Optical Solitons for a Concatenation Model by Trial Equation Approach

Abstract

This paper addresses the newly proposed concatenation model by the usage of trial equation approach. The concatenation is a chain model that is a combination of the nonlinear Schrodinger’s equation, Lakshmanan–Porsezian–Daniel model as well as the Sasa–Satsuma equation. The recovered solutions are displayed in terms of dark solitons, singular solitons, cnoidal waves and singular periodic waves. The trial equation approach enables to recover a wide spectrum of solutions to the governing model. The numerical schemes give a visual perspective to the solutions derived analytically.

1. Introduction

The dynamics of optical solitons has been sustained for half a century since its first inception in 1973. There are enumerable advances to soliton science, especially in the field of optoelectronics and telecommunications engineering [1,2,3,4,5,6,7,8,9,10]. There are several forms of solitons that have been addressed in this context in the field of quantum optics. They are non-Kerr law solitons, dispersion-managed solitons, quiescent solitons and many others. While several governing models exist to study the propagation of solitons through optical fibers, a new trend is to concatenate some of the pre-existing models to establish novel models that would govern the dynamics of soliton propagation across inter-continental distances. A first attempt was proposed during 2014 as a concatenation of the pre-existing nonlinear Schrodinger’s equation (NLSE), Lakshmanan–Porsezian–Daniel (LPD) model and the Sasa–Satsuma (SS) equation [1,2]. This gave way to the newly proposed concatenation model [1,2,3,4,5].

This concatenation model has been studied earlier, where many of its other features have been addressed. These include Painleve analysis [5], conservation laws [4], undetermined coefficients [4], reduction to ordinary differential equations by traveling wave hypothesis [3] and other such analytical methodologies. The numerical issues have also been addressed where bright and dark solitons have been recovered numerically by the application of the Laplace–Adomian decomposition scheme [10]. The rogue wave solutions have also been addressed earlier [2]. The current paper gives a fresher perspective to the model by the aid of trial equation approach where new solutions in terms of cnoidal waves and singular periodic solutions are enumerated. The numerical scheme and the surface plots of such solutions are also illustrated.

Governing Model

$$\begin{array}{cc}& i{q}_t+a{q}_{xx}+{b\left|q\right|}^{2}q+c_1[{\delta }_1{q}_{xxxx}+{\delta }_2{(q_x)}^2{q}^{*}+{\delta }_3|q_x{|}^{2}q+{\delta }_4{\left|q\right|}^2{q}_{xx}+{\delta }_5{q}^2{q}_{xx}^{*}+{\delta }_6{\left|q\right|}^{4}q]\hfill \\ & +i{c}_2[{\delta }_7{q}_{xxx}+{\delta }_8{\left|q\right|}^2{q}_x+{\delta }_9{q}^2{q}_x^{*}]=0,\hfill \end{array}$$

(1)

where $q=q(x,t)$ purports the wave profile, while x and t stand for the spatial and temporal variables in sequence. The first term signifies the linear temporal evolution, while a comes from the chromatic dispersion. $c_1$ and $c_2$ yield the dispersion terms, while b gives the Kerr of nonlinearity. Setting $c_1=c_2=0$ reduces (1) to the NLSE, while taking $c_2=0$ decreases (1) to the LPD model. Additionally, setting $c_1=0$ simplifies (1) to the Sasa–Satsuma (SS) equation.

The novelty of the model is truly unique. This is a combination of three well-known models that describe the soliton propagation dynamics through optical fibers across intercontinental distances. Thus, the model is triply beneficial. With $c_1=c_2=0$, the model is purely NLSE that describes the soliton propagation across inter-continental distances. However, with $c_1\ne 0$ and $c_2=0$, the dynamics of the propagation of dispersive solitons is modeled and with $c_1=0$ and $c_2\ne 0$, we have the Sasa–Satsuma equation that governs the perturbed soliton propagation with third–order dispersion and self-frequency shift. Thus, the current model with $c_1\ne 0$ and $c_2\ne 0$, is unique in the sense that all of the perturbation and dispersive effects collectively describe the soliton propagation dynamics.

2. Trial Equation Method

Step 1. Consider a model equation

$$M(u,u_x,u_t,u_{xx},u_{xt},u_{tt},\cdots )=0,$$

(2)

where $u=u(x,t)$ is the wave profile, while x and t stand for the spatial and temporal variables.

Step 2. The wave transformation

$$u(\xi )=u(x,t),\xi =k(x-wt),$$

(3)

simplifies (2) to

$$N(u,u^{\prime },u^{″},\cdots )=0,$$

(4)

where k is the wave width, while w is the wave velocity.

Step 3. Take the trial equation

$${\left(u^{\prime }\right)}^2=F(u)=\sum _{i=0}^n{a}_i{u}^i(\xi ),$$

(5)

where $a_i$ are constants, while n is a positive integer.

Step 4. Rewrite Equation (5) as an integral form

$$\pm (\xi -{\xi }_0)=\int \frac{du}{\sqrt{F(u)}}.$$

(6)

On solving the integral (6), we arrive at the exact solutions of Equation (2).

The novelty of this integration approach is that the algorithm retrieves a variety of solution structures from it with this single algorithm. They range from soliton solutions, singular-periodic solutions as well as doubly periodic functions. This is not the case with a variety of other integration algorithms, such as inverse scattering transform, method of undetermined coefficients, semi-inverse variation, Kudryashov’s approach, and others. However, one major limitation for this scheme is that it fails to recover soliton radiation as with other methods except for the Inverse scattering transform that obtains pure solitons and radiation when we do not have reflectionless potentials.

3. Mathematical Analysis

We set the traveling wave hypotheses as

$$\begin{array}{c}\hfill q(x,t)=Q(\xi )e^{i\varphi (x,t)},\xi =h(x-vt),\varphi (x,t)=-kx+\omega t+\theta ,\end{array}$$

(7)

where v, k, $\omega$, $\theta$, $\varphi (x,t)$ and $Q\left(\xi \right)$ stand for the velocity, frequency, wave number, phase constant, phase component and amplitude component of the soliton in sequence. Substituting Equation (7) into Equation (1) leads to the real part

$$\begin{array}{cc}\hfill & c_1{\delta }_1{h}^{4}Q{}^{″}{}^{″}+(c_1{\delta }_4{h}^2+c_1{\delta }_5{h}^2)Q^{2}Q{}^{″}+(c_1{\delta }_2{h}^2+c_1{\delta }_3{h}^2)Q{(Q^{{}^{\prime }})}^2+(a{h}^2-6c_1{\delta }_1{h}^2{k}^2+3c_2{\delta }_{7}k{h}^2)Q^{{}^{″}}+c_1{\delta }_6{Q}^5\hfill \\ \hfill & +(b-c_1{\delta }_2{k}^2+c_1{\delta }_3{k}^2-c_1{\delta }_4{k}^2-c_1{\delta }_5{k}^2+c_2{\delta }_{8}k-c_2{\delta }_{9}k)Q^3+(-\omega -a{k}^2+c_1{\delta }_1{k}^4-c_2{\delta }_7{k}^3)Q=0,\hfill \end{array}$$

(8)

and the imaginary part

$$\begin{array}{cc}\hfill & (-4c_1{\delta }_{1}k{h}^3+c_2{\delta }_7{h}^3)Q{}^{″}{}^{\prime }+(-hv-2akh+4c_1{\delta }_1{k}^{3}h-3c_2{\delta }_{7}h{k}^2)Q^{{}^{\prime }}\hfill \\ \hfill & +(-2c_1{\delta }_{2}hk-2c_1{\delta }_{4}kh+2c_1{\delta }_{5}kh+c_2{\delta }_{8}h+c_2{\delta }_{9}h)Q^2{Q}^{{}^{\prime }}=0.\hfill \end{array}$$

(9)

Form (9), we yield the constraints

$$\begin{array}{cc}& -4c_1{\delta }_{1}k{h}^3+c_2{\delta }_7{h}^3=0,\hfill \\ \hfill & -hv-2akh+4c_1{\delta }_1{k}^{3}h-3c_2{\delta }_{7}h{k}^2=0,\hfill \\ \hfill & -2c_1{\delta }_{2}hk-2c_1{\delta }_{4}kh+2c_1{\delta }_{5}kh+c_2{\delta }_{8}h+c_2{\delta }_{9}h=0.\hfill \end{array}$$

(10)

Simplify Equation (8) as

$$\begin{array}{cc}\hfill & A_{1}Q{}^{″}{}^{″}+A_2{Q}^2{Q}^{{}^{″}}+A_{3}Q{(Q^{{}^{\prime }})}^2+A_4{Q}^{{}^{″}}+A_5{Q}^5+A_6{Q}^3+A_{7}Q=0,\hfill \end{array}$$

(11)

where

$$\begin{array}{cc}& A_1=c_1{\delta }_1{h}^4,\hfill \\ \hfill & A_2=c_1{\delta }_4{h}^2+c_1{\delta }_5{h}^2,\hfill \\ \hfill & A_3=c_1{\delta }_2{h}^2+c_1{\delta }_3{h}^2,\hfill \\ \hfill & A_4=a{h}^2-6c_1{\delta }_1{h}^2{k}^2+3c_2{\delta }_{7}k{h}^2,\hfill \\ \hfill & A_5=c_1{\delta }_6,\hfill \\ \hfill & A_6=b-c_1{\delta }_2{k}^2+c_1{\delta }_3{k}^2-c_1{\delta }_4{k}^2-c_1{\delta }_5{k}^2+c_2{\delta }_{8}k-c_2{\delta }_{9}k,\hfill \\ \hfill & A_7=-\omega -a{k}^2+c_1{\delta }_1{k}^4-c_2{\delta }_7{k}^3.\hfill \end{array}$$

(12)

Substituting Equation (5) into Equation (11), we obtain

$${(Q^{{}^{\prime }})}^2=a_4{Q}^4+a_3{Q}^3+a_2{Q}^2+a_{1}Q+a_0,$$

(13)

where

$$\begin{array}{cc}& a_4=\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{48A_1},\hfill \\ & a_3=0,\hfill \\ & a_2=\frac{-(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})A_4-2A_2{A}_4+24A_1{A}_6}{2A_1(5(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})-2A_2-12A_3)},\hfill \\ & a_1=0,\hfill \\ & a_0=(\frac{A_4(-(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})A_4-2A_2{A}_4+24A_1{A}_6)}{2A_1(5(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})-2A_2-12A_3)}-\hfill \\ & \frac{{(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}{A}_4-2A_2{A}_4+24A_1{A}_6)}^2}{4A_1{(5(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})-2A_2-12A_3)}^2}-A_7)\hfill \\ \hfill & ÷\frac{1}{4}(-(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})-2A_2)+A_3.\hfill \end{array}$$

(14)

We take transformation

$$U={(4a_4)}^{\frac{1}{3}}{Q}^2,{\xi }_1={(4a_4)}^{\frac{1}{3}}\xi .$$

(15)

Equation (13) becomes

$${(U_{{\xi }_1})}^2=U^3+b_2{U}^2+b_{1}U,$$

(16)

where

$$\begin{array}{cc}& b_2=(\frac{-2(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})A_4-4A_2{A}_4+48A_1{A}_6}{A_1(5(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})-2A_2-12A_3)}){(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{2}{3}},\hfill \\ & b_1=((\frac{4A_4(-(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})A_4-2A_2{A}_4+24A_1{A}_6)}{2A_1(5(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})-2A_2-12A_3)}-\hfill \\ & \frac{{(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}{A}_4-2A_2{A}_4+24A_1{A}_6)}^2}{A_1{(5(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})-2A_2-12A_3)}^2}-4A_7)\hfill \\ & ÷\frac{1}{4}(-(A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5})-2A_2)+A_3)\hfill \\ & \times {(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}.\hfill \end{array}$$

(17)

Rewrite Equation (16) as

$$\pm ({\xi }_1-{\xi }_0)=\int \frac{dU}{\sqrt{F(U)}},$$

(18)

where

$$F(U)=U(U^2+b_{2}U+b_1).$$

(19)

The second order polynomial discriminant system is

$$\Delta =b_2^2-4b_1.$$

(20)

The roots of polynomial $U^2+b_{2}U+b_1$ are classified by the discriminant system, then all possible solutions of integral (18) are obtained.

4. Exact Solutions

Case 1.$\Delta =0$. For $U>0$, if $b_2<0$, we have the dark and singular solitons

$$\begin{array}{cc}\hfill q_1=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ & \times (-\frac{b_2}{2}{tanh}^2(\frac{1}{2}\sqrt{-\frac{b_2}{2}}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0))){\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill q_2=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ & \times (-\frac{b_2}{2}{coth}^2(\frac{1}{2}\sqrt{-\frac{b_2}{2}}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0))){\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )},\hfill \end{array}$$

(22)

if $b_2>0$, we have the singular periodic wave

$$\begin{array}{cc}\hfill q_3=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ & \times (\frac{b_2}{2}{tan}^2(\frac{1}{2}\sqrt{\frac{b_2}{2}}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0))){\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )},\hfill \end{array}$$

(23)

if $b_2=0$, we have the rational wave

$$\begin{array}{cc}\hfill q_4=& {\{\frac{4{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}}{{({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0)}^2}\}}^{ }{ }^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )}.\hfill \end{array}$$

(24)

Case 2.$\Delta >0$ and $b_1=0$. For $U>-b_2$, if $b_2>0$, we have the dark and singular solitons

$$\begin{array}{cc}\hfill q_5=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ & \times (\frac{b_2}{2}{tanh}^2(\frac{1}{2}\sqrt{\frac{b_2}{2}}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0))-b_2){\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill q_6=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ & \times (\frac{b_2}{2}{coth}^2(\frac{1}{2}\sqrt{\frac{b_2}{2}}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0))-b_2){\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )},\hfill \end{array}$$

(26)

if $b_2<0$, we have the singular periodic wave

$$\begin{array}{cc}\hfill q_7=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ & \times (-\frac{b_2}{2}{tan}^2(\frac{1}{2}\sqrt{-\frac{b_2}{2}}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0))-b_2){\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )}.\hfill \end{array}$$

(27)

Case 3.$\Delta >0$ and $b_1\ne 0$. Suppose that ${\gamma }_1<{\gamma }_2<{\gamma }_3$, one of them is zero, and others two are roots of $U^2+b_{2}U+b_1$. For ${\gamma }_1<U<{\gamma }_2$, we have the snoidal wave

$$\begin{array}{cc}\hfill q_8=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ & \times [{\gamma }_1+({\gamma }_2-{\gamma }_1)s{n}^2(\frac{\sqrt{{\gamma }_3-{\gamma }_1}}{2}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0),m)]{\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )},\hfill \end{array}$$

(28)

and for $U>{\gamma }_3$, we have the combo snoidal and cnoidal wave

$$\begin{array}{cc}\hfill q_9=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ & \times [\frac{{\gamma }_3-{\gamma }_{2}s{n}^2(\frac{\sqrt{{\gamma }_3-{\gamma }_1}}{2}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0),m)}{c{n}^2(\frac{\sqrt{{\gamma }_3-{\gamma }_1}}{2}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0),m)}]{\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )},\hfill \end{array}$$

(29)

where $m^2=\frac{{\gamma }_2-{\gamma }_1}{{\gamma }_3-{\gamma }_1}$.

Case 4.$\Delta <0$. For $U>0$, we have the cnoidal wave

$$\begin{array}{cc}\hfill q_{10}=& \{{(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{-\frac{1}{3}}\hfill \\ \hfill & \times (\frac{2\sqrt{b_1}}{1+cn(b_1^{\frac{1}{4}}({(\frac{-2A_2-A_3\pm \sqrt{{(2A_2+A_3)}^2-96A_1{A}_5}}{12A_1})}^{\frac{1}{3}}\xi -{\xi }_0),m)}-\sqrt{b_1}){\}}^{\frac{1}{2}}{e}^{i(-kx+\omega t+\theta )},\hfill \end{array}$$

(30)

where $m^2=\frac{1-\frac{b_2}{2\sqrt{b_1}}}{2}.$

The Figure 1 and Figure 2 represent the surface plots of a dark soliton and cnoidal wave respectively. The selected parameter values are indicated in the captions.

5. Conclusions

This paper displays a wider variety of solutions to the concatenation model than what has been reported earlier. The cnoidal waves and singular periodic solutions are a few of the new solutions that are being reported here in this paper for the first time. The surface plots are also included so that an illustrative effect is also displayed beside the analytically located solutions. It needs to be noted that the dark and singular solitons that are derived in this work are comparable to the ones that have been reported earlier [3,4,5]. While no other analytical methods are applicable since an exhaustive set of solutions are recovered, at least as far as single soliton solutions are concerned. The next move for this model would be to move to other devices.

The solutions are going to be of great asset in the optoelectronics field. They would provide the fundamental structure of the solutions when the model would later be studied with differential group delay and also with dispersion-flattened fibers. Later, this scalar model would be studied in the context of metamaterials, magneto-optic waveguides and optical couplers. The perturbation terms would later be incorporated, and the extended version of this concatenation model would be addressed successfully. The results of such extended studies are currently awaited.