Jamilu Sabi'u^,1, Eric Tala-Tebue^,2, Hadi Rezazadeh^,3, Saima Arshed^,4, Ahmet Bekir^,5,∗1Department of Mathematics, Northwest University, Kano, Nigeria
²Laboratoire d' Automatiqueetd' InformatiqueAppliquee (LAIA), IUT-FV of Bandjoun, The University of Dschang, BP 134, Bandjoun, Cameroon
³Faculty of Engineering Technology, Amol University of Special Modern Technologies, Amol, Iran
⁴Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
⁵Neighbourhood of Akcaglan, Imarli Street, Number: 28/4, 26030, Eskisehir, Turkey

First author contact: ∗Author to whom any correspondence should be addressed.
Received:2020-12-02Revised:2021-04-18Accepted:2021-04-29Online:2021-06-01


Abstract
Most of the important aspects of soliton propagation through optical fibers for transcontinental and transoceanic long distances can best be described using the nonlinear Schrödinger equation. Optical solitons are electromagnetic waves that span in nonlinear dispersive media and permit the stress and intensity to stay unaltered as a result of the delicate balance between dispersion and nonlinearity effects. However, this study exploited the Jacobi elliptic method and obtained different soliton solutions of the decoupled nonlinear Schrödinger equation with ease. Discussions about the obtained solutions were made with the aid of some 3D graphs.
Keywords： Jacobi elliptic method;optical fibers;decoupled NLSE;optical solitons


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Jamilu Sabi'u, Eric Tala-Tebue, Hadi Rezazadeh, Saima Arshed, Ahmet Bekir. Optical solitons for the decoupled nonlinear Schrödinger equation using Jacobi elliptic approach. Communications in Theoretical Physics, 2021, 73(7): 075003- doi:10.1088/1572-9494/abfcb1

1. Introduction

In recent years, there are some substantial advancements in the field of communication systems due to the origination and advance of nonlinear effects in optical fiber. This makes conveying information through optical fibers for millions of kilometers much easier than before. Therefore the needs for high capacity ultra-band communication networks become a real challenge to the world of science and technology. However, the study of soliton propagation through optical fibers through long distances is also becoming one of the best attractive areas of research in optical fibers and general mathematical physics. The majority of the systems in optical fibers are commonly described in the time domain with different frequencies. It's vital to mention that, dynamical systems are usually illustrated in nonlinear complex partial differential equations [1–15]. Quite a several methods were proposed in the literature to investigate the exact solutions of nonlinear wave problems due to their contributions in describing the physical meaning of different mathematical models see [16–23]. In this work, the Jacobi elliptic scheme [24] is employed to find new soliton solutions of the decoupled nonlinear Schrödinger equation. The considered equation plays an important role in dual-core optical fibers.

The dynamics of soliton propagation for dual-core optical fibers is governed by the decoupled nonlinear Schrödinger equation as follows(1)$\begin{eqnarray}\begin{array}{l}{\rm{i}}({u}_{x}+{\lambda }_{1}{q}_{t})+{\lambda }_{2}{u}_{tt}+{\lambda }_{3}{\left|u\right|}^{2}u+{\lambda }_{4}q=0,\\ {\rm{i}}({q}_{x}+{\lambda }_{1}{u}_{t})+{\lambda }_{2}{q}_{tt}+{\lambda }_{3}{\left|q\right|}^{2}q+{\lambda }_{4}u=0.\end{array}\end{eqnarray}$The dependent variables $u(x,t)$ and $q(x,t)$ are complex-valued functions that denote wave profile. The independent variable x represents the distance along with the fiber and the independent variable t designates time in dimensionless form. The coefficient $\tfrac{1}{{l}_{1}}$ represents the group velocity mismatch, ${l}_{2}$ stands for group velocity dispersion, ${l}_{4}$ is the linear coupling coefficient and ${l}_{3}=\tfrac{2\pi {m}_{2}}{\kappa \,{B}_{{\rm{e}}{\rm{f}}{\rm{f}}}},$ where ${m}_{2}$ is the nonlinear refractive index, $\kappa $ is the wavelength and ${B}_{{\rm{e}}{\rm{f}}{\rm{f}}}$ is effective mode area of each wavelength [25].

In recent years, the nonlinear Schrödinger equation has gotten remarkable attention among researchers. Younis et al [26] applied ansatz techniques to acquire bright, dark, and singular soliton solutions for equation (1) with Kerr law and power-law nonlinearities. The conditions for the existence of the solitons are also given in [27]. Haci et al [25] employed extended Sinh-Gordon equation expansion method to obtain dark, bright, combined dark–bright, singular, and combined singular soliton solutions to the decoupled nonlinear Schrödinger equation. However, for the nonlinear evolution equations, the Jacobi Elliptic function method has been among the most outstanding methods for obtaining the traveling wave solutions, see [28, 29] for more details.

In this paper, we are going to derive some more general solutions of the decoupled nonlinear Schrödinger equation via the Jacobi Elliptic approach. We will utilize this method to construct more general solutions concerning the model under consideration. According to our knowledge, no paper has yet treated this equation using the method of Jacobi. In the continuation, we describe the method used in section 2; the application of this method is the aim of section 3; in section 4, we discuss the results found. We end the work with a conclusion.

2. Basic idea

In this section, we survey the main steps of the Jacobi elliptic method as follows;

Step 1.A general form of the nonlinear equation is considered as:

(2)$\begin{eqnarray}F\left(q,{q}_{t},{q}_{x},{q}_{y},{q}_{z},{q}_{xt},{q}_{xx},{q}_{tt},{q}_{yy},{q}_{zz},\mathrm{...}\right).\end{eqnarray}$

Step 2.The traveling wave solution of the nonlinear equation is found by using the wave variable

(3)$\begin{eqnarray}\eta =k\left(x+y+z+{v}_{0}t\right),\end{eqnarray}$where k and ${v}_{0}$ are constants. Therefore(4)$\begin{eqnarray}q\left(x,y,z,t\right)=q\left(\eta \right).\end{eqnarray}$Thus, the nonlinear equation becomes an ordinary differential equation given by(5)$\begin{eqnarray}F\left(q,q^{\prime} ,q^{\prime\prime} ,\mathrm{...}\right),\end{eqnarray}$by which the prime stands for derivative with respect to $\eta .$

Step 3.We introduce a new ansatz and then, the solution $q\left(\eta \right)$ has the following form:

(6)$\begin{eqnarray}q\left(\eta \right)=\displaystyle \sum _{j=0}^{N}{a}_{j}{\left(Y\left(\eta \right)\right)}^{j},\end{eqnarray}$where $\left(Y\left(\eta \right)\right)=\mathrm{sn}\left(\alpha \eta ,m\right)$ or $\left(F\left(\eta \right)\right)=\mathrm{cn}\left(\alpha \eta ,m\right)$ or $\left(F\left(\eta \right)\right)=\mathrm{dn}\left(\alpha \eta ,m\right).$

Step 4.The constant N is determined using the balancing process. We then collect all the coefficients of powers of Y in the resulting equation, where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters ${a}_{i},k\,{\rm{and}}\,{v}_{0}.$

3. Application

Applying the following transformation on equation (1)(7)$\begin{eqnarray}\begin{array}{l}u={\rm{\Phi }}(\eta ){{\rm{e}}}^{{\rm{i}}Y},\,q=\psi (\eta ){{\rm{e}}}^{{\rm{i}}Y},\\ \eta =\vartheta (x-ct),\,{\rm{\Psi }}=-\mu x+\omega t+p,\end{array}\end{eqnarray}$where Y is the phase component, m is the soliton frequency, w is the soliton wave number, p is the phase constant, and $c$ is the soliton velocity.

Substituting equation (7) into equation (1), we have(8)$\begin{eqnarray}\begin{array}{l}(m-{w}^{2}{l}_{2})F+{l}_{3}{F}^{3}+{l}_{4}y-{l}_{1}wy+{c}^{2}{J}^{2}{l}_{2}F^{\prime\prime} =0,\\ (m-{w}^{2}{l}_{2})y+{l}_{3}{y}^{3}+{l}_{4}F-{l}_{1}wF+{c}^{2}{J}^{2}{l}_{2}y^{\prime\prime} =0.\end{array}\end{eqnarray}$From the real part, and(9)$\begin{eqnarray}\begin{array}{l}(2cw{l}_{2}-1)F^{\prime} +{l}_{1}cy^{\prime} =0,\\ (2cw{l}_{2}-1)y^{\prime} +{l}_{1}cF^{\prime} =0.\end{array}\end{eqnarray}$From the imaginary part. Integrating equation (9) once, yields(10)$\begin{eqnarray}\begin{array}{l}(2c\omega {\lambda }_{2}-1){\rm{\Phi }}+{\lambda }_{1}c\psi =0,\\ (2c\omega {\lambda }_{2}-1)\psi +{\lambda }_{1}c{\rm{\Phi }}=0.\end{array}\end{eqnarray}$F and y functions of h satisfying equations (8)–(10), we get the following relation from equation (10)(11)$\begin{eqnarray}\displaystyle \frac{2c\omega {\lambda }_{2}-1}{{\lambda }_{1}c}=\displaystyle \frac{{\lambda }_{1}c}{2c\omega {\lambda }_{2}-1}.\end{eqnarray}$Solving for $c$ in equation (11) gives(12)$\begin{eqnarray}c=\displaystyle \frac{1}{2\omega {\lambda }_{2}-{\lambda }_{1}}.\end{eqnarray}$Balancing the terms ${F}^{3},F^{\prime\prime} ,{y}^{3}$ and $y^{\prime\prime} $ equation (8), yields $N=1.$

${\rm{\Phi }}$ and $\psi $ respectively can be expressed as a finite series of Jacobi sine elliptic functions as follows:(13)$\begin{eqnarray}\begin{array}{l}{\rm{\Phi }}(\eta )=\displaystyle \sum _{j=0}^{N}{A}_{j}{F}^{j}\left(\eta \right)\\ \psi (\eta )=\displaystyle \sum _{j=0}^{N}{B}_{j}{F}^{j}\left(\eta \right),\end{array}\end{eqnarray}$where ${F}^{j}\left(\eta \right)=sn\left(\alpha \eta ,m\right)$ or ${F}^{j}\left(\eta \right)=cn\left(\alpha \eta ,m\right)$ or ${F}^{j}\left(\eta \right)=dn\left(\alpha \eta ,m\right);$ $\alpha $ is a constant will be determined later. The integer N is obtained by using the balancing process. The balancing process leads to $N=1.$ Thus, equation (13) becomes(14)$\begin{eqnarray}\left\{\begin{array}{l}\,{\rm{\Phi }}\left(\eta \right)={A}_{1}F(\eta )+{A}_{0}\\ \psi \left(\eta \right)={B}_{1}F(\eta )+{B}_{0}\end{array}\right..\end{eqnarray}$Inserting (14) into (8), we have

Family 1.$F(\eta )=\mathrm{sn}(\alpha \eta ,m)$

(15)$\begin{eqnarray}\begin{array}{l}{A}_{0}\,=\,0,\,{A}_{1}=mvc\alpha \sqrt{-\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}},\,{B}_{0}=0,\,{B}_{1}=mvc\alpha \sqrt{-\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}},\\ \,\omega \,=\,\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}},\,\mu =\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}({m}^{2}+1)+{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}.\end{array}\end{eqnarray}$Thus the solution is(16)$\begin{eqnarray}\begin{array}{lll}u(x,t) & = & mvc\alpha \sqrt{-\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\mathrm{sn}\left(\alpha v(x-ct),m\right)\\ & & \times \exp \left({\rm{i}}\left(-\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}({m}^{2}+1)+{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{array}\end{eqnarray}$(17)$\begin{eqnarray}\begin{array}{lll}q(x,t) & = & mvc\alpha \sqrt{-\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\mathrm{sn}\left(\alpha v(x-ct),m\right)\\ & & \times \,\exp \left({\rm{i}}\left(-\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}({m}^{2}+1)+{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{array}\end{eqnarray}$where $c=\tfrac{{\lambda }_{1}}{4{\lambda }_{2}{\lambda }_{4}-{\lambda }_{1}^{2}}.$ In these solutions, the following condition must be respected ${\lambda }_{2}{\lambda }_{3}\lt 0.$

If $m\to 1,$ the above solutions become(18)$\begin{eqnarray}u(x,t)=vc\alpha \sqrt{-\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\,\tanh \left(\alpha v(x-ct)\right)\exp \left({\rm{i}}\left(-\displaystyle \frac{{\lambda }_{2}\left[2{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}+{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{eqnarray}$(19)$\begin{eqnarray}q(x,t)=vc\alpha \sqrt{-\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\,\tanh \left(\alpha v(x-ct)\right)\exp \left({\rm{i}}\left(-\displaystyle \frac{{\lambda }_{2}\left[2{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}+{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right).\end{eqnarray}$

Family 2.$F(\eta )=\mathrm{cn}(\alpha \eta ,m)$

(20)$\begin{eqnarray}\begin{array}{l}{A}_{0}=0,\,{A}_{1}=mvc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}},\,{B}_{0}=0,\,{B}_{1}=mvc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}},\\ \,\omega =\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}},\,\mu =-\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}(2{m}^{2}-1)-{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}.\end{array}\end{eqnarray}$Thus the solution is(21)$\begin{eqnarray}\begin{array}{lll}u(x,t) & = & mvc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\mathrm{cn}\left(\alpha v(x-ct),m\right)\\ & & \times \,\exp \left({\rm{i}}\left(\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}(2{m}^{2}-1)-{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{array}\end{eqnarray}$(22)$\begin{eqnarray}\begin{array}{lll}q(x,t) & = & mvc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\mathrm{cn}\left(\alpha v(x-ct),m\right)\\ & & \times \,\exp \left({\rm{i}}\left(\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}(2{m}^{2}-1)-{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{array}\end{eqnarray}$where $c=\tfrac{{\lambda }_{1}}{4{\lambda }_{2}{\lambda }_{4}-{\lambda }_{1}^{2}}.$ In these solutions, the following condition must be respected ${\lambda }_{2}{\lambda }_{3}\gt 0.$ If $m\to 1,$ the above solutions become(23)$\begin{eqnarray}u(x,t)=vc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\text{sech}\left(\alpha v(x-ct)\right)\exp \left({\rm{i}}\left(\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}-{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{eqnarray}$(24)$\begin{eqnarray}q(x,t)=vc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\text{sech}\left(\alpha v(x-ct)\right)\exp \left({\rm{i}}\left(\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}-{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right).\end{eqnarray}$

Family 3.$F(\eta )=\mathrm{dn}(\alpha \eta ,m)$

(25)$\begin{eqnarray}\begin{array}{lll}{A}_{0} & = & 0,\,{A}_{1}=vc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}},\,{B}_{0}=0,\,{B}_{1}=vc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}},\\ \omega & = & \displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}},\,\mu =\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}({m}^{2}-2)+{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}.\end{array}\end{eqnarray}$Thus the solution is(26)$\begin{eqnarray}\begin{array}{lll}u(x,t) & = & vc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\mathrm{dn}\left(\alpha v(x-ct),m\right)\\ & & \times \,\exp \left({\rm{i}}\left(-\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}({m}^{2}-2)+{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{array}\end{eqnarray}$(27)$\begin{eqnarray}\begin{array}{lll}q(x,t) & = & vc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\mathrm{dn}\left(\alpha v(x-ct),m\right)\\ & & \times \,\exp \left({\rm{i}}\left(-\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}({m}^{2}-2)+{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{array}\end{eqnarray}$where $c=\tfrac{{\lambda }_{1}}{4{\lambda }_{2}{\lambda }_{4}-{\lambda }_{1}^{2}}.$ In these solutions, the following condition must be respected ${\lambda }_{2}{\lambda }_{3}\gt 0.$

If $m\to 1,$ the above solutions become(28)$\begin{eqnarray}u(x,t)=vc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\text{sech}\left(\alpha v(x-ct)\right)\exp \left({\rm{i}}\left(\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}-{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right),\end{eqnarray}$(29)$\begin{eqnarray}q(x,t)=vc\alpha \sqrt{\displaystyle \frac{2{\lambda }_{2}}{{\lambda }_{3}}}\text{sech}\left(\alpha v(x-ct)\right)\exp \left({\rm{i}}\left(\displaystyle \frac{{\lambda }_{2}\left[{\alpha }^{2}{c}^{2}{v}^{2}{\lambda }_{1}^{2}-{\lambda }_{4}^{2}\right]}{{\lambda }_{1}^{2}}x+\displaystyle \frac{{\lambda }_{4}}{{\lambda }_{1}}t+p\right)\right).\end{eqnarray}$

4. Results and discussion

This part is dedicated to the discussion of the results found. We have many solutions given by equations (16)–(19), (21)–(24) and (26)–(29). We plotted these solutions in figures 1–3.

Figure 1.

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Figure 1.The solution $u(x,t)$ for ${\lambda }_{1}=1,{\lambda }_{2}=2,{\lambda }_{3}=-2,{\lambda }_{4}=1,\alpha =1,v=-2$ and $p=1.$ (c) Modulus.

Figure 2.

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Figure 2.The solution $q(x,t)$ for ${\lambda }_{1}=1,{\lambda }_{2}=2,{\lambda }_{3}=1,{\lambda }_{4}=1,\alpha =5,v=-2$ and $p=1.$

Figure 3.

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Figure 3.The solution $u(x,t)$ for ${\lambda }_{1}=1,{\lambda }_{2}=2,{\lambda }_{3}=2,{\lambda }_{4}=1,\alpha =1,v=-2$ and $p=1.$


These solutions are Jacobi elliptic functions and hyperbolic solutions when the modulus $m\to 1.$ Comparing our solutions with those found in the literature, we observe that ours are new solutions not yet met. We then deduce that the method used is more general than those encountered in the literature for the resolution of the model considered. These solutions have many applications in the study of solitons based optical communication, particularly indual-coreoptical fibers.

5. Conclusion

This study provided different varieties of solitary wave solutions to the decoupled nonlinear Schrödinger equation by employing the powerful Jacobi elliptic method. The three families of the Jacobi elliptic method are taken into consideration in the process of constructing solitary wave solutions of the decoupled nonlinear Schrödinger equation. Some of the solutions were graphed in 3D using appropriate values to display the robustness and importance of the proposed scheme. The discovered results will be useful in explaining the physical meaning of the considered model (nonlinear Schrödinger).

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