=3.5, b_1=3, b_2=-2.5, k_1=3, c_2=1, c_5=3, p_1=4, v=0.1$ in Figure 10. The 3D M-shape soliton profiles for the solution in Equation (52) are attained with $d_3=0.5, d_5=5, c_2=3, c_3=0.5, c_4=3.5, d_5=3, a=-2.5$ in Figure 11. Similarly, the 2D M-shape soliton profiles for the solution in Equation (52) are computed with $d_3=0.5, d_5=5, c_2=3, c_3=0.5, c_4=3.5, d_5=3, a=-2.5$ in Figure 12. The one-kink M-shape soliton interaction slots via $d_3=0.5, c_2=3, c_3=0.5, c_4=3.5, k_1=3, k_2=1, d_7=1, c_5=1, v=1$ are given in Figure 13 and Figure 14 respectively. Finally the 3D M-shape soliton interaction profiles for Equation (60) are plotted with $a_2=-6, a_4=3, b=0.5, c_3=3.5, a=3, b_1=1, b_2=4, b=1$ in Figure 15.

13. Conclusions

This article derived the grey and black optical solitons to the IP-NLSE with and dual power law nonlinearity via a complex envelope ansatz solution which arises in optical fibers and photovoltaic-photo-refractive materials. We thereafter applied the CLT to compute new forms of COS. Furthermore, the propagation of homoclinic breathers, multiwaves and M-shape solitons is analytically deliberated. All new analytical solutions are evaluated by ansatz functions techniques and transformation involving logarithmic functions. Multiwave solitons are successfully constructed by using the three-waves method. Two categories of interactions for M-shape solitons through exponential functions are successfully examined in Figure 13, Figure 14 and Figure 15.