Optical Solitons in Magneto-Optic Waveguides Having Kudryashov’s Law of Nonlinear Refractive Index by Trial Equation Approach

Abstract

The paper addresses optical solitons in magneto-optic waveguides that are studied using Kudryashov’s law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation terms. The trial solution approach yielded a variety of soliton solutions, which are listed in this paper.

1. Introduction

Optical soliton dynamics is an essential feature in the telecommunications industry. The current world is paralyzed without Internet activity, which means with no solitons, life comes to a pure standstill. Therefore, it is imperative to address the soliton science dynamics to the fullest extent [1,2,3,4,5,6,7,8,9]. Incidentally, it is always wise to take into account magneto-optic waveguides as opposed to regular waveguides. Such forms of waveguides are always helpful considering the data load that is transmitted across trans-continental and trans-oceanic distances. The presence of a magnetic field can loosen up the solitons from their state of clutter. This only protects the spill-over of information between the solitons when they are very close together. Thus, it is of paramount importance to address the soliton science dynamics in nonlinear optics.

Incidentally, a variety of models that have been studied in the context of magneto-optic solitons. The various forms of the non-Kerr law of nonlinearity, apart from the Kerr law itself, studied in this context are parabolic law, power law, quadratic-cubic law, dual-power law and various others. The current work addresses the same issue, but with Kudryashov’s proposed form of nonlinear refractive index [10,11,12,13,14,15]. The trial solutions approach will be implemented to secure the solitons. The analytical results are displayed, along with the respective parameter constraints, and the corresponding numerical simulations are also exhibited. The derivation and display of the solitons and their numerics are presented after an overview of the model.

Governing Model

The model equations are introduced below [10,11,12,13,14,15]:

$$\begin{array}{cc}& i{u}_t+a_1{u}_{xx}+\left(\frac{b_1}{{\left|u\right|}^n}+\frac{c_1}{{\left|u\right|}^{2n}}+d_1{\left|u\right|}^n+e_1{\left|u\right|}^{2n}+\frac{f_1}{{\left|v\right|}^n}+\frac{g_1}{{\left|v\right|}^{2n}}+h_1{\left|v\right|}^n+k_1{\left|v\right|}^{2n}\right)u\hfill \\ & =Q_{1}v+i\left[{\beta }_1{u}_x+{\lambda }_1{\left({\left|u\right|}^{2n}u\right)}_x+{\gamma }_1{\left({\left|u\right|}^{2n}\right)}_{x}u+{\theta }_1{\left|u\right|}^{2n}{u}_x\right],\hfill \end{array}$$

(1)

and

$$\begin{array}{cc}& i{v}_t+a_2{v}_{xx}+\left(\frac{b_2}{{\left|v\right|}^n}+\frac{c_2}{{\left|v\right|}^{2n}}+d_2{\left|v\right|}^n+e_2{\left|v\right|}^{2n}+\frac{f_2}{{\left|u\right|}^n}+\frac{g_2}{{\left|u\right|}^{2n}}+h_2{\left|u\right|}^n+k_2{\left|u\right|}^{2n}\right)v\hfill \\ \hfill & =Q_{2}u+i\left[{\beta }_2{v}_x+{\lambda }_2{\left({\left|v\right|}^{2n}v\right)}_x+{\gamma }_2{\left({\left|v\right|}^{2n}\right)}_{x}v+{\theta }_2{\left|v\right|}^{2n}{v}_x\right],\hfill \end{array}$$

(2)

where ${\gamma }_l$ and ${\theta }_l$ for $l=1,2$ come from the nonlinear dispersions, while $Q_l$ stem from the magneto-optic parameters. $f_l$, $g_l$, $h_l$ and $k_l$ arise from the cross-phase modulation, while ${\lambda }_l$ emerge from the self-steepening terms. $b_l$, $c_l$, $d_l$ and $e_l$ yield the self-phase modulation, while ${\beta }_l$ give the inter-modal dispersions. x and t are the spatial and temporal variables in sequence, while $a_l$ stem from the chromatic dispersion. $u(x,t)$ and $v(x,t)$ depict the soliton wave profiles, while n comes from the full nonlinearity. Lastly, the first terms stem from the linear temporal evolution of the solitons, where $i=\sqrt{-1}$.

2. Mathematical Analysis

The soliton wave profiles shape up as

$$u(x,t)=U\left(\xi \right)e^{i\psi (x,t)},$$

(3)

$$v(x,t)=V\left(\xi \right)e^{i\psi (x,t)},$$

(4)

where $U\left(\xi \right)$ and $V\left(\xi \right)$ are the soliton amplitude components, while the wave variable and soliton phase component evolve as

$$\xi =x-ct,\psi (x,t)=-\kappa x+\omega t+{\theta }_0.$$

(5)

Here c, $\kappa$, $\omega$ and ${\theta }_0$ represent the velocity, frequency, wave number and phase constant in the sequence. Inserting (3) and (4) into (1) and (2), the real parts stick out as

$$\begin{array}{cc}& a_1{U}^{″}-\left[\omega +{\beta }_1\kappa +a_1{\kappa }^2\right]U-\kappa \left({\lambda }_1+{\theta }_1\right)U^{2n+1}-Q_{1}V+\frac{b_1}{U^{n-1}}+\frac{c_1}{U^{2n-1}}+d_1{U}^{n+1}\hfill \\ & +e_1{U}^{2n+1}+\left(\frac{f_1}{V^n}+\frac{g_1}{V^{2n}}+h_1{V}^n+k_1{V}^{2n}\right)U=0,\hfill \end{array}$$

(6)

and

$$\begin{array}{cc}& a_2{V}^{″}-\left[\omega +{\beta }_2\kappa +a_2{\kappa }^2\right]V-\kappa \left({\lambda }_2+{\theta }_2\right)V^{2n+1}-Q_{2}U+\frac{b_2}{V^{n-1}}+\frac{c_2}{V^{2n-1}}+d_2{V}^{n+1}\hfill \\ & +e_2{V}^{2n+1}+\left(\frac{f_2}{U^n}+\frac{g_2}{U^{2n}}+h_2{U}^n+k_2{U}^{2n}\right)V=0,\hfill \end{array}$$

(7)

while the imaginary parts stand out as

$$\left(c+2a_1\kappa +{\beta }_1\right)U^{\prime }+\left[(2n+1){\lambda }_1+2n{\gamma }_1+{\theta }_1\right]U^{2n}{U}^{\prime }=0,$$

(8)

and

$$\left(c+2a_2\kappa +{\beta }_2\right)V^{\prime }+\left[(2n+1){\lambda }_2+2n{\gamma }_2+{\theta }_2\right]V^{2n}{V}^{\prime }=0.$$

(9)

Equations (8) and (9) leave us with the following restrictions:

$$c=-2a_1\kappa -{\beta }_1,$$

(10)

$$(2n+1){\lambda }_1+2n{\gamma }_1+{\theta }_1=0,$$

(11)

$$c=-2a_2\kappa -{\beta }_2,$$

(12)

and

$$(2n+1){\lambda }_2+2n{\gamma }_2+{\theta }_2=0,$$

(13)

while Equations (10) and (12) provide the frequency

$$\kappa =\frac{{\beta }_2-{\beta }_1}{2\left(a_1-a_2\right)},a_1\ne a_2,{\beta }_1\ne {\beta }_2.$$

(14)

Set

$$V\left(\xi \right)=\varpi U\left(\xi \right),\varpi \ne 0,\varpi \ne 1.$$

(15)

Thus, Equations (6) and (7) come out as

$$\begin{array}{cc}& a_1{U}^{2n-1}{U}^{″}+\left(c_1+g_1{\varpi }^{-2n}\right)+\left(b_1+f_1{\varpi }^{-n}\right)U^n-\left[\omega +{\beta }_1\kappa +a_1{\kappa }^2+Q_1\varpi \right]U^{2n}+\left(d_1+h_1{\varpi }^n\right)U^{3n}\hfill \\ & +\left[\left(e_1+k_1{\varpi }^{2n}\right)-\kappa \left({\lambda }_1+{\theta }_1\right)\right]U^{4n}=0,\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}& a_2\varpi U^{2n-1}{U}^{″}+\left(c_2{\varpi }^{1-2n}+g_2\varpi \right)+\left(b_2{\varpi }^{1-n}+f_2\varpi \right)U^n-\left[\left(\omega +{\beta }_2\kappa +a_2{\kappa }^2\right)\varpi +Q_2\right]U^{2n}+\hfill \\ & \left(d_2{\varpi }^{n+1}+h_2\varpi \right)U^{3n}+\left[e_2{\varpi }^{2n+1}+k_2\varpi -\kappa \left({\lambda }_2+{\theta }_2\right){\varpi }^{2n+1}\right]U^{4n}=0.\hfill \end{array}$$

(17)

Considering Equations (16) and (17), one can be addressed by the aid of the constraint relations

$$a_1=a_2\varpi ,$$

(18)

$$c_1+g_1{\varpi }^{-2n}=c_2{\varpi }^{1-2n}+g_2\varpi ,$$

(19)

$$b_1+f_1{\varpi }^{-n}=b_2{\varpi }^{1-n}+f_2\varpi ,$$

(20)

$$Q_1\varpi +a_1{\kappa }^2+{\beta }_1\kappa +\omega =Q_2+\left[a_2{\kappa }^2+{\beta }_2\kappa +\omega \right]\varpi ,$$

(21)

$$d_1+h_1{\varpi }^n=d_2{\varpi }^{n+1}+h_2\varpi ,$$

(22)

and

$$\left(e_1+k_1{\varpi }^{2n}\right)-\kappa \left({\lambda }_1+{\theta }_1\right)=e_2{\varpi }^{2n+1}+k_2\varpi -\kappa \left({\lambda }_2+{\theta }_2\right){\varpi }^{2n+1}.$$

(23)

Equations (18) and (21) give way to the wave number

$$\omega =\frac{\varpi Q_1-Q_2-\kappa \left(\varpi {\beta }_2-{\beta }_1\right)}{(\varpi -1)}.$$

(24)

With this restriction, $U=H^{\frac{1}{n}}$ simplifies (16) to

$$n{a}_{1}H{H}^{″}+(1-n)a_1{\left(H^{\prime }\right)}^2+n^2\left(A_0+A_{1}H+A_2{H}^2+A_3{H}^3+A_4{H}^4\right)=0,$$

(25)

where

$$\begin{array}{cc}& A_0=c_1+g_1{\varpi }^{-2n},\hfill \\ & A_1=b_1+f_1{\varpi }^{-n},\hfill \\ \hfill & A_2=-\left(\omega +{\beta }_1\kappa +a_1{\kappa }^2+Q_1\varpi \right),\hfill \\ \hfill & A_3=d_1+h_1{\varpi }^n,\hfill \\ & A_4=\left(e_1+k_1{\varpi }^{2n}\right)-\kappa \left({\lambda }_1+{\theta }_1\right).\hfill \end{array}$$

(26)

We take the trial equation

$${\left(H^{\prime }\right)}^2=\sum _{i=0}^n{s}_i{H}^i,$$

(27)

then we derive

$$H^{″}=\sum _{i=1}^n\frac{i{s}_i}{2}{H}^{i-1}.$$

(28)

Substituting (27) and (28) into (25) and the next balancing ${\left(H^{\prime }\right)}^2$ with $H^4$ decreases (27) to

$${\left(H^{\prime }\right)}^2=s_4{H}^4+s_3{H}^3+s_2{H}^2+s_1{H}^1+s_0,$$

(29)

where

$$\begin{array}{cc}& s_4=\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1},\hfill \\ & s_3=\frac{2n^2{A}_3}{-2a_1+2n{a}_1-3n{a}_1},\hfill \\ & s_2=\frac{n^2{A}_2}{-a_1+n{a}_1-n{a}_1},\hfill \\ \hfill & s_1=\frac{2n^2{A}_1}{-2a_1+2n{a}_1-n{a}_1},\hfill \\ & s_0=\frac{n^2{A}_0}{(-1+n)a_1}.\hfill \end{array}$$

(30)

Take transformation

$$G={\left(s_4\right)}^{\frac{1}{4}}\left(H+\frac{s_3}{4s_4}\right),{\xi }_1={\left(s_4\right)}^{\frac{1}{4}}\xi .$$

(31)

Thus, Equation (29) turns into

$${\left(G_{{\xi }_1}\right)}^2=G^4+z_2{G}^2+z_{1}G+z_0,$$

(32)

where

$$\begin{array}{cc}\hfill & z_2=-\frac{3}{8}{\left(s_3\right)}^2{\left(s_4\right)}^{-\frac{3}{2}}+s_2{\left(s_4\right)}^{-\frac{1}{2}},\hfill \\ \hfill & z_1=(\frac{{\left(s_3\right)}^3}{8{\left(s_4\right)}^2}-\frac{s_2{s}_3}{2s_4}+s_1){\left(s_4\right)}^{-\frac{1}{4}},\hfill \\ \hfill & z_0=-\frac{3{\left(s_3\right)}^4}{256{\left(s_4\right)}^3}+\frac{s_2{\left(s_3\right)}^2}{16{\left(s_4\right)}^2}-\frac{s_1{s}_3}{4s_4}+s_0.\hfill \end{array}$$

(33)

Rewrite Equation (32) as

$$\pm \left({\xi }_1-{\xi }_0\right)=\int \frac{dG}{\sqrt{F\left(G\right)}},$$

(34)

where $F\left(G\right)=G^4+z_2{G}^2+z_{1}G+z_0$. According to Liu [16], the fourth-order complete discrimination system for the polynomial $F\left(G\right)$ is given as

$$\begin{array}{cc}& D_1=1,\hfill \\ & D_2=-z_1,\hfill \\ & D_3=-2z_1^3+8z_1{z}_3-9z_2^2,\hfill \\ & D_4=-z_1^3{z}_2^2+4z_1^4{z}_3+36z_1{z}_2^2{z}_3-32z_1^2{z}_3^2-\frac{27}{4}{z}_2^4+64z_3^3,\hfill \\ & E_2=9z_2^2-32z_1{z}_3.\hfill \end{array}$$

(35)

We can classify the roots of $F\left(G\right)$ [16,17,18,19,20,21,22,23,24] and solve the integral (34).

3. Exact Solutions

(1) $D_4=0,D_3=0,D_2=0$, then $F\left(G\right)=G^4$, we get the rational wave

$$u_1={\{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}{[-({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0)]}^{-1}-\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}\}}^{\frac{1}{n}}{e}^{i(-\kappa x+\omega t+{\theta }_0)}.$$

(36)

(2) $D_2<0,D_3=D_4=0,E_2<0$, then $F\left(G\right)={[{(G-{\tau }_1)}^2+{\tau }_2^2]}^2$, we get the singular periodic wave

$$u_2={\{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}[{\tau }_{2}tan({\tau }_2{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0)+{\tau }_1]-\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}\}}^{\frac{1}{n}}{e}^{i(-\kappa x+\omega t+{\theta }_0)},$$

(37)

where ${\tau }_1$ and ${\tau }_2$ are real constants.

(3) $E_2>0,D_4=D_3=0,D_2>0$, then $F\left(G\right)={(G-{\tau }_1)}^2{(G-{\tau }_2)}^2$, we get the singular soliton

$$\begin{array}{ccc}\hfill u_3& =& {\{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}[\frac{{\tau }_2-{\tau }_1}{2}(\coth \frac{({\tau }_1-{\tau }_2)[{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0]}{2}-1)+{\tau }_2]-\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}\}}^{\frac{1}{n}}\hfill \\ & & \times e^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(38)

and the dark soliton

$$\begin{array}{ccc}\hfill u_4& =& {\{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}[\frac{{\tau }_2-{\tau }_1}{2}(\tanh \frac{({\tau }_1-{\tau }_2)[{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0]}{2}-1)+{\tau }_2]-\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}\}}^{\frac{1}{n}}\hfill \\ & & \times e^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(39)

where ${\tau }_1$ and ${\tau }_2$ are real constants.

(4) $E_2=0,D_4=D_3=0,D_2>0$, then $F\left(G\right)={(G-{\tau }_1)}^3(G-{\tau }_2)$, we get the rational wave

$$u_5={\{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}[{\tau }_1+\frac{4({\tau }_1-{\tau }_2)}{{({\tau }_2-{\tau }_1)}^2{[{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0]}^2-4}]-\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}\}}^{\frac{1}{n}}{e}^{i(-\kappa x+\omega t+{\theta }_0)},$$

(40)

where ${\tau }_1$ and ${\tau }_2$ are real constants.

(5) $D_4=0,D_3>0,D_2>0$, then $F\left(G\right)={(G-{\tau }_1)}^2(G-{\tau }_2)(G-{\tau }_3)$, we get the bright soliton

$$\begin{array}{ccc}\hfill u_6& =& {\{\frac{2{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}({\tau }_1-{\tau }_2)({\tau }_1-{\tau }_3)}{({\tau }_2-{\tau }_3)\cosh [\sqrt{({\tau }_1-{\tau }_2)({\tau }_1-{\tau }_3)}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0)]-(2{\tau }_1-{\tau }_2-{\tau }_3)}-\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}\}}^{\frac{1}{n}}\hfill \\ & & \times e^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(41)

and the singular soliton

$$\begin{array}{ccc}\hfill u_7& =& {\{\frac{2{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}({\tau }_1-{\tau }_2)({\tau }_1-{\tau }_3)}{\pm ({\tau }_2-{\tau }_3)\sinh [\sqrt{-({\tau }_1-{\tau }_2)({\tau }_1-{\tau }_3)}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0)]-(2{\tau }_1-{\tau }_2-{\tau }_3)}-\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}\}}^{\frac{1}{n}}\hfill \\ & & \times e^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(42)

where ${\tau }_1$, ${\tau }_2$ and ${\tau }_3$ are real constants.

(6) $D_3<0,D_4=0$, then $F\left(G\right)={(G-{\tau }_1)}^2[{(G-{\tau }_2)}^2+{\tau }_3^2]$, we get the rational wave

$$\begin{array}{cc}\hfill u_8=& \{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}\frac{e^{\pm \sqrt{{({\tau }_1-{\tau }_2)}^2+{\tau }_3^2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0)}-\frac{{\tau }_1-2{\tau }_2}{\sqrt{{({\tau }_1-{\tau }_2)}^2+{\tau }_3^2}}+2\sqrt{{({\tau }_1-{\tau }_2)}^2+{\tau }_3^2}-({\tau }_1-2{\tau }_2)}{{(e^{\pm \sqrt{{({\tau }_1-{\tau }_2)}^2+{\tau }_3^2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0)}-\frac{{\tau }_1-2{\tau }_2}{\sqrt{{({\tau }_1-{\tau }_2)}^2+{\tau }_3^2}})}^2-1}\hfill \\ & -\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}{\}}^{\frac{1}{n}}{e}^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(43)

where ${\tau }_1$, ${\tau }_2$ and ${\tau }_3$ are real constants.

(7) $D_4>0,D_3>0,D_2>0$, then $F\left(G\right)=(G-{\tau }_4)(G-{\tau }_3)(G-{\tau }_2)(G-{\tau }_1)$, we get the snoidal periodic waves

$$\begin{array}{cc}\hfill u_9=& \{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}\frac{{\tau }_2({\tau }_1-{\tau }_4)s{n}^2(\frac{\sqrt{({\tau }_1-{\tau }_3)({\tau }_2-{\tau }_4)}}{2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),m)-{\tau }_1({\tau }_2-{\tau }_4)}{({\tau }_1-{\tau }_4)s{n}^2(\frac{\sqrt{({\tau }_1-{\tau }_3)({\tau }_2-{\tau }_4)}}{2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),m)-({\tau }_2-{\tau }_4)}\hfill \\ & -\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}{\}}^{\frac{1}{n}}{e}^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(44)

and

$$\begin{array}{cc}\hfill u_{10}=& \{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}\frac{{\tau }_4({\tau }_2-{\tau }_3)s{n}^2(\frac{\sqrt{({\tau }_1-{\tau }_3)({\tau }_2-{\tau }_4)}}{2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),m)-{\tau }_3({\tau }_2-{\tau }_4)}{({\tau }_2-{\tau }_3)s{n}^2(\frac{\sqrt{({\tau }_1-{\tau }_3)({\tau }_2-{\tau }_4)}}{2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),m)-({\tau }_2-{\tau }_4)}\hfill \\ & -\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}{\}}^{\frac{1}{n}}{e}^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(45)

where ${\tau }_1$, ${\tau }_2$, ${\tau }_3$ and ${\tau }_4$ are real constants, and $m^2=\frac{({\tau }_1-{\tau }_4)({\tau }_2-{\tau }_3)}{({\tau }_1-{\tau }_3)({\tau }_2-{\tau }_4)}$.

(8) $D_4<0\&\left(\right(D_2<0\&D_3<0)\parallel (D_2=0\&D_3\le 0)\parallel D_2>0)$, then $F\left(G\right)=(G-{\tau }_1)(G-{\tau }_2)[{(G-{\tau }_3)}^2+{\tau }_4^2]$, we get the cnoidal periodic wave

$$\begin{array}{ccc}\hfill u_{11}& =& {\{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}\frac{{ϵ}_{1}c{n}^2(\frac{\sqrt{\mp 2{\tau }_4{o}_1({\tau }_1-{\tau }_2)}}{2o_{1}o}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),o)+{ϵ}_2}{{ϵ}_{3}c{n}^2(\frac{\sqrt{\mp 2{\tau }_4{o}_1({\tau }_1-{\tau }_2)}}{2o_{1}o}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),o)+{ϵ}_4}-\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}\}}^{\frac{1}{n}}\hfill \\ & & \times e^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(46)

where ${\tau }_1$, ${\tau }_2$, ${\tau }_3$ and ${\tau }_4$ are real constants, and

$$\begin{array}{cc}\hfill & {ϵ}_1=\frac{1}{2}({\tau }_1+{\tau }_2){ϵ}_3-\frac{1}{2}({\tau }_1-{\tau }_2){ϵ}_4,\hfill \\ \hfill & {ϵ}_2=\frac{1}{2}({\tau }_1+{\tau }_2){ϵ}_4-\frac{1}{2}({\tau }_1-{\tau }_2){ϵ}_3,\hfill \\ \hfill & {ϵ}_3={\tau }_1-{\tau }_3-\frac{{\tau }_4}{o_1},\hfill \\ \hfill & {ϵ}_4={\tau }_1-{\tau }_3-{\tau }_4{o}_1,\hfill \\ \hfill & E=\frac{{\tau }_4^2+({\tau }_1-{\tau }_3)({\tau }_2-{\tau }_3)}{{\tau }_4({\tau }_1-{\tau }_2)},\hfill \\ & o_1=E\pm \sqrt{E^2+1},\hfill \\ \hfill & o^2=\frac{1}{1+o_1^2}.\hfill \end{array}$$

(47)

(9) $D_4>0\&((D_2>0\&D_3\le 0)\parallel D_2\le 0)$, then $F\left(G\right)=[{(G-{\tau }_1)}^2+{\tau }_2^2][{(G-{\tau }_3)}^2+{\tau }_4^2]$, we get the combo snoidal and cnoidal periodic wave

$$\begin{array}{cc}\hfill u_{12}=& \{{(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{-\frac{1}{4}}\hfill \\ \hfill & \times \frac{{ϵ}_{1}sn(\frac{{\tau }_2\sqrt{({ϵ}_3^2+{ϵ}_4^2)(o_1^2{ϵ}_3^2+{ϵ}_4^2)}}{{ϵ}_3^2+{ϵ}_4^2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),o)+{ϵ}_{2}cn(\frac{{\tau }_2\sqrt{({ϵ}_3^2+{ϵ}_4^2)(o_1^2{ϵ}_3^2+{ϵ}_4^2)}}{{ϵ}_3^2+{ϵ}_4^2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),o)}{{ϵ}_{3}sn(\frac{{\tau }_2\sqrt{({ϵ}_3^2+{ϵ}_4^2)(o_1^2{ϵ}_3^2+{ϵ}_4^2)}}{{ϵ}_3^2+{ϵ}_4^2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),o)+{ϵ}_{4}cn(\frac{{\tau }_2\sqrt{({ϵ}_3^2+{ϵ}_4^2)(o_1^2{ϵ}_3^2+{ϵ}_4^2)}}{{ϵ}_3^2+{ϵ}_4^2}({(\frac{n^2{A}_4}{-a_1+n{a}_1-2n{a}_1})}^{\frac{1}{4}}\xi -{\xi }_0),o)}\hfill \\ & -\frac{A_3(-1+n-2n)}{2A_4(-2+2n-3n)}{\}}^{\frac{1}{n}}{e}^{i(-\kappa x+\omega t+{\theta }_0)},\hfill \end{array}$$

(48)

where ${\tau }_1$, ${\tau }_2$, ${\tau }_3$ and ${\tau }_4$ are real constants, and

$$\begin{array}{cc}& {ϵ}_1={\tau }_1{ϵ}_3+{\tau }_2{ϵ}_4,\hfill \\ \hfill & {ϵ}_2={\tau }_1{ϵ}_4-{\tau }_2{ϵ}_3,\hfill \\ \hfill & {ϵ}_3=-{\tau }_2-\frac{{\tau }_4}{o_1},\hfill \\ \hfill & {ϵ}_4={\tau }_1-{\tau }_3,\hfill \\ \hfill & E=\frac{{({\tau }_1-{\tau }_3)}^2+{\tau }_2^2+{\tau }_4^2}{2{\tau }_2{\tau }_4},\hfill \\ & o_1=E+\sqrt{E^2-1},\hfill \\ & o=\sqrt{\frac{o_1^2-1}{o_1^2}}.\hfill \end{array}$$

(49)

Figure 1 and Figure 2 exhibit the plots of dark and bright magneto-optic solitons (39) and (41), respectively.

Figure 1. Surface plots of dark magneto-optic solitons (39) with $e_1=k_1={\lambda }_1={\theta }_1=\kappa =n=d_1={\xi }_0=c=\omega ={\theta }_0=1$, $a_1=-1$, $h_1={\tau }_2=\varpi =2$ and ${\tau }_1=3$.

Figure 2. Surface plots of bright magneto-optic solitons (41) with $e_1=k_1={\lambda }_1={\theta }_1=\kappa =n=d_1={\xi }_0=c=\omega ={\theta }_0={\tau }_3=1$, $a_1=-1$, $h_1={\tau }_2=\varpi =2$ and ${\tau }_1=3$.

4. Conclusions

The current paper displays solitons in magneto-optic waveguides that are studied using Kudryashov’s proposed law of refractive index. The trial solutions approach gave way to the solitons. The results are displayed, along with the numerical simulations. These results give a further insight into such waveguides with the form of self-phase modulation, as studied. These results give way to a number of future avenues of research. An immediate thought would be to handle the conservation laws. The conservation laws, when computed, would lead to further openings. The issue of quasi-monochromatic dynamics of optical solitons in a magneto-optic waveguide would subsequently follow through. The stochastically perturbed version of the model can also be pursued and studied once the conservation laws are in place. This would lead to the Langevin equations to finally depict the mean-free velocity of the soliton. Such studies are all underway and the results would be sequentially reported.