Modulation instability analysis, optical and other solutions to the modified nonlinear Schr【-逻*辑*与-】
本站小编 Free考研考试/2022-01-02
Muhammad Younis1, Tukur Abdulkadir Sulaiman2,3, Muhammad Bilal1, Shafqat Ur Rehman1, Usman Younas,11Punjab University College of Information Technology, University of the Punjab, Lahore 54000, Pakistan 2Faculty of Science, Federal University Dutse, Jigawa, Nigeria 3Department of Computer Engineering, Biruni University, Istanbul, Turkey
Abstract This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrödinger equation, which models the propagation of rogue waves in ocean engineering. The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions. It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters. This method is beneficial for solving nonlinear partial differential equations, because it is not only useful for finding the new exact traveling wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first-kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. Moreover, by means of the concept of linear stability, we prove that the governing model is stable. 3D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions. Keywords：optical soliton;MNLSE;stability analysis;generalized elliptic equation;extended Fan sub-equation method
PDF (1857KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Muhammad Younis, Tukur Abdulkadir Sulaiman, Muhammad Bilal, Shafqat Ur Rehman, Usman Younas. Modulation instability analysis, optical and other solutions to the modified nonlinear Schrödinger equation. Communications in Theoretical Physics, 2020, 72(6): 065001- doi:10.1088/1572-9494/ab7ec8
1. Introduction
Nonlinear partial differential equations (NLPDEs) have remarkable importance because of their broad range of usages and applications. Nonlinear phenomena have become one of the most impressive fields for researchers in this modern era of science. NLPDEs are largely used in diverse scientific fields such as biology, physics, geochemistry, ocean engineering, fluid mechanics, solid-state physics, geophysics, optical fibers, plasma physics and many other fields, to describe the physical mechanisms of natural phenomena and dynamical processes. NLPDEs are often used to explain the behavior of waves in diverse fields [1–5]. In order to understand these intricate phenomena, it is key to extract more exact solutions of NLPDEs. By using the obtained exact solutions we can understand the complex structure of physical phenomena. It is notable that many NLPDEs in diverse fields like biology, physics and chemistry consist of unknown functions and parameters, and the study of exact solutions provides guidance to researchers to maintain and design the experiments, by producing a suitable natural environment to obtain these unknown functions and parameters. The betterment of mathematical approaches for finding out a general and compact class of exact solutions is one of the most basic tasks necessary to understand the whole dynamical process modeling by complicated NLPDEs from recent years. Finding the exact solutions of NLPDEs is important in order to discuss the stability of numerical solutions, and also for the development of a wide range of new mathematical solvers to simplify the routine calculation. Therefore, the foremost concern for researchers is to find the exact solutions of NLPDEs. For this reason, various powerful techniques have been developed for finding the analytic solutions of NLPDEs by using various symbolic computations like Mathematica, Matlab and Maple [6–31].
Therefore, the goal of this article is to seek new traveling wave solutions which are in the form of soliton, single and combined non-degenerate Jacobi elliptic function, triangular type solutions. These different types of solutions will be beneficial in the area of ocean engineering and modern science. The extended Fan sub-equation method [32–34] is utilized as an integrating approach to find out different kind of solutions. The basic feature of this technique is to discuss some elementary relationships between NLPDEs and other simple NLODEs. It has been discovered that with the aid of simple solutions and solvable ODEs, different kinds of traveling wave solutions of some complicated NLPDEs can be easily constructed. This is the key concept of Fan’s technique. The primary benefit of applying this technique is that we have succeeded in a single move, to gather various types of new solutions. An important aspect of this technique is to provide us with a guideline for how to organize these solutions. Such types of solutions have a significant role in the formation of deep water waves, as these are created by modulation instability; they appear from nowhere.
The conventional form of the modified Schrödinger equation [35] is given below$\begin{eqnarray}\begin{array}{l}{\rm{i}}{q}_{t}+{\alpha }_{1}{q}_{{xx}}+{\alpha }_{2}| q{| }^{2}q={\rm{i}}{r}_{1}{q}_{{xxx}}\\ \quad +\,{\rm{i}}{r}_{2}{q}^{2}{q}_{x}^{* }-{\rm{i}}{r}_{3}| q{| }^{2}{q}_{x}+{r}_{4}q,\end{array}\end{eqnarray}$where, ${\alpha }_{1}={\omega }_{0}/8{k}_{0}^{2}(-3\cos (\alpha )+2)$, ${\alpha }_{2}=-{\omega }_{0}{k}_{0}^{2}/2,{r}_{1}\,={\omega }_{0}\cos (\alpha )/16{k}_{0}^{3}(-5{\cos }^{2}(\alpha )-6)$, ${r}_{2}={\omega }_{0}{k}_{0}\cos (\alpha )/4,{r}_{3}\,=3{\omega }_{0}{k}_{0}/2$, ${r}_{4}={k}_{0}| q{| }_{x}^{2}\,{| }_{x=0}$ and ω0, k0 are the frequency and the wave number of the carrier wave, respectively, [35].
This article is discussed in sequence: in section 2, we discuss the application of this method to solve MNLSE. Section 3 discusses modulation instability analysis. We present a graphical description of solutions in section 4, and finally, we present our conclusions in section 5.
2. Mathematical analysis
In the following section, the extended Fan sub-equation method will be utilized for the extraction of the solutions. For solving the above equation, we suppose the traveling wave transformation to be as follows:$\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & Q(\xi ){{\rm{e}}}^{{\rm{i}}\psi },\\ {\rm{where,}}\,\,\,\xi & = & \eta (x-\nu t)\,\,{\rm{and}}\,\,\psi =-{kx}+\omega t+\theta ,\end{array}\end{eqnarray}$and $\theta ,\omega $ and k are parameters, representing the phase constant, frequency and wave number respectively. Substituting equation (2) into equation (1), we get the real part of the form.$\begin{eqnarray}\begin{array}{l}{\eta }^{2}({\alpha }_{1}-3{r}_{1}k)Q^{\prime\prime} +({\alpha }_{2}+({r}_{2}+{r}_{3})k){Q}^{3}\\ \quad +\,(-\omega -{\alpha }_{1}{k}^{2}+{r}_{1}{k}^{3}-{r}_{4})Q=0.\,\,\end{array}\end{eqnarray}$The above equation takes the following form$\begin{eqnarray}{\zeta }_{1}Q^{\prime\prime} +{\zeta }_{2}{Q}^{3}+{\zeta }_{3}Q=0,\,\,\end{eqnarray}$where ${\zeta }_{1}={\eta }^{2}({\alpha }_{1}-3{r}_{1}k)$, ${\zeta }_{2}={\alpha }_{2}+({r}_{2}+{r}_{3})k$ and ${\zeta }_{3}\,=(-\omega -{\alpha }_{1}{k}^{2}+{r}_{1}{k}^{3}-{r}_{4})$.
By making a balance between the linear term${Q}^{{\prime\prime} }$ and the nonlinear termQ3to determine the value of n by simple calculation, we get n=1. So the equation (4) has the following solution$\begin{eqnarray}Q(\xi )=\sum _{i=0}^{n}{\beta }_{i}{\phi }^{i}(\xi ),\end{eqnarray}$$\begin{eqnarray*}Q(\xi )={\beta }_{0}+{\beta }_{1}\phi (\xi ),\end{eqnarray*}$where β0 and β1 will be calculated later and φ(ξ) holds the generalized elliptic equation and adjusting equation (2) by the assistant of generalized elliptic equation into equation (4); we have equations of an algebraic system for β0, β1, ζ1, ζ2 and ζ3 as follows:$\begin{eqnarray}\displaystyle \frac{{\beta }_{1}{\zeta }_{1}{h}_{1}}{2}+{\beta }_{0}^{3}{\zeta }_{2}+{\zeta }_{3}{\beta }_{0}=0,\end{eqnarray}$$\begin{eqnarray}{\beta }_{1}{\zeta }_{1}{h}_{2}+3{\zeta }_{2}{\beta }_{0}^{2}{\beta }_{1}+{\beta }_{1}{\zeta }_{3}=0,\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{3{\beta }_{1}{\zeta }_{1}{h}_{3}}{2}+3{\beta }_{0}{\zeta }_{2}{\beta }_{1}^{2}=0,\end{eqnarray}$$\begin{eqnarray}2{\beta }_{1}{\zeta }_{1}{h}_{4}+{\zeta }_{2}{\beta }_{1}^{3}=0.\end{eqnarray}$For getting the maximum of hi (i=0, 1, 2, 3, 4) arbitrary, it is necessary to select variables properly. Due to this, β0, β1, ζ2 and ζ3 are considered as variables. Solving equations (6)–(9), we have the solutions as follows:$\begin{eqnarray}{\beta }_{0}=\left(\displaystyle \frac{-{h}_{3}{\zeta }_{1}}{2{\zeta }_{2}{\beta }_{1}}\right),\end{eqnarray}$$\begin{eqnarray}{\beta }_{1}=\left(\pm \sqrt{\displaystyle \frac{-2{\zeta }_{1}{h}_{4}}{{\zeta }_{2}}}\right),\,\,\,\,\,\,{\zeta }_{1}{h}_{4}{\zeta }_{2}\lt 0,\end{eqnarray}$$\begin{eqnarray}{\zeta }_{2}=-\left(\displaystyle \frac{{\zeta }_{1}{\beta }_{1}{h}_{1}}{2{\beta }_{0}^{3}}+\displaystyle \frac{{\zeta }_{3}}{{\beta }_{0}^{2}}\right),\end{eqnarray}$$\begin{eqnarray}{\zeta }_{3}=-\left({\zeta }_{1}{h}_{2}+3{\beta }_{0}^{2}{\zeta }_{2}\right).\end{eqnarray}$The new traveling wave solutions of equation (1) are determined as$\begin{eqnarray}q(x,t)=Q(\xi ){{\rm{e}}}^{{\rm{i}}\psi }=[{\beta }_{0}+{\beta }_{1}\phi (\xi )]{{\rm{e}}}^{{\rm{i}}\psi },\end{eqnarray}$where$\begin{eqnarray}\xi =\eta (x-\nu t),\,\,\,\,\psi =-{kx}+\omega t+\theta .\end{eqnarray}$It may also be noted that$\begin{eqnarray}{\beta }_{0}=\left(\displaystyle \frac{-{h}_{3}{\eta }^{2}({\alpha }_{1}-3{r}_{1}k)}{2({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2}){\beta }_{1}}\right),\end{eqnarray}$$\begin{eqnarray}{\beta }_{1}=\pm \eta \left(\sqrt{\displaystyle \frac{2(3{r}_{1}k-{\alpha }_{1}){h}_{4}}{{\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2}}}\right),\end{eqnarray}$$\begin{eqnarray}{\zeta }_{2}=-\left(\displaystyle \frac{{\eta }^{2}({\alpha }_{1}-3{r}_{1}k){\beta }_{1}{h}_{1}}{2{\beta }_{0}^{3}}+\displaystyle \frac{(-\omega -{\alpha }_{1}{k}^{2}+{k}^{3}{r}_{1}-{r}_{4})}{{\beta }_{0}^{2}}\right),\end{eqnarray}$$\begin{eqnarray}{\zeta }_{3}=-\left({\eta }^{2}({\alpha }_{1}-3{r}_{1}k){h}_{2}+3{\beta }_{0}^{2}({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2}\right).\end{eqnarray}$We can get different types of solutions by considering ζ1,ζ2,ζ3,h0,h1,h2,h3 and h4 as arbitrary constants.
Case I:φ will be the one solution of the 24 ${\phi }_{l}^{I}$ (l=1, 2, 3, …, 24). When we take$\begin{eqnarray*}{h}_{0}={r}^{2},\,\,{h}_{1}=2{rp},\,\,{h}_{2}=2{rq}+{p}^{2},\,\,{h}_{3}=2{pq},\,\,{h}_{4}={q}^{2},\end{eqnarray*}$In this case, we get more two types of solutions.
Type I:φ will be the one solution from ${\phi }_{l}^{I}$ (l=1, 2, …, 1 2). When we take$\begin{eqnarray*}{p}^{2}-4{qr}\gt 0\,{\rm{and}}\,{pq}\ne 0,{qr}\ne 0,\end{eqnarray*}$For example, if by considering l=1, 2, 3, 4, 5, 9 then we get following soliton solutions:
Here p, q and r are arbitrary constants. Moreover, the above solutions are valid for$\begin{eqnarray*}(3{r}_{1}k-{\alpha }_{1})({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})\gt 0\end{eqnarray*}$and$\begin{eqnarray*}\xi =\eta (x-\nu t),\,\,\,\psi =-{kx}+\omega t+\theta .\end{eqnarray*}$We cannot express each solution here due to the limit of length.
Case II:φ will be one solution from ${\phi }_{l}^{{II}}(l=1,2,3,\ldots ,12)$. When we take$\begin{eqnarray*}\begin{array}{rcl}{h}_{0} & = & {r}^{2},\,{h}_{1}=2{rp},\,{h}_{2}=0,\\ {h}_{3} & = & 2{pq},\,{h}_{4}={q}^{2},\,{p}^{2}=-2{rq},\end{array}\end{eqnarray*}$only one type will be discussed in this case.
Type I:
When$\begin{eqnarray*}{qr}\lt 0\,\,{\rm{and}}\,\,{qr}\ne 0,\end{eqnarray*}$for instance, take l=1, 2, 3, 4, 9 then we have the following soliton solutions:
Here p, q and r are arbitrary constants. Moreover, the above solutions are valid for$\begin{eqnarray*}(3{r}_{1}k-{\alpha }_{1})({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})\gt 0\end{eqnarray*}$and$\begin{eqnarray*}\xi =\eta (x-\nu t),\,\,\,\psi =-{kx}+\omega t+\theta .\end{eqnarray*}$Case III:φ has solution one of the 10 ${\phi }_{l}^{{III}}(l=1,2,3,\ldots ,10)$. If we take h0=0, h1=0, and h2, h3, h4 are arbitrary constants.
For instance, if we take l=1, then$\begin{eqnarray*}{h}_{2}=1,\,\,{h}_{3}=\displaystyle \frac{-2c}{a},\,\,{h}_{4}=\displaystyle \frac{{c}^{2}-{b}^{2}}{{a}^{2}},\end{eqnarray*}$The traveling wave solution will take the form as$\begin{eqnarray}\begin{array}{l}{q}_{17}(x,t)=\left[\left(\displaystyle \frac{{\eta }^{2}c({\alpha }_{1}-3{r}_{1}k)}{a({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2}){\beta }_{1}}\right)\right.\\ \left.\pm \,\displaystyle \frac{\eta }{a}\left(\sqrt{\displaystyle \frac{2(3{r}_{1}k-{\alpha }_{1})({c}^{2}-{b}^{2})}{({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})}}\right)\left\{\displaystyle \frac{a{\rm{sech}} \xi }{b+c{\rm{sech}} \xi }\right\}\right]{{\rm{e}}}^{{\rm{i}}\psi },\end{array}\end{eqnarray}$here a, b and c are arbitrary constants. Moreover, above solutions are valid for$\begin{eqnarray*}({c}^{2}-{b}^{2})({\alpha }_{1}-3{r}_{1}k)({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})\gt 0\end{eqnarray*}$and$\begin{eqnarray*}\xi =\eta (x-\nu t),\,\,\,\psi =-{kx}+\omega t+\theta .\end{eqnarray*}$Similarly for l=3, and ${h}_{2}=4,\,\,{h}_{3}=-\tfrac{4(2b+{\rm{d}})}{a},\,{h}_{4}\,=\tfrac{{c}^{2}+4{b}^{2}+4b{\rm{d}}}{{a}^{2}},$ then we have$\begin{eqnarray}\begin{array}{rcl}{q}_{18}(x,t) & = & \left[\left(\displaystyle \frac{\left.2{\eta }^{2}({\alpha }_{1}-3{r}_{1}k)(2b+{\rm{d}}\right)}{a({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2}){\beta }_{1}}\right)\right.\\ & & \pm \displaystyle \frac{\eta }{a}\left(\sqrt{\displaystyle \frac{\left.2(3{r}_{1}k-{\alpha }_{1})({c}^{2}+4{b}^{2}+4b{\rm{d}}\right)}{\left({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2}\right)}}\right)\\ & & \left.\times \left\{\displaystyle \frac{a{{\rm{{\rm{sech}} }}}^{2}\xi }{b{{\rm{{\rm{sech}} }}}^{2}\xi +c\tanh \xi +{\rm{d}}}\right\}\right]{{\rm{e}}}^{{\rm{i}}\psi }.\,\,\end{array}\end{eqnarray}$Here a, b, c and d are arbitrary constants. Moreover, the above solutions are valid for$\begin{eqnarray*}({c}^{2}+4{b}^{2}+4b{\rm{d}})(3{r}_{1}k-{\alpha }_{1})({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})\gt 0\end{eqnarray*}$and$\begin{eqnarray*}\xi =\eta (x-\nu t),\,\,\,\psi =-{kx}+\omega t+\theta .\end{eqnarray*}$Case IV:φ has solution one of the 16 ${\phi }_{l}^{{IV}}(l\,=1,2,3,\ldots ,16)$. If we take h1=0, h3=0 and h0,h2,h4 as arbitrary constants.
Type 1-
For instance, by choosing l=13, and$\begin{eqnarray*}{h}_{0}=\displaystyle \frac{1}{4},\,\,{h}_{2}=\displaystyle \frac{1-2{m}^{2}}{2},\,\,{h}_{4}=\displaystyle \frac{1}{4},\end{eqnarray*}$We get the wave solution as$\begin{eqnarray}{q}_{19}(x,t)=[{\beta }_{0}+{\beta }_{1}\left\{\mathrm{ns}\xi \pm \mathrm{cs}\xi \right\}]{{\rm{e}}}^{{\rm{i}}\psi }.\end{eqnarray}$In the limiting case, if we take $m\to 1$ and ns(ξ)=coth(ξ), cs(ξ)=csch(ξ) then equation (38) gains the form of the Jacobi wave function degenerate as combined soliton-like solutions as$\begin{eqnarray}\begin{array}{rcl}{q}_{20}(x,t) & = & \pm \eta \left(\sqrt{\displaystyle \frac{(3{r}_{1}k-{\alpha }_{1})}{2({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})}}\right)\\ & & \times [\left\{\coth (\xi )\pm \mathrm{csch}(\xi )\right\}]{{\rm{e}}}^{{\rm{i}}\psi }.\end{array}\end{eqnarray}$From the above equation, one may obtain the singular and dark soliton solutions by using the identities $\coth (\xi )+\mathrm{csch}(\xi )\,=\coth \xi /2$, $\coth (\xi )-\mathrm{csch}(\xi )$ =$\,\tanh \xi /2$ respectively.$\begin{eqnarray}{q}_{{20}_{1}}(x,t)=\pm \eta \left(\sqrt{\displaystyle \frac{(3{r}_{1}k-{\alpha }_{1})}{2({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})}}\right)\coth [\xi /2]{{\rm{e}}}^{{\rm{i}}\psi }\end{eqnarray}$$\begin{eqnarray}{q}_{{20}_{2}}(x,t)=\pm \eta \left(\sqrt{\displaystyle \frac{(3{r}_{1}k-{\alpha }_{1})}{2({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})}}\right)\tanh [\xi /2]{{\rm{e}}}^{{\rm{i}}\psi }\end{eqnarray}$When $m\to 0$ and ns(ξ)=csc(ξ), cs(ξ)=cot(ξ) in this case equation (38) takes the periodic singular solutions.$\begin{eqnarray}\begin{array}{rcl}{q}_{21}(x,t) & = & \pm \eta \left(\sqrt{\displaystyle \frac{(3{r}_{1}k-{\alpha }_{1})}{2({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})}}\right)\\ & & \times [\left\{\csc (\xi )\pm \cot (\xi )\right\}]{{\rm{e}}}^{{\rm{i}}\psi }\end{array}\end{eqnarray}$Type 2-
If we take l=16 and ${h}_{0}=\tfrac{{m}^{2}}{4}$, ${h}_{2}=\tfrac{{m}^{2}-2}{2}$ and ${h}_{4}=\tfrac{{m}^{2}}{4};$ then we have the following solution as$\begin{eqnarray}{q}_{22}(x,t)=[{\beta }_{0}+{\beta }_{1}\left\{\mathrm{sn}\xi \pm \mathrm{ics}\xi \right\}]{{\rm{e}}}^{{\rm{i}}\psi }.\end{eqnarray}$For limiting cases when $m\to 1$ and sn(ξ)=tanh(ξ), cs(ξ)=csch(ξ) then equation (43) gains the form of the Jacobi wave function degenerate as combined soliton-like solutions as$\begin{eqnarray}\begin{array}{rcl}{q}_{23}(x,t) & = & \pm \eta \left(\sqrt{\displaystyle \frac{(3{r}_{1}k-{\alpha }_{1})}{2({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})}}\right)\\ & & \times [\left\{\tanh (\xi )\pm \mathrm{icsch}(\xi )\right\}]{{\rm{e}}}^{{\rm{i}}\psi }\end{array}\end{eqnarray}$when $m\to 0$ and sn(ξ)=sin(ξ), cs(ξ)=cot(ξ) then equation (43) has the periodic singular solution$\begin{eqnarray}\begin{array}{rcl}{q}_{24}(x,t) & = & \pm \eta \left(\sqrt{\displaystyle \frac{(3{r}_{1}k-{\alpha }_{1})}{2({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})}}\right)\\ & & \times [\left\{\sin (\xi )\pm \mathrm{icot}(\xi )\right\}]{{\rm{e}}}^{{\rm{i}}\psi }.\end{array}\end{eqnarray}$Here a, b and c are arbitrary constants. Moreover, the above solutions are valid for$\begin{eqnarray*}(3{r}_{1}k-{\alpha }_{1})({\alpha }_{2}+{{kr}}_{3}+{{kr}}_{2})\gt 0\end{eqnarray*}$and$\begin{eqnarray*}\xi =\eta (x-\nu t),\,\,\,\psi =-{kx}+\omega t+\theta .\end{eqnarray*}$Case V:System does not admit a solution of this group if h2=h4=0, h0, h1, h3, are arbitrary constants.
3. Modulation instability analysis
Several nonlinear phenomena display an instability that results in modulation of the steady state as a result of a coaction between the nonlinear and dispersive effects [36]. In this section, we derive modulation instability of the modified nonlinear Schrödinger equation by utilizing the standard linear stability analysis [36–38].
Suppose the steady-state solutions of the modified nonlinear Schrödinger equation to be of the form$\begin{eqnarray}q(x,t)=\left(\sqrt{{k}_{0}}+Q(x,t\right){{\rm{e}}}^{{\rm{i}}{k}_{0}t},\end{eqnarray}$where k0 is the normalized optical power.
Placing equation (46) into equation (1) and linearizing, provides$\begin{eqnarray}\begin{array}{l}{\rm{i}}{Q}_{t}+{\alpha }_{1}{Q}_{{xx}}-({k}_{0}+{r}_{4}-3{k}_{0}{\alpha }_{2})(Q+{Q}^{* })\\ \quad -\,{\rm{i}}{r}_{1}{Q}_{{xxx}}-{\rm{i}}{k}_{0}{r}_{2}{Q}_{x}^{* }+{\rm{i}}{k}_{0}{r}_{3}{Q}_{x}=0,\end{array}\end{eqnarray}$where ∗ indicates the conjugate of the unknown complex function Q(x, t).
Suppose the solution equation (47) to be of the form$\begin{eqnarray}Q(x,t)={b}_{1}{{\rm{e}}}^{{\rm{i}}({kx}-\omega t)}+{b}_{2}{{\rm{e}}}^{-{\rm{i}}({kx}-\omega t)},\end{eqnarray}$where k and ω are the normalized wave number and frequency of perturbation, respectively.
Placing equation (48) into equation (47), splitting the coefficients of ${{\rm{e}}}^{{\rm{i}}({kx}-\omega t)}$ and ${{\rm{e}}}^{-{\rm{i}}({kx}-\omega t)}$, and solving the determinant of the coefficient matrix, we obtain the following dispersion relation:$\begin{eqnarray}\begin{array}{l}2{k}^{3}\omega {r}_{1}-{\omega }^{2}-{k}^{6}{r}_{1}^{2}+{k}^{2}{k}_{0}^{2}{r}_{2}^{2}+2k\omega {k}_{0}{r}_{3}\\ \quad -\,2{k}^{4}{k}_{0}{r}_{1}{r}_{3}-{k}^{2}{k}_{0}^{2}{r}_{3}^{2}+2{k}^{2}{k}_{0}{\alpha }_{1}+2{k}^{2}{r}_{4}{\alpha }_{1}\\ \quad +\,{k}^{4}{\alpha }_{1}^{2}-6{k}^{2}{k}_{0}{\alpha }_{1}{\alpha }_{2}=0.\end{array}\end{eqnarray}$Solving the dispersion relation (49) for ω, provides$\begin{eqnarray}\begin{array}{l}\omega ={k}^{3}{r}_{1}+{{kk}}_{0}{r}_{3}\\ +\,\sqrt{{k}^{2}{k}_{0}^{2}{r}_{2}^{2}+2{k}^{2}{k}_{0}{\alpha }_{1}+2{k}^{2}{r}_{4}{\alpha }_{1}+{k}^{4}{\alpha }_{1}^{2}-6{k}^{2}{k}_{0}{\alpha }_{1}{\alpha }_{2}}.\end{array}\end{eqnarray}$In a situation whereby ${k}^{2}{k}_{0}^{2}{r}_{2}^{2}+2{k}^{2}{k}_{0}{\alpha }_{1}+2{k}^{2}{r}_{4}{\alpha }_{1}\,+{k}^{4}{\alpha }_{1}^{2}-6{k}^{2}{k}_{0}{\alpha }_{1}{\alpha }_{2}\geqslant 0$, the wave number ω is real for all real values of ${k}_{0},\ {r}_{1},\ {r}_{2},\ {r}_{3}\ {r}_{4},\ {\alpha }_{1},\ {\alpha }_{2}$ and the steady state is stable against small perturbations. Moreover, in contrary to the above statement, the steady-state solution turns to be unstable in the situation whereby ${k}^{2}{k}_{0}^{2}{r}_{2}^{2}+2{k}^{2}{k}_{0}{\alpha }_{1}\,+2{k}^{2}{r}_{4}{\alpha }_{1}+{k}^{4}{\alpha }_{1}^{2}\lt 6{k}^{2}{k}_{0}{\alpha }_{1}{\alpha }_{2}$, the wave number ω turns to be imaginary, and the perturbation grows exponentially. Under this condition, the growth rate of modulation stability gain spectrum G(κ) may be given as follows:$\begin{eqnarray}\begin{array}{l}G(k)=2\mathrm{Im}(\omega )=2\mathrm{Im}\left({k}^{3}{r}_{1}+{{kk}}_{0}{r}_{3}\right.\\ \left.+\,\sqrt{{k}^{2}{k}_{0}^{2}{r}_{2}^{2}+2{k}^{2}{k}_{0}{\alpha }_{1}+2{k}^{2}{r}_{4}{\alpha }_{1}+{k}^{4}{\alpha }_{1}^{2}-6{k}^{2}{k}_{0}{\alpha }_{1}{\alpha }_{2}}\right).\end{array}\end{eqnarray}$
4. Graphical description of solutions
The graphical descriptions of derived exact traveling wave solutions have been expressed in the mentioned figures by allotting the different values of the parameters. Figure 1 depicts the physical features of the gain spectrum. Figures 2 and 9 represent two families of dark solitons solutions of the equation (20) and equation (31) respectively, for the values w=3, k=2, θ=−1, η=2, ν=1, p=2, q=1, and r=−2, whereas figures 3 and 10 depict singular soliton solutions for the equation (21) and equation (32) respectively, with the values w=3, k=1, θ=1, η=−2, ν=1, p=−2, q=−4 and r=3. Moreover, figures 4 and 11 show the bright-dark soliton solutions for the equation (22) and equation (33) while figures 5 and 12 illustrate combined singular soliton solutions for the equation (23) and equation (39) with the values w=3, k=2, θ=−1, η=−2, ν=1, p=−1, q=−3 and r=1. Furthermore, the graphical depiction of equations (26), (29) and (30) describe singular periodic solutions which are shown in the figures 6, 7, 8 respectively and figure 13 shows a periodic wave solution in a combined form of equation (42) for the values of w=3, k=1, θ=1, η=2, ν=1, p=2, q=2, r=1 2 and m=.01.
Figure 1.
New window|Download| PPT slide Figure 1.Gain spectrum of modulation instability for three distinct values of the parameters.
Dark soliton describes the solitary waves with lower intensity than the background, bright soliton describes the solitary waves whose peak intensity is larger than the background, and the singular soliton solution is a solitary wave with discontinuous derivatives; examples of such solitary waves include compactions, which have finite (compact) support, and peakons, whose peaks have a discontinuous first derivative [39].
5. Conclusions
In this research article, a model has been considered which propagates the dynamics of water waves. Mathematically, this model is explained by the MNLS equation. We derived different families of exact traveling wave solutions with the assistance of an extended Fan sub-equation method for the range of five parameters, and it also may be noted that during the derivation of the solutions the constraint conditions fall out naturally. During the calculations, we found that our solutions take the forms of soliton-type solutions, triangular type solutions, doubly periodic-like solutions, single and combined non-degenerate Jacobi elliptic function like solutions, and observed the hyperbolic function solutions as well as trigonometric function solutions respectively, when $m\to 0,1$. The calculations also reveal to us the importance of this method to find the analytical solutions in a more general way. Furthermore, the stability analysis for the governing model was also reported in a well organized way.
At least, we can say that this method is very consistent, efficient and reliable, and much more practical for obtaining the exact traveling wave solutions for complex PDEs and other NLPDEs that appear in many fields such as mathematical biology, physics and chemistry, and vice versa.
,11Punjab University College of Information Technology, 
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