Nonparaxial pulse propagation to the cubic–quintic nonlinear Helmholtz equation

1. Introduction

In this paper, we are concerned with a semiconductor and doped fiber for studying dynamics of a broad range of physical systems to the cubic–quintic Helmholtz (CQH) equation. The following dimensionless CQH equation can be utilized to specify the nonparaxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities^1,2,3,4 as follows :

where $\Re =\Re (x,t)$ depicts nonparaxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities.¹ Subsequently, researchers also provided theoretical support for this equation and proved it,^5,6,7 which made great progress in the solution of solitary wave and attracted the attention of a large number of researchers.

The nonlinear Schrödinger equation naturally describes the pulse propagations in optical fibers, as explained in Ref. 8. The symmetry reductions of the NLH equation have been reached by the Lie symmetry method.⁹ In Ref. 10, author analytically studied the Schrödinger-type nonlinear evolution equations by improved $tan(\mathrm{\Phi }∕2)$-expansion method. Moreover, the propagation of nonparaxial solitons has been studied in diverse nonlinear optical environments including Kerr,¹¹ cubic–quintic,¹² power-law¹³ and saturable nonlinear media.¹⁴

The vector soliton is produced when the nonlinearity of the fiber causes coupling between diverse optical modes during the propagation in the multimode optical fiber, which vector soliton provides an efficient way to a variety of practical applications such as channel wavelength division-multiplexing, pulse generation and high-speed optical switching.¹⁵ Soliton interactions can be divided into two types: coherent interactions and incoherent interactions.¹⁶

In the last decades, researchers have developed numerous methods such as the multiple Exp-function scheme,¹⁷ the variety interaction between k-lump and k-kink solutions,¹⁸ Hirota’s bilinear method,^19,20,21 a two-step singularly P-stable method,²² a deep learning for Feynman’s path integral,²³ a nonlinear vibration isolator supported on a flexible plate,²⁴ the extended homoclinic breather wave solutions,²⁵ a global stabilization of the full attraction-repulsion Keller–Segel system,²⁶ a Lie symmetry analysis,²⁷ the inverse scattering transformation method,²⁸ one-dimensional (1D) BBM equation and 2D coupled BBM-system by finite element method,²⁹ a ($3+1$)-dimensional variable coefficient generalized nonlinear wave equation in a Liquid with Gas Bubbles,³⁰ a generalized Hietarinta equation with Hirota’s bilinear operator,³¹ the ($3+1$)-dimensional variable coefficients generalized shallow water wave equation,³² the modulation instability analysis for a metamaterials model via the integration schemes³³ and so forth.^34,35,36

Biswas–Milovic model with variable coefficients comprising Kerr law and damping effect was studied by Kaur and Wazwaz.³⁷ The exact solutions of space–time fractional Calogero–Degasperis equation and space–time fractional Sharma–Tasso–Olver equation were obtained by using the generalized $G^{\prime }∕G$-expansion method in Ref. 38. A new nonlinear integrable fifth-order equation has been developed and was examined to explore the movable critical points by using Painlevé test.³⁹ The exact solutions for coupled nonlinear Hirota equation and coupled nonlinear Helmholtz equation were found via the $exp(-\mathrm{\Phi }$)-expansion method.⁴⁰ Einstein’s vacuum field equation for exploring movable critical points was investigated by using the Painlevé analysis and the auto-Bäcklund transformation.⁴¹ There are some recent developments in the field of the soliton solutions as well as their applications including Hirota bilinear method⁴²; Exponential function approach⁴³; robust optimization technique⁴⁴; Robust optimization-based optimal chiller loading⁴⁵; demand response and improved water wave optimization algorithm⁴⁶; and optimal modeling of combined cooling⁴⁷; the correlation of random variables with Copula theory⁴⁸; probabilistic decomposition-based security constrained transmission.⁴⁹ In order to compute the favorite solutions and better understand the fundamental characteristics of physical structures in varied contexts, a variety of techniques, as a result, the analytical or numerical methods, have been developed and it has been shown that no single technique can be used to solve all types of nonlinear problems with precision. Therefore, many different methods have emerged, some of which are the estimation of seismic wave attenuation,⁵⁰ compressional and shear wave attenuations,⁵¹ propagation of waves with a wide range,⁵² sector beam synthesis in linear antenna arrays,⁵³ design of environmental monitoring system,⁵⁴ design and simulation of grid-connected photovoltaic system,⁵⁵ a comparative study of the hardness and force analysis methods,⁵⁶ prediction of motion simulator signals,⁵⁷ a linear time-varying model predictive control-based motion cueing algorithm,⁵⁸ derivation of optimized equations for estimation of dispersion coefficient in natural streams.⁵⁹

In the context of water wave theory, a complete study on the related physical systems is executed by exploring several integrable as well as nonintegrable evolution equations in one and higher dimensions.⁶⁰ Integrability is a fascinating property to characterize any dynamical models in addition to the existence of Lax pair and infinitely many conserved quantities. The ancillary techniques containing direct algebraic techniques, auxiliary equation method, Kudryashov expansion technique, Riccati–Bernoulli sub-ordinary differential equation (ODE) scheme, sinh-Gordon expansion method, cosh–tanh method, simplest equation method, and so on are utilized widely to get different classes of traveling wave solutions.⁶¹ Especially, these methodologies provide a variety of exotic wave patterns, including solitons, breathers, lumps, dromions, rogue waves and elliptic waves. On the advantageous part, the Hirota bilinear method is an intermediate tool which can be utilized to extract the localized nonlinear wave solution to most of the integrable as well as a few class of nonintegrable soliton models and it becomes a widely used tool to obtain several localized nonlinear wave solutions.⁶²

For extracting accurate solutions for nonlinear evolution equations, the generalized $G^{\prime }∕G$-expansion method has been considered the most appropriate approach. These strategies are simple, easy, effective and they may be used to find solitary wave solutions for variety of nonlinear partial differential equations (NLPDEs). These methods are beneficial because they may be used to solve problems with a big balance number. As a result of their importance, a range of solutions have been developed and implemented in this area. In recent decades, the nonlinear optics had been relying on soliton solutions, which are novel and may be utilized to explain physical and pictorial structures.

The main purpose of this study is to systematically construct a set of new exact solutions that have the physical properties of this new cubic–quintic nonlinear Helmholtz model, which passed the $G^{\prime }∕G$-expansion method. Moreover, in this paper, some solutions including soliton, bright soliton, singular soliton, periodic wave solutions by a generalized $G^{\prime }∕G$-expansion method were also obtained.

Inspired by the previous work, the aim of the paper is to investigate nonparaxial solitons and other forms of solutions. The outline of the paper is as follows. In Sec. 2, we transform the cubic–quintic nonlinear Helmholtz equation to a system of the nonlinear ordinary differential equation. Furthermore, in Sec. 3, different forms of solitary wave solutions have been established by a generalized $G^{\prime }∕G$-expansion technique. Finally, the conclusions are provided in Sec. 4.

2. Transforming PDE to ODE

In Eq. (1), $x,t$ show the longitudinal and transverse coordinates, respectively.¹ Also, the parameters ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ are illustrated and are given in Ref. 2. The defined parameters can be stated with the values ${\alpha }_1=1∕{\kappa }^2\omega$, ${\alpha }_2=\pm 1$, ${\alpha }_3=\pm 1$, ${\alpha }_4=-{\epsilon }_0^2{n}_4∕n_2$, where $\kappa ,\omega ,{\epsilon }_0,n_4,n_2$ are described in Refs. 3 and 4. Employing the next wave transformation $\Re =\Re (x,t)=exp(i{\eta }_2)\lambda ({\eta }_1)$, ${\eta }_1={\theta }_{3}x+{\theta }_{4}t+p_2$, ${\eta }_2={\theta }_{1}x+{\theta }_{2}t+p_1$, where ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,p_1,p_2$ are free values, to Eq. (1), changes the PDE into the following ODE :

Thus, by solving the second equation in the above, we get ${\theta }_3=-{\alpha }_2{\theta }_2{\theta }_4∕(1+2{\alpha }_1{\theta }_1)$. Exchanging the value of ${\theta }_3$ into real Eq. (1) in Eqs. (2) takes one to where $s_1=\frac{{\alpha }_4}{{\alpha }_3}$, $s_2=\frac{-2{\alpha }_1{{\theta }_1}^2-{\alpha }_2{{\theta }_2}^2-2{\theta }_1}{2{\alpha }_3}$, $s_3=\frac{2{{\theta }_3}^2{\alpha }_1+{{\theta }_4}^2{\alpha }_2}{2{\alpha }_3}$. Deponing up the balance principle to the terms of Eq. (3) gets to $k=1∕2$. Hence, we must use the another function such as $\lambda =F^{\frac{1}{2}}({\eta }_1)$. Inserting the new reached function into Eq. (3) takes one to Then, balancing the terms $F{F}^{\prime \prime }$ or $F^{\prime 2}$ with $F^4$ of Eq. (3) gets $k=1$. On this basis, we can get the following corresponding solitary solution :

3. Generalized G-Expansion Technique

Here, we further consider the generalized G-expansion technique to get the following steps as mentioned earlier:

	Step 1. $${{𝒮}}_1(\Re ,{\Re }_x,{\Re }_t,{\Re }_{xx},{\Re }_{tt},\dots )=0,$$(6) where ${𝒮}$ is a polynomial of $\Re$ and its their partial derivatives.
	Step 2. First, utilize the traveling wave transformation as follows : $$\xi ={\theta }_{1}x+{\theta }_{2}t+p_1,$$(7) where ${\theta }_1,{\theta }_2$ are the nonzero arbitrary values, which will allow us to diminish Eq. (6) to an ODE of $\Re =\Re (\xi )$ in the following form : $${{𝒮}}_2(\Re ,{\theta }_1{\Re }^{\prime },{\theta }_2{\Re }^{\prime },{\theta }_1^2{\Re }^{\prime \prime },{\theta }_2^2{\Re }^{\prime \prime },\dots )=0.$$(8)
	Step 3. The generated solutions of (6) are $$\Re (\xi )=\sum _{l=0}^k{A}_l\mathrm{\Phi }{(\xi )}^l,$$(9) where $A_m\ne 0$, and $\mathrm{\Phi }(\xi )=G^{\prime }(\xi )∕G(\xi )$ are satisfied as follows : $$m_{1}G{G}^{\prime \prime }-m_{2}G{G}^{\prime }-m_3{(G^{\prime })}^2-m_4{G}^2=0.$$(10) The particular solutions of Eq. (10) will be read as
	Group 1. When $m_2\ne 0$, $g=m_1-m_3$ and $\mathrm{\Delta }=m_2^2+4m_{4}g>0$, $\mathrm{\Phi }(\xi )=\frac{m_2}{2g}+\frac{\sqrt{\mathrm{\Delta }}}{2g}\frac{C_{1}sinh(\frac{\sqrt{\mathrm{\Delta }}}{2m_1}\xi )+C_{2}cosh(\frac{\sqrt{\mathrm{\Delta }}}{2m_1}\xi )}{C_{1}cosh(\frac{\sqrt{\mathrm{\Delta }}}{2m_1}\xi )+C_{2}sinh(\frac{\sqrt{\mathrm{\Delta }}}{2m_1}\xi )}.$
	Group 2. When $m_2\ne 0$, $g=m_1-m_3$ and $\mathrm{\Delta }=m_2^2+4m_{4}g<0$, $\mathrm{\Phi }(\xi )=\frac{m_2}{2g}+\frac{\sqrt{-\mathrm{\Delta }}}{2g}\frac{-C_{1}sin(\frac{\sqrt{-\mathrm{\Delta }}}{2m_1}\xi )+C_{2}cos(\frac{\sqrt{-\mathrm{\Delta }}}{2m_1}\xi )}{C_{1}cos(\frac{\sqrt{-\mathrm{\Delta }}}{2m_1}\xi )+C_{2}sin(\frac{\sqrt{-\mathrm{\Delta }}}{2m_1}\xi )}.$
	Group 3. When $m_2\ne 0$, $g=m_1-m_3$ and $\mathrm{\Delta }=m_2^2+4m_{4}g=0$, $\mathrm{\Phi }(\xi )=\frac{m_2}{2g}+\frac{C_2}{C_1+C_2\xi }.$
	Group 4. When $m_2=0$, $g=m_1-m_3$ and ${\mathrm{\Delta }}_1=4m_{4}g>0$, $\mathrm{\Phi }(\xi )=\frac{\sqrt{{\mathrm{\Delta }}_1}}{g}\frac{C_{1}sinh(\frac{\sqrt{{\mathrm{\Delta }}_1}}{2m_1}\xi )+C_{2}cosh(\frac{\sqrt{{\mathrm{\Delta }}_1}}{2m_1}\xi )}{C_{1}cosh(\frac{\sqrt{{\mathrm{\Delta }}_1}}{2m_1}\xi )+C_{2}sinh(\frac{\sqrt{{\mathrm{\Delta }}_1}}{2m_1}\xi )}.$
	Group 5. When $m_2=0$, $g=m_1-m_3$ and ${\mathrm{\Delta }}_1=4m_{4}g<0$, $\mathrm{\Phi }(\xi )=\frac{\sqrt{-{\mathrm{\Delta }}_1}}{g}\frac{-C_{1}sin(\frac{\sqrt{-{\mathrm{\Delta }}_1}}{2m_1}\xi )+C_{2}cos(\frac{\sqrt{-{\mathrm{\Delta }}_1}}{2m_1}\xi )}{C_{1}cos(\frac{\sqrt{-{\mathrm{\Delta }}_1}}{2m_1}\xi )+C_{2}sin(\frac{\sqrt{-{\mathrm{\Delta }}_1}}{2m_1}\xi )}.$
	Group 6. When $m_4=0$ and $g=m_1-m_3,$$\mathrm{\Phi }(\xi )=\frac{C_1{m}_2^{2}exp(\frac{-m_2}{m_1}\xi )}{g{m}_1+C_1{m}_1{m}_{2}exp(\frac{-m_2}{m_1}\xi )}.$
	Group 7. When $m_2\ne 0$ and $g=m_1-m_3=0,$$\mathrm{\Phi }(\xi )=-\frac{m_4}{m_2}+C_{1}exp(\frac{m_2}{m_1}\xi ).$
	Group 8. When $m_1=m_3$, $m_2=0$ and $g=m_1-m_3=0,$$\mathrm{\Phi }(\xi )=C_1+\frac{m_4}{m_1}\xi .$
	Group 9. When $m_3=2m_1$, $m_2=0$ and $m_4=0,$$\mathrm{\Phi }(\xi )=-\frac{1}{C_1+(\frac{m_3}{m_1}-1)\xi }$, where $A_l(l=1,\dots ,k),m_1,m_2,m_3$ and $m_4$ are constants to be determined later.
	Step 4. By balancing the nonlinear ODE, we can obtain the value k.
	Step 5. By solving the algebraic equations, we can get to the mentioned values in the above.

4. Nonparaxial Soliton Solution

It can be seen that the above governing differential equation is highly nonlinear, and such nonlinearity imposes some difficulties in the development of exact analytical methods to generate closed form solution for the equation. Therefore, a generalized G-expansion technique is used in this work. The Generalized G-expansion technique, which is an analytical scheme for providing analytical solutions to the nonlinear ordinary differential equations, is adopted in generating solutions to the ordinary nonlinear differential equations. Upon constructing the transformation and a new function, the following categories of solutions can be expressed as

The set of category of solutions:

4.1. Set I

As a result by (Groups 6, 9, 4), the kink soliton solution is given by

Figure 1 depicts the impact of analysis bright soliton solution where graphs of $\Re$ are given with the following values :

for Eq. (12). We investigate the dynamics of general kink solitons received from the mentioned method, which is presented in Fig. 1. From the figure, it is apparent that the solitons exhibit a stable propagation in both components of Helmholtz equation, as shown in Fig. 1. Also, Fig. 2 depicts the impact of analysis bright soliton solution where graphs of $\Re$ are given with the following values : for Eq. (14).

4.2. Set II

As a result by (Group 9), the capson solution is given by

4.3. Set III

As a result by (Groups 1, 3, 6, 7), the soliton, kink and singular solutions are produced by

Moreover, Fig. 3 depicts the impact of analysis soliton solution where graphs of $\Re$ are given with the following values :

for Eq. (22).

4.4. Set IV

As a result by (Groups 1, 6), the soliton solutions are produced by

4.5. Set V

As a result by (Groups 1, 2, 7), the soliton solutions are produced by when $m_2^2-4m_4{m}_3>0$ and when $m_2^2-4m_4{m}_3<0$ and $\xi =-{\alpha }_2{\theta }_2{\theta }_4∕(1+2{\alpha }_1{\theta }_1)x+{\theta }_{4}t+p_2$.

4.6. Set VI

As a result by (Group 9), the capson solution is given by

4.7. Set VII

As a result by (Groups 1, 2, 3), the soliton and periodic wave solutions are produced by

when $A_0{\alpha }_3{\alpha }_2({\alpha }_1{A}_0{\alpha }_3-1)>0$ and when $A_0{\alpha }_3{\alpha }_2({\alpha }_1{A}_0{\alpha }_3-1)<0$ and $\xi =-{\alpha }_2{\theta }_2{\theta }_4∕(1+2{\alpha }_1{\theta }_1)x+{\theta }_{4}t+p_2$. Figure 4 depicts the impact of analysis soliton solution where plots of $\Re$ are given with the following values : for Eq. (38). Moreover, Fig. 5 depicts the impact of analysis periodic wave solution where plots of $\Re$ are offered with the following values : for Eq. (39).

The analytical solutions are coded in Maple and the parametric and sensitivity analyses are carried out using the codes. The parametric results are presented in Figs. 1–5. The present results from the simulations show an inherent property of auxiliary parameters for the adjustment and control of region and rate of convergence of approximate series solutions. Also, a generalized $G^{\prime }∕G$-expansion technique has been proven, which is an efficient and capable technique in handling nonlinear engineering problems in wider ranges of parameters. The importance of this study lies in the fact that it can serve as a base for the experimental work that we undertake on the nonparaxial pulse propagation.

5. Conclusion

To conclude, we have analytically constructed the nonparaxial solitary wave solutions by utilizing the generalized $G^{\prime }∕G$-expansion technique in which the systems was nonintegrable. The impact of nonparaxiality on the physical parameters including speed and amplitudes of solitary waves was focused. In particular, we studied four forms of function solution including soliton, bright soliton, singular soliton, periodic wave solutions. To get this case, the demonstrative pattern of the cubic–quintic nonlinear Helmholtz equation was supplied to demonstrate the probability and dependability of the protocol utilized in this research. The effect of the free variables on the behavior of obtained plots of a few procured solutions for the nonlinear rational exact solutions was also analyzed. For a better understanding on the resulting dynamics, we provided a categorical discussion and clear graphical demonstration for solitons and periodic wave on both constant and spatially-varying backgrounds. Many solutions were obtained and represented in 2D, 3D, density and contour plots.