Abstract
In this study, we acquired some optical solitons of (3 + 1) dimensional nonlinear Schrödinger equation (3DNLSE) and coupled nonlinear Helmholtz equations (CNLHE) by using generalized \(\left(\frac{{G}^{\prime}}{G}\right)\)-expansion method (GEM). We are able to derive exponential, trigonometric, and hyperbolic solutions. We notice that Mathematica 11.2 offer the equations for these answers. Aside from that, we display some of the solution’s graphic performance. Recently, accurate traveling-wave solutions for nonlinear partial differential equations (NLPDEs) have been acquired using this technique. Through linear stability analysis, we derive an analytical expression for the instability gain and analyze its main characteristics. Finally, we assess the robustness and stability of the solitary waves through numerical simulations and compare the accuracy of the analytical and numerical results. The obtained ultra-short optical solitons can be widely used as a guide for practical applications in telecommunications domain.
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1 Introduction
In applied mathematics and physics, nonlinear phenomena are important. In mathematical physics, exact and numerical solutions of nonlinear equations, particularly the computation of traveling wave and soliton solutions, play a crucial role in the field of soliton theory. Providing precise solutions for NLPDEs using symbolic computer tools like Mathematica, Matlab, and Maple, which simplify intricate algebraic computations, has grown more appealing in recent times. It is very important to find exact solutions for NLPDEs. These mathematical representations of intricate physical events can be found in a variety of fields, including engineering (Zobeiry and Humfeld 2021), chemistry (Sgura et al. 2019), biology (Jornet 2021; Centenera et al. 2018), mechanics (Shi and Zhou 2022), and physics (Gepreel 2020; Arora et al. 2022). Many useful techniques have been created to comprehend the workings of these physical models in order to support engineers and medical professionals, as well as to gain knowledge about physical issues and their applications. The literature has a number of analytical techniques (Shang 2007; Bock and Kruskal 1979; Matveev and Salle 1991; Abourabia and El Horbaty 2006; Malfliet 1992; Chuntao 1996; Cariello and Tabor 1989; Fan 2000a, b; Clarkson 1989). In addition to these approaches, there are numerous other approaches that use an auxiliary equation to arrive at a solution. These methods are given in (Malfliet 1992; Fan 2000a, b; Elwakil et al. 2002; Chen and Zhang 2004; Fu et al. 2001; Shen and Pan 2003; Chen and Hong-Qing 2004; Chen et al. 2004; Chen and Yan 2006; Wang et al. 2008; Guo and Zhou 2010; Lü et al. 2010; Li et al. 2010; Manafian 2016; Khater 2015; Youssoufa et al. 2019; Youssoufa et al. 2022). Many researchers have applied such methods to various equations, (Khater and Zahran 2016; Wazwaz 2007; Manafian et al. 2017; Mirzazadeh et al. 2015; Wazwaz and Mehanna 2021; Biswas, Aceves et al. 2001; Singh et al. 2020; El-Ganaini et al. 2015; Ali Faridi et al. 2024; Ali Faridi and AlQahtani 2023; Ali Faridi et al. 2023; Ali Faridi et al. 2023a; Ali Faridi et al. 2023b; Zain Majida et al. 2023; Hamza Rafiq et al. 2023; Kaplan et al. 2018; Shakeel et al. 2014; Ahmad and Mustafa 2023; Hussain Tipu et al. 2024; Rani et al. 2022; Rani et al. 2020; Aniqa and Ahmad 2022; Ali et al. 2023a, b; Ali et al. 2023b; Ali et al. 2023c; Wazwaz and Triki 2010; Khater and Zahran 2015; Manafian Heris and Lakestani 2013; Manafian Heris, Lakestani 2014, Feng and Zheng 2010; Zayed and Hoda Ibrahim 2013; Bekir 2008). In this context, Biswas et al. (2001) constructed dark, bright and singular optical solitons of 3DNLSE. Recently, Wazwaz and Mehanna (2021) found bright and dark optical solitons of 3DNLSE. In 2020, Singh et al. (2020) derived solitary wave solitons of CNLHE. The 3DNLSE is essential for studying the evolution of water waves and other nonlinear waves. Moreover, the nonlinear Schrödinger equation physically illustrates the pulse propagation in the polarization-preserving nonlinear optical fiber. Unlike the ideal uniform fiber material, a single nonlinear equation cannot appropriately describe the behavior of solitons in real optical fiber. So, the CNLHE is the most prominent technological model to describe transverse effects in nonlinear optical systems and also models the transmission of coupled wave packets and optical solitons in nonlinear optical fibers (Abraham et al. 1990).
In the present manuscript, we will use the GEM (Manafian et al. 2017) in order to find the some optical and other solitons of the 3DNLSE and the CNLHE. In addition, we will present the modulation instability analysis and numerical simulations for the two models.
2 Analysis of the method
Assume for the moment that a particular NLPDE has the following generic form:
we can convert this equation to an ODE by using the \(\xi =x-\mu t\) conversion as follows:
In the final equation, \(\mu\) is a constant that will be found later. We agree that Eq. (2) can be solved as follows:
The constants \({d}_{k}\) and \({e}_{k}\) in the above-mentioned solution are not equal to zero. It is calculated using the homogeneous balance method, where “m” is a positive integer. Furthermore, the ODE given by \(\phi \left(\xi \right)=\left(\frac{{G}^{\prime}\left(\xi \right)}{G\left(\xi \right)}\right)\) equality is as follows:
The following exceptional solutions to Eq. (4) will be examined:
Solution (1) \({k}_{2}\ne 0, \, f={k}_{1}-{k}_{3}, \, {s}_{1}={k}_{2}^{2}+4{k}_{4}\left({k}_{1}-{k}_{3}\right)>0,\) then
Solution (2) \({k}_{2}\ne 0, \, f={k}_{1}-{k}_{3}, \, {s}_{1}={k}_{2}^{2}+4{k}_{4}\left({k}_{1}-{k}_{3}\right)<0,\) then
Solution (3) \({k}_{2}\ne 0, \, f={k}_{1}-{k}_{3}, \, {s}_{1}={k}_{2}^{2}+4{k}_{4}\left({k}_{1}-{k}_{3}\right)=0,\) then \(\phi \left(\xi \right)=\frac{{k}_{2}}{2f}+\frac{{C}_{2}}{{C}_{1}+{C}_{2}\xi }.\)
Solution (4) \({k}_{2}=0, \, f={k}_{1}-{k}_{3}, \, g=f{k}_{4}>0,\) then \(\phi \left(\xi \right)=\frac{\sqrt{g}}{f}\frac{{C}_{1}sinh\left(\frac{\sqrt{g}}{{k}_{1}}\xi \right)+{C}_{2}cosh\left(\frac{\sqrt{g}}{{k}_{1}}\xi \right)}{{C}_{1}cosh\left(\frac{\sqrt{g}}{{k}_{1}}\xi \right)+{C}_{2}sinh\left(\frac{\sqrt{g}}{2{k}_{1}}\xi \right)}.\)
Solution (5) \({k}_{2}=0, \, f={k}_{1}-{k}_{3}, \, g=f{k}_{4}<0,\) then \(\phi \left(\xi \right)=\frac{{k}_{2}}{2f}+\frac{\sqrt{-g}}{f}\frac{-{C}_{1}sin\left(\frac{\sqrt{-g}}{{k}_{1}}\xi \right)+{C}_{2}cos\left(\frac{\sqrt{-g}}{{k}_{1}}\xi \right)}{{C}_{1}cos\left(\frac{\sqrt{-g}}{{k}_{1}}\xi \right)+{C}_{2}sin\left(\frac{\sqrt{-g}}{{k}_{1}}\xi \right)}.\)
Solution (6) \({k}_{4}=0, \, f={k}_{1}-{k}_{3},\) then \(\phi \left(\xi \right)=\frac{{C}_{1}{k}_{2}^{2}exp\left(\frac{-{k}_{2}}{{k}_{1}}\xi \right)}{f{k}_{1}+{C}_{1}{k}_{1}{k}_{2}exp\left(\frac{-{k}_{2}}{{k}_{1}}\xi \right)}.\)
Solution (7) \({k}_{2}\ne 0, \, f={k}_{1}-{k}_{3}=0,\) then \(\phi \left(\xi \right)=-\frac{{k}_{4}}{{k}_{2}}+{C}_{1}exp\left(\frac{{k}_{2}}{{k}_{1}}\xi \right).\)
Solution (8) \({k}_{2}=0, \, {k}_{1}={k}_{3}, \, f={k}_{1}-{k}_{3}=0,\) then \(\phi \left(\xi \right)={C}_{1}+\frac{{k}_{4}}{{k}_{1}}\xi .\)
Solution (9) \({k}_{2}=0, \, {k}_{3}={2k}_{1}, \, {k}_{4}=0,\) then \(\phi \left(\xi \right)=-\frac{1}{{C}_{1}+\left(\frac{{k}_{3}}{{k}_{1}}-1\right)\xi }.\)
When solution (3) is written in solution (2), a linear algebraic equation system is obtained; then, the system is solved by equalizing the coefficients of each of \({\left(p+\phi \left(\xi \right)\right)}^{k}\) and \({\left(p+\phi \left(\xi \right)\right)}^{-k}\) \(\left(k=\mathrm{0,1},2,\dots \right)\) in the system; and \({d}_{0},{d}_{k},{e}_{k}\) \(\left(k=\mathrm{1,2},\dots \right),\) \({k}_{1},{k}_{2},{k}_{3},{k}_{4}\) and \(p\) values are found. If these coefficients are written in the solutions of Eq. (4), then the solutions of Eq. (1) is obtained.
3 Applications
Example 1
In order to apply the GEM to a single nonlinear equation, let’s firstly consider the 3DNLSE (Wazwaz and Mehanna 2021).
Upon applying the subsequent transformation to this equation, whereby
we acquire the following ODE
Here \({T}_{1}=\left(-{d}_{4}{h}_{1}+{h}_{1}^{2}-{d}_{5}{h}_{2}-{d}_{1}{h}_{1}{h}_{2}+{h}_{2}^{2}-{d}_{6}{h}_{3}-{d}_{3}{h}_{1}{h}_{3}-{d}_{2}{h}_{2}{h}_{3}+{h}_{3}^{2}-{h}_{4}\right), {T}_{2}=\left({d}_{1}+{d}_{2}+{d}_{3}-3\right)\) and \({h}_{5}=\left(-{d}_{4}-{d}_{5}-{d}_{6}+2{h}_{1}-{d}_{1}{h}_{1}-{d}_{3}{h}_{1}+2{h}_{2}-{d}_{1}{h}_{2}-{d}_{2}{h}_{2}+2{h}_{3}-{d}_{2}{h}_{3}-{d}_{3}{h}_{3}\right)\). If \({U}^{{\prime}^{\prime}}\) and \({U}^{3}\) are balanced in Eq. (7), \(m=1\) is obtained. The solution function in this instance is as follows:
In the solution expressed by (8) \({d}_{0}, \, {d}_{1}\) and \({e}_{1}\) are constants to be found. If solution (8) are substituted in Eq. (7), the system of algebraic equations that follows is acquired:
The following coefficients can be acquired by solving this system of algebraic equations.
Case 1:
Case 2:
Case 3:
Case 4:
Case 5:
The solutions of Eq. (5) are acquired as follows if the coefficients discovered by (9)–(13) are substituted in (6) together with the solutions of (8):
Solutions for Case 1:
When \({k}_{2}\ne 0, \, f={k}_{1}-{k}_{3}, \, {s}_{1}={k}_{2}^{2}+4{k}_{4}\left({k}_{1}-{k}_{3}\right)>0,\)
If we take \({C}_{1}\ne 0, {C}_{2}=0\) in (14), then we discover an additional equation solution, which is as follows:
Again, if \({C}_{1}=0, {C}_{2}\ne 0\) in (15), then we discover an additional equation solution, which is as follows:
When \({k}_{2}\ne 0, f={k}_{1}-{k}_{3}, {s}_{1}={k}_{2}^{2}+4{k}_{4}\left({k}_{1}-{k}_{3}\right)<0,\)
In another instance, if \({C}_{1}\ne 0, \, {C}_{2}=0\) in (17), we find another equation solution in the manner described below:
Again, if \({C}_{1}=0, \, {C}_{2}\ne 0\) in (17), we find another equation solution in the manner described below:
Solutions for Case 2:
When \({k}_{2}\ne 0, \, f={k}_{1}-{k}_{3}=0,\)
Solutions for Case 3:
When \({k}_{2}=0, \, f={k}_{1}-{k}_{3}, \, g=f{k}_{4}>0,\)
If \({C}_{1}\ne 0, \, {C}_{2}=0\) in (21), then we discover an additional equation solution, which is as follows:
Again, if \({C}_{1}=0, \, {C}_{2}\ne 0\) in (21), we discover an additional equation solution, which is as follows:
When \({k}_{2}=0, f={k}_{1}-{k}_{3}, g=f{k}_{4}<0,\)
In another instance, if \({C}_{1}\ne 0, {C}_{2}=0\) in (24), we acquire another equation solution in the manner described below:
Again, if \({C}_{1}=0, {C}_{2}\ne 0\) in (24), we acquire another solution of the equation below:
Solutions for Case 4:
When \({k}_{2}=0, f={k}_{1}-{k}_{3}, g=f{k}_{4}>0,\)
If \({C}_{1}\ne 0, {C}_{2}=0\) in (27), we acquire another solution of the equation as follows:
Again, if \({C}_{1}=0, {C}_{2}\ne 0\) in (27), we acquire another solution of the equation below:
When \({k}_{2}=0, f={k}_{1}-{k}_{3}, g=f{k}_{4}<0,\)
In another instance, if \({C}_{1}\ne 0, {C}_{2}=0\) in (30), we acquire another solution of the equation as follows:
Again, if \({C}_{1}=0, {C}_{2}\ne 0\) in (30), we acquire another solution of the equation below:
Solutions for Case 5:
When \({k}_{2}=0, f={k}_{1}-{k}_{3}, g=f{k}_{4}>0,\)
If \({C}_{1}\ne 0, {C}_{2}=0\) in (33), we acquire another solution of the equation as follows:
Again, if \({C}_{1}=0, {C}_{2}\ne 0\) in (33), we acquire another solution:
When \({k}_{2}=0, f={k}_{1}-{k}_{3}, g=f{k}_{4}<0,\)
In another instance, if \({C}_{1}\ne 0, {C}_{2}=0\) in (36), we acquire other solution:
Again, if \({C}_{1}=0, {C}_{2}\ne 0\) in (36), we acquire another solution:
Note: In the solutions above;
Example 2
Now, let’s consider the CNLHE (Biswas, Aceves 2001).
The nonlinear Helmholtz equation play a vital role in modeling a progressive miniaturization of photonic devices and plasmonics; affecting also the amplitude and phase modulation of a continuous wave. Physically, the CNLHE Eq. (39) describes transverse effects in nonlinear optical systems and also models the transmission of coupled wave packets and optical solitons in nonlinear optical fibers (Abraham et al. 1990). In this context, Yu et al. (2019) considered a (3 + 1)-dimensional coupled nonlinear Helmholtz equation with constant coefficients, to construct soliton solutions of the model using the Hirota method. In this paper, we apply the GEM to Eq. (39) in order to derive traveling- and solitary-wave solutions when this equation undergoes the subsequent transformation, in which
Substituting (40) in (39), separating the real and imaginary parts, we get
If the \(V=\beta U\) transform is made for (41), \(\beta\) is a constant and equals one. Equation (40) converts to
From (43), \({m}_{4}=\frac{-{m}_{1}}{1+2\alpha {m}_{2}}.\) Also, \({\Gamma }_{1}=\left({m}_{1}^{2}+2{m}_{2}+2\alpha {m}_{2}^{2}\right)\), \({\Gamma }_{2}=\left(-{m}_{3}^{2}-2\alpha {m}_{3}^{2}{m}_{4}^{2}\right)\), \({\Gamma }_{3}=\left(-2{\sigma }_{1}-2{\sigma }_{2}\right).\) When \({U}^{{\prime}^{\prime}}\) and \({U}^{3}\) in (42) are balanced, \(m=1\) is achieved. The solution function in this instance is as follows:
In the solution expressed by (44) \({d}_{0},{d}_{1}\) and \({e}_{1}\) are constants to be found. Substituting solution (44) into problem (42) produces the algebraic equation system that follows:
From this system we get the following coefficients:
Case 1:
Case 2:
Case 3:
Solutions for Case 1: When \({k}_{2}\ne 0, f={k}_{1}-{k}_{3}=0,\)
Solutions for Case 2: When \({k}_{2}=0, f={k}_{1}-{k}_{3}, g=f{k}_{4}>0,\)
If \({C}_{1}\ne 0, {C}_{2}=0\) in the solution expressed by (47) and (48), here is how we arrive to a different equation solution:
If \({C}_{1}=0, {C}_{2}\ne 0\) in the solution expressed by (47) and (48), here is how we arrive to a different equation solution:
When \({k}_{2}=0, f={k}_{1}-{k}_{3}, g=f{k}_{4}<0,\)
If \({C}_{1}\ne 0, {C}_{2}=0\) in the solution expressed by (53) and (54), here is how we arrive to a different equation solution:
If \({C}_{1}=0, {C}_{2}\ne 0\) in the solution expressed by (53) and (54), here is how we arrive to a different equation solution:
Solutions for Case 3: When \({k}_{2}\ne 0, f={k}_{1}-{k}_{3}, {s}_{1}={k}_{2}^{2}+4{k}_{4}\left({k}_{1}-{k}_{3}\right)>0,\)
If \({C}_{1}\ne 0, {C}_{2}=0\) in the solution expressed by (59) and (60), another solution to the equation can be acquired as follows:
If \({C}_{1}=0, {C}_{2}\ne 0\) in the solution expressed by (59) and (60), another solution to the equation can be acquired as follows:
When \({k}_{2}\ne 0, f={k}_{1}-{k}_{3}, {s}_{1}={k}_{2}^{2}+4{k}_{4}\left({k}_{1}-{k}_{3}\right)<0,\)
If \({C}_{1}=0, {C}_{2}\ne 0\) in the solution expressed by (65) and (66), another solution to the equation can be acquired as follows:
If \({C}_{2}=0, {C}_{1}\ne 0\) in the solution expressed by (65) and (66), another solution to the equation can be acquired as follows:
where
4 Numerical simulations
In this section, graphs of some acquired solutions are given. When drawing the 3D-graphic and contour graphic for solution (15), values are in the range 0 \(\le x,t\le 3\); while for the 2D-graphic drawings, values in the range of \(0\le x\le 10\) are taken into account [see Fig. 1]. The 3D-graphic and contour graphic for solution (23), have values in the range of − 5 \(\le x,t\le 5\) and \(-10\le x\le 10\) [see Fig. 2]. The range − 3 \(\le x,t\le 3\) were taken into account when drawing 2D- and 3D-graphics for solution (28), solution (33) and solution (38) [see Figs. 3, 4 and 5, respectively]. When illustrating 3D-graphic and contour graphic for solutions (45), (53) and (61), values in the range of − 2 \(\le x,t\le 2,\) and \(-10\le x\le 10\) for the 2D-graphic, were taken into account while drawing [see Figs. 6, 7 and 8, respectively].
The 3D surfaces of Eq. (15) for \({h}_{1}=-1, \, {h}_{2}=-2, \, {h}_{3}=-1, \, {h}_{4}=-3, \, {d}_{1}=1, \, {d}_{2}=-1, \, {d}_{3}=-1, \, {d}_{4}=-3, \, {d}_{5}=2, \, {d}_{6}=-2, \, {k}_{1}=1, \, {k}_{2}=-1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, y=0, \, z=0.\) The 2D surfaces of Eq. (15) for \({h}_{1}=-1, \, {h}_{2}=-2, \, {h}_{3}=-1, \, {h}_{4}=-3, \, {s}_{1}=1, \, {d}_{1}=1, \, {d}_{2}=-1, \, {d}_{3}=-1, \, {d}_{4}=-3, \, {d}_{5}=2, \, {d}_{6}=-2, \, {k}_{1}=1, \, {k}_{2}=-1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, t=1, \, y=0, \, z=0.\) The Contour surfaces (CS) of Eq. (15) for \({h}_{1}=-1, \, {h}_{2}=-2, \, {h}_{3}=-1, \, {h}_{4}=-3, \, {s}_{1}=1, \, {d}_{1}=1, \, {d}_{2}=-1, \, {d}_{3}=-1, \, {d}_{4}=-3, \, {d}_{5}=2, \, {d}_{6}=-2, \, {k}_{1}=1, \, {k}_{2}=-1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, y=0, \, z=0\)
The 3D surfaces of Eq. (23) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=-3, \, {k}_{4}=2, \, s=4, \, y=0, \, z=0.\) The 2D surfaces of Eq. (23) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=-3, \, {k}_{4}=2, \, s=4, \, t=1, \, y=0, \, z=0.\) The CS of Eq. (23) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=-3, \, {k}_{4}=2, \, s=4, \, y=0, \, z=0\)
The 3D surfaces of Eq. (28) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=-2, \, s=4, \, {C}_{1}=1, \, {C}_{2}=2, \, y=0, \, z=0.\) The 2D surfaces of Eq. (28) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=-4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=-2, \, s=4, \, {C}_{1}=1, \, {C}_{2}=2, \, t=1, \, y=0, \, z=0.\) The CS of Eq. (28) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=-4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=-2, \, s=4, \, {C}_{1}=1, \, {C}_{2}=2, \, y=0, \, z=0\)
The 3D surfaces of Eq. (33) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, {C}_{1}=1, \, {C}_{2}=2, \, y=0, \, z=0.\) The 2D surfaces of Eq. (33) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4 \, , {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, {C}_{1}=1, \, {C}_{2}=2,, \, t=1, \, y=0, \, z=0.\) The CS of Eq. (33) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, {C}_{1}=1, \, {C}_{2}=2, \, y=0, \, z=0\)
The 3D surfaces of Eq. (38) for \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, y=0, \, z=0.\) The 2D surfaces of Eq. (38) \({h}_{1}=1, \, {h}_{2}=2, \, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, 3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, y=0, \, z=0, \, t=1, \, y=0, \, z=0.\) The CS of Eq. (38) \({h}_{1}=1, \, 2, {h}_{3}=1, \, {h}_{4}=4, \, {d}_{1}=-3, \, {d}_{2}=1, \, {d}_{3}=1, \, {d}_{4}=3, \, {d}_{5}=2, \, {d}_{6}=2, \, {k}_{1}=1, \, {k}_{2}=1, \, {k}_{3}=3, \, {k}_{4}=2, \, s=4, \, y=0, \, z=0\)
The 3D surfaces of Eq. (45) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4}=4, \, {k}_{1}=1, \, {k}_{4}=-2, \, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, {C}_{1}=1, \, \alpha =1, \, \beta =1.\) The 2D surfaces of Eq. (45) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4}=4, \, {k}_{1}=1, \, {k}_{4}=-2, \, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, {C}_{1}=1, \, \alpha =1, \, \beta =1.t=1.\) The CS of Eq. (45) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4}=4, \, {k}_{1}=1, \, {k}_{4}=-2, \, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, {C}_{1}=1, \, \alpha =1, \, \beta =1\)
The 3D surfaces of Eq. (53) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4} \, =-2, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, {C}_{1}=1, \, {C}_{2}=2, \, \alpha =1, \, \beta =1.\) The 2D surfaces of Eq. (53) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4}=4, \, {{k}_{3}=1, \, k}_{4}=-2, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, {C}_{1}=1, \, {C}_{2}=2, \, \alpha =1, \, \beta =1, \, t=1.\) The CS of Eq. (53) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4}=4, \, {{k}_{3}=1, \, k}_{4}=-2, \, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, {C}_{1}=1, \, {C}_{2}=2, \, \alpha =1, \, \beta =1\)
The 3D surfaces of Eq. (61) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4}=4, \, {k}_{2}=2, { \, {k}_{3}=1, \, k}_{4}=-2, \, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, \alpha =1, \, \beta =1.\) The 2D surfaces of Eq. (61) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4}=4, \, {k}_{2}=2, {{k}_{3}=1, \, k}_{4}=-2, \, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, \alpha =1 \, , \beta =1, \, t=1.\) The CS of Eq. (61) for \({m}_{1}=1, \, {m}_{2}=-2, \, {m}_{3}=1, \, {m}_{4}=4, \, {k}_{2}=2, \, {{k}_{3}=1, \, k}_{4}=-2, \, {\sigma }_{1}=2, \, {\sigma }_{2}=2, \, \alpha =1, \, \beta =1\)
We can notice from the 3D-graphic in Figs. 2, 3, 7 and 8, the singular kink type soliton, the kink/anti-kink profile and singular bell shape solitons, respectively. This observation implies that solution (23) and (28) are evidently solitary-wave solutions. It is obvious to notice that, the kink soliton can be used in practical applications, as polarization switches between two different domains or optical logic units (Huang 2012).
5 Modulation instability
In this section, we will apply the standard linear stability analysis (Agrawal 2013) on the 3DNLSE (5) and the CNLHE (39):
5.1 The (3 + 1)-dimensional Schrödinger differential equation
We consider for the 3DNLSE (5) solutions in the following form:
where \({U}_{0}\) denotes the power of the stationary solution and \({\phi }_{0}\) its phase shift given by \({\phi }_{0}=s{{U}_{0}}^{2}\). In order to perform the stability analysis of the steady-state solution, we introduce small disturbance in Eq. (71) as follows:
where \(\eta \left(x,y,z,t\right)\) represents the complex amplitude of the perturbation with \(\left|\eta \left(x,y,z,t\right)\right|\ll {U}_{0}\). Thus, if the perturbed field grows exponentially, the steady-state becomes unstable.
Plugging Eq. (72) into Eq. (5) yields the following linearized equation:
where \({\eta }^{*}\) indicates the complex conjugate of \(\eta\). We assume for Eq. (73) the following wave plane ansatz
where \({K}_{x}\), \({K}_{y}\) and \({K}_{z}\) are real perturbation wave-numbers, \(\Omega\) stands for the complex frequency and \({F}_{1}\), \({F}_{2}\) are complex numbers. Inserting Eq. (74) into Eq. (73) yields the following homogeneous equations
To get a nontrivial solution of Eq. (75), we must have
Equation (76) determines the stability of the steady-state. If \(\Omega\) is real, the steady-state is stable against small disturbances. When \(\Omega\) has an imaginary part, steady-state solution is unstable because the disturbance rises exponentially. As we can see from Eq. (76), the MI depends on parameters \(s\), \({U}_{0}\), \({d}_{j}\) and \({K}_{l}\) and it occurs when
MI gain spectrum is obtained as
In Fig. 9, we illustrate the MI gain behaviours under various cobinations of system parameters. We first set \({d}_{1}=1, {d}_{2}=1, {d}_{3}=1, {d}_{4}=1,{d}_{5}=1, {d}_{6}=1, {K}_{x}={K}_{y}={K}_{x}=K\) and plot the MI gain versus the power \({U}_{0}\) under various combinations of the wave-number \(K\). One can see in Fig. 9a, as the wave number \(K\) increases (\(K=0.5, K=1\), and \(K=2\)), the magnitude of instability side lobe increases with the power \({U}_{0}\) for \(s=0.1\). When we fix \(K=2\) and manage the cubic nonlinearity parameter as \(s=0.1, s=0.3,\) and \(s=1\), the MI spectrum in Fig. 9b illustrates side-lobes with increase width in \({U}_{0}\), and constant maximum gain \(g\). This implies that the cubic nonlinearity impact is related to the power \({U}_{0}\). When we consider MI gain behaviors versus the wave number \(K\), and varying the cubic nonlinearity coefficient as \(s=0.01, s=0.05,\) and \(s=0.1\), Fig. 9c exhibits the monotonous increasing of gain spectrum with the increase \(s\). Figure 9d illustrates the MI gain as function of the cubic nonlinearity \(s\) for three values of the wave number \(K\) notably \(K=0.5, K=1\), and \(K=3\). We notice that, as the wave number \(K\) increases, the maximum also gain increases.
The 2D plot showing the variation of MI gain (78) for \({d}_{1}=1, {d}_{2}=1, {d}_{3}=1, {d}_{4}=1,{d}_{5}=1, {d}_{6}=1, {K}_{x}={K}_{y}={K}_{x}=K\): a The MI gain versus the power \({U}_{0}\), under the variation of the wave-number \(K\) (\(K=0.5, K=1\), and \(K=2\)) with \(s=0.1\); b The MI gain versus the power \({U}_{0}\) under the variation of the cubic nonlinearity \(s\) (\(s=0.1, s=0.3,\) and \(s=1\)) with \(K=1\); c The MI gain versus the wave-number \(K\), under the influence of the cubic nonlinearity \(s\) (\(s=0.01, s=0.05,\) and \(s=0.1\)) with \({U}_{0}=10\) and \(K=1\); d The MI gain versus the cubic nonlinearity \(s\), under the influence of the wave-number \(K\) (\(K=0.5, K=1\), and \(K=3\)) with \({U}_{0}=10\)
In order to compare the accuracy of the analytical and numerical results, we illustrate the numerical evolution of some obtained solutions of the 3DNLSE and we assess them to the cubic nonliearity influence. In Fig. 10, the numerical evolution of the periodic-wave \({u}_{7}\) (20) is established when \(y=z=0\), \({h}_{1}=1, {h}_{2}=2, {h}_{3}=1, {h}_{4}=4, {d}_{1}=-3, {d}_{2}=1, {d}_{3}=1, {d}_{4}=3,{d}_{5}=2,{d}_{6}=2,{k}_{1}=1,{k}_{2}=1,{k}_{3}=3,{k}_{4}=-2,s=4,{C}_{1}=1,\) and \({C}_{2}=2\). For the same parametric values, we plot in Fig. 11 the Dark solitary-wave solution \({u}_{15}\) (28) of the 3DNLSE.
Considering three values of the cubic nonlinearity \(s\), notably \(s=0.1\), \(s=0.3\), and \(s=1\), one can deduct from Fig. 12 fact that, as the cubic nonlinearity coefficient increases, the soliton intensity decreases. This implies that, cubic nonlinearity coefficient \(s\) perturbs significantly soliton propagation and consequently promotes instability, as noticed in the MI investigations.
5.2 The coupled nonlinear Helmholtz equation
We consider now solutions of Eq. (39) as the following steady-state anstatz:
where \(P\) and \(Q\) are input powers and the phase of the steady-state solutions depends on the powers and reads as \(\varphi =\frac{1\pm \sqrt{1+4\alpha \left(P{\sigma }_{1}+Q{\sigma }_{2}\right)}}{2\alpha }\). Introducing an infinitesimal disturbances \({a}_{1}\left(t,x\right)\) and \({a}_{2}\left(t,x\right)\), we perturb the steady-state solutions (79) as
with \(\left|{a}_{1}\left(t,x\right)\right|\ll \sqrt{P}\) and \(\left|{a}_{2}\left(t,x\right)\right|\ll \sqrt{Q}\).
Plugging Eq. (80) into Eq. (39), we get after linearization, the following linearized equation
Assuming for the perturbations plane wave ansatz solutions as
and inserting Eq. (82) into Eq. (81) yields the following \(4\times 4\) coupled equations satisfed by \({G}_{j}\) and \({H}_{j}\):
where:
The solvability condition of the previous equations is obtained by requiring the determinant of the associated matrix Eq. (82) to be zero. It results in the following dispersion relation satisfayed by \(K\) as:
The eight roots of the polynomial in \(K\) acquired in Eq. (83) determine the stability of the continuous wave solution. The MI process is measured by a power gain given by
where \(Im\left({K}_{max}\right)\) denotes imaginary part of the root of the polynomial with the largest value \({K}_{max}\).
In Fig. 13, we plot in a 3D-graph the MI gain (84) as function of frequency \(\Omega\) and power \(P\), for the CNLHE (39). We start the discussion by variying the nonparaxial parameter \(\alpha\). As we can see from Fig. 13a to Fig. 10e (for a \(\alpha =0.01\), b \(\alpha =0.1\), c \(\alpha =0.5\), d \(\alpha =1.5\), and e \(\alpha =2\) respectively), as the coefficent \(\alpha\) increases, the instability sidebands diminishes. So, the system becomes more stable.
The 3D plot showing the variation of MI gain (84) versus the frequency \(\Omega\) and power \(P\), under the influence of \(\alpha\) for \({\sigma }_{1}=1, {\sigma }_{2}=2, Q=10\): a \(\alpha =0.01\), b \(\alpha =0.1\), c \(\alpha =0.5\), d \(\alpha =1.5\), and e \(\alpha =2\)
Now, we manage the cubic nonlinearity coefficients \({\sigma }_{1}\) and \({\sigma }_{2}\) in order to analyze their impact on MI gain spectrum. When \({\sigma }_{1}=1, {\sigma }_{2}=0.2\), the MI gain in Fig. 14a exhibits symmetrical side lobes of instability who are present for all values of the power \(P\). As the power increases, the maximum gain increases. For a focusing cubic nonlinearity \({\sigma }_{1}<0\) and defocusing cubic nonlinearity \({\sigma }_{2}>0\), the is present only near the central frequency \(\Omega =0\) as found in Fig. 14b. Considering the case of defocusing cubic nonlinearities, the MI gain in Fig. 14c is not quite different as that found in Fig. 13c. For \({\sigma }_{1}>0\) and \({\sigma }_{2}<0\), Fig. 14d illustrates two symmetrical side lobes of instability with a no nil value at the central frequency \(\Omega =0\). When \({\sigma }_{1}=0.1, {\sigma }_{2}=1\), Fig. 14e is not quite different to the case in Fig. 14a for \({\sigma }_{1}>{\sigma }_{2}\), but here, the maximum gain remains constant with the increasing power \(P\). We conclude that, the development of MI can be controlled by the management of the cubic nonlinearity coefficients.
The 3D plot showing the variation of MI gain (84) versus the power \(P\) and frequency \(\Omega\), under the influence of cubic nonlinearities (\({\sigma }_{1}\) and \({\sigma }_{2}\)) for \(\alpha =0.01, Q=10\): a \({\sigma }_{1}=1, {\sigma }_{2}=0.2\), b \({\sigma }_{1}=-1, {\sigma }_{2}=0.2\), c \({\sigma }_{1}=-1, {\sigma }_{2}=-0.2\), d \({\sigma }_{1}=1, {\sigma }_{2}=-0.2\), and e \({\sigma }_{1}=0.1, {\sigma }_{2}=1\)
The profile evolution of the CNLHE Dark-solitary wave solution \({u}_{2}\) (47), when \(y=z=0\), \({m}_{1}=1, {m}_{2}=-2, {m}_{3}=1, {m}_{4}=4, {{k}_{3}=1,k}_{4}=-2, {\sigma }_{1}=0.1,{\sigma }_{2}=0.5,{C}_{1}=1,{C}_{2}=2,\alpha =1.5, \beta =1\)
Now, in order to compare the accuracy of the analytical and numerical results of the CNLHE (39), we illustrate the numerical evolution of the Dark-solitary wave \({u}_{2}\) (47) and we assess it to the non-paraxial parameter \(\alpha\) and the cubic nonliearities (\({\sigma }_{1}\) and \({\sigma }_{2})\) influences. In Fig. 15, the numerical evolution of the periodic-wave \({u}_{7}\) (20) is illustrated when \(y=z=0\), \({m}_{1}=1, {m}_{2}=-2, {m}_{3}=1, {m}_{4}=4, {{k}_{3}=1,k}_{4}=-2, {\sigma }_{1}=0.1,{\sigma }_{2}=0.5,{C}_{1}=1,{C}_{2}=2,\alpha =1.5,\) and \(\beta =1\). We observe a quasi-similar profile as that in Fig. 11 for the 3DNLSE Dark-solitary wave \({u}_{15}\) (28).
Considering three values of the non-paraxial parameter \(\alpha\), particularly \(\alpha =0.5\), \(\alpha =1.5\), and \(\alpha =2\). We can see that in Fig. 16a, as the non-paraxial coefficent \(\alpha\) increases, the soliton intensity increases whereas the instability sidebands diminishes as established by the stability analysis in Fig. 13. In Fig. 16b for three combinations of cubic nonlinearities \({\sigma }_{1}\) and \({\sigma }_{2}\), namely \({\sigma }_{1}=1, {\sigma }_{2}=0.2\), \({\sigma }_{1}=1, {\sigma }_{2}=-0.2\), and \({\sigma }_{1}=0.1, {\sigma }_{2}=1\) we can see the promotion of system stability’s by the increasing dark soliton power, when \({\sigma }_{1}{\sigma }_{2}<0\) (see the solid green line). From the previous observations, we can assure the accuracy between the analytical and numerical results.
Influence of the system parameters on the CNLHE Dark-solitary wave \({u}_{2}\) given by Eq. (47), under various combinations of the non-paraxial parameter \(\alpha\) and the cubic nonlinearities \({\sigma }_{1}\) and\({\sigma }_{2}\). The other parameter values are the same as those in Fig. 15. a Influence of the non-paraxial parameter \(\alpha\) (\(\alpha =0.5\),\(\alpha =1.5\), and\(\alpha =2\)); b (a) Influence of the cubic nonlinearities \({\sigma }_{1}\) and \({\sigma }_{2}\) (\({\sigma }_{1}=1,{\sigma }_{2}=0.2\);\({\sigma }_{1}=1,{\sigma }_{2}=-0.2\); and\({\sigma }_{1}=0.1,{\sigma }_{2}=1\))
6 Conclusion
In this study, we used the GEM to provide some exact traveling wave solutions of 3DNLSE and CNLHE. We were able to derive optical, exponential, trigonometric, and hyperbolic solutions. Then, using Mathematica 11.2 we saw that these answers gave the equations. Aside from that, we’ve displayed some of the solutions’ graphic performance. Numerous other nonlinear equations and coupled equations can be solved using this technique. The numerical simulations we conducted in the current work also demonstrate the robustness and stable propagation of the generated solitons in a perturbed environment, hence proving the accuracy of the analytical and numerical results. The outcomes may be used as a guide for the production of solitons and ultra-short optical pulses in the telecom sector.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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Acknowledgements
The authors would like to extend their sincere appreciation to Researchers Supporting Project number (RSP2024R472), King Saud University, Riyadh, Saudi Arabia.
Funding
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HA, MY: software, data curation, writing, formal analysis. MY, MI: data curation, software, formal analysis, writing. IEI: writing—reviewing editing, ınvestigation. HB: conceptualization, supervision, writing—reviewing editing, validation.
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Alsaud, H., Youssoufa, M., Inc, M. et al. Some optical solitons and modulation instability analysis of (3 + 1)-dimensional nonlinear Schrödinger and coupled nonlinear Helmholtz equations. Opt Quant Electron 56, 1138 (2024). https://doi.org/10.1007/s11082-024-06851-4
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DOI: https://doi.org/10.1007/s11082-024-06851-4