A generalized trial equation scheme: A tool for solving thin films constructed from the ferroelectric materials

1. Introduction

Numerous branches of nonlinear sciences including plasma physics, geochemistry, solid-state physics, fluid mechanics, optical fibers, nuclear physics and chemical physics have been studied through nonlinear evolution equations (NLEEs). According to the objectives of many researchers, traveling wave solutions for NLEEs can be investigated using a number of analytical and numerical techniques to get the exact solutions for them, such as in the fabrication of a composite membrane,¹ the generalized Burgers equation with variable coefficients,² the modulation of electronic properties in spintronic interfaces,³ the characteristic values for fiber/matrix adhesion,⁴ the (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation,⁵ the creep behavior of single-lap adhesive joints,⁶ the random decrement signature technique and artificial neural network algorithm,⁷ the ferroelectricity due to symmetry-breaking phase transition,⁸ the nonlinear vibration and dispersive wave systems,⁹ a hybrid power system,¹⁰ a Ferrotoroidic with well-separated spin chains,¹¹ the Konopelchenko–Dubrovsky equation,¹² the (3+1)D Burgers system,¹³ He’s variational direct methods for the KP–BBM equation,¹⁴ the cubic–quintic nonlinear Helmholtz equation,¹⁵ the probabilistic decomposition-based security,¹⁶ the extended sinh-Gordon equation expansion method,¹⁷ ranking extreme efficient decision-making units method,¹⁸ the new ($G^{\prime }/G$)-expansion method,¹⁹ the finite element method,²⁰ generalized nonlinear wave equation in a liquid with gas bubbles,²¹ the generalized Hietarinta equation,²² the design and analysis of a torsional-mode microelectromechanical system²³ and generalized shallow water wave equation.²⁴

Ferroelectric thin films have become a potential choice in the field of detection with ultraviolet photodetectors due to their wide bandgap and unique photovoltaic aspects. Additionally, ferroelectric thin films have excellent dielectric, piezoelectric, pyroelectric, acousto-optic effects, etc.²⁵ Reference 26 showed that the growth of ferroelectric layer on the original perovskite grains can reduce the formation of grain boundaries and hence minimize the recombination of electron and hole at the grain boundaries. The solitary wave dynamics of the thin-film ferroelectric material equation (TFFME) was investigated in Ref. 27. Umoh et al. designed materials exhibiting the combined properties such as ferroelectricity, ferromagnetism and ferroelasticity at the same phase for nanoelectronic devices based on oxide films.²⁸ An analysis of the thermal and ferroelectric properties of an Ising thin film in a transverse field extended for a higher spin within the quantum Monte Carlo method, was provided in Ref. 29. Ferroelectric thin films have demonstrated great potential in electrocaloric solid-state refrigeration on account of large adiabatic temperature changes.²⁸ Reference 30 demonstrated ferroelectric tunnel junctions whose conductivity varies linearly and symmetrically by judiciously combining ferroelectric domain switching and oxygen vacancy migration.

First, we give the solitary wave behavior of polarization for ferroelectric materials concerning the TFFME¹¹ in one-dimensional form as follows :

where mass and charge density are m and u, $T,p_2,p_4$ and $p_6$ are the temperature and pressures. Also, J denotes the space inhomogeneity coefficient and $\mu$ is the reciprocal of electric susceptibility.³¹ The ferroelectric material was subjected to a standing electric field which inhibits remanent polarization and facilitates the access to the instantaneous polarization.³¹ Two distinct techniques have been applied to get the accurate solutions for the thin-film ferroelectric material equation which plays a vital role in optics with respect to waves propagating through ferroelectric materials.³² Moreover, the Paul–Painlevé approach that discovered surprising results was used to achieve the optical soliton solutions of the thin-film ferroelectric material equation. ³³

Soliton theory is a very efficient and competent way to describe nonlinear features. Soliton theory involves two basic routes to study and explain nonlinear features. Soliton solutions are visible in the analysis of numerous nonlinear subjects. Soliton is an extremely thin, high-intensity light pulse. Solitons have the most remarkable properties of both particles and waves that can reflect the nonlinear features in a well-organized and competent way. Studying the nature by framing nonlinear evolution equations along with their soliton solutions is quicker and unquestionable. Solitons keep their velocities, shapes and amplitudes unchanged even after interacting with others due to their perfectly elastic interaction. In this paper, some solutions, including soliton, bright soliton, singular-soliton and periodic wave solutions, by the generalized trial equation scheme (GTES) were also obtained. These results show that the auxiliary methods are powerful mathematical tools to handle the nonlinear integrable equations from nature. Finding precise solutions is crucial for understanding the fundamental properties of these phenomena, particularly for soliton and energy storage systems³⁴ and a combined energy system.³⁵ Numerous researchers lately discovered the analytical traveling-wave formulations of NLPDEs which play an important function in different methods such as image processing and flow field reconstruction algorithm,³⁶ the van der Waals equation for the acid–base theory of surfaces,³⁷ the probabilistic decomposition-based security,¹⁶ robust optimization technique,³⁸ the demand response and improved water wave optimization algorithm,³⁹ the hybrid forecast engine-based intelligent algorithm,⁴⁰ the hybrid convolutional neural network and extreme learning machine,⁴¹ amended Dragon Fly optimization algorithm,⁴² a hybrid robust–stochastic approach⁴³ and correlation of random variables with Copula theory.⁴⁴

The nonlinear effects on dynamical features of soliton waves in a nonlinear Schrödinger equation were analyzed with dark soliton solutions.⁴⁵ Authors⁴⁶ investigated the dynamics of soliton waves in a generalized nonlinear Schrödinger equation applying the modified Jacobi elliptic expansion method. The nonsingular multi-complexion wave and multi-shock wave for a generalized KdV equation were studied using the principle of linear superposition with the help of symbolic computations.⁴⁷ Both compressive and rarefactive subsonic solitary waves were found depending on the wave speeds in various directions of propagation.⁴⁸ The hyperbolic, exponential, trigonometric functions, other soliton solutions and their combinations for the cold bosonic atoms in a zigzag optical lattice model have been obtained based on the generalized Riccati equation mapping method and generalized Kudryashov method.⁴⁹ Abundant exact solitary solutions including multiple-soliton, bell-shaped soliton, traveling wave, trigonometric and rational solutions have been constructed by applying the generalized exponential rational function method to the strain wave equation in microstructured solids.⁵⁰ The Lie group of point transformation method to construct the generalized invariant solutions for the (2+1)-dimensional dispersive long-wave equations under some constraints was studied in Ref. 51. This work successfully applied the generalized trial equation approach with a homogeneous balancing principle to TFFME in (1+1) dimensions with constant coefficients for obtaining the spatiotemporal soliton solutions and exact extended traveling-wave solutions.

Inspired by the previous works, the aim of this paper is to investigate the solitons and other forms of solutions by a generalized trial equation scheme. The outline of the paper is as follows. In Sec. 2, the TFFME for the nonlinear ordinary differential equation is obtained by transformation. Furthermore, in Secs. 3 and 4, different forms of solitary wave solutions are established by the generalized trial equation scheme. Finally, the conclusions are provided in Sec. 5.

2. Transforming PDE to ODE

For Eq. (1), let x and t be the longitudinal and transverse coordinates. Using the next wave transformation $u(x,t)=u(\eta ),\eta =x-\lambda t$, where $\lambda$ is an arbitrary constant to be determined through the method’s steps, leads to the following ODE :

where the wave speed is denoted by $\lambda$. By using the balancing principle on the terms of Eq. (3) between ${\psi }^{\prime \prime }$ and ${\psi }^5$ leads to $k+2=5k$, then $k=1/2$. By utilizing the transformation $u(\eta )=\sqrt{\psi (\eta )}$, Eq. (2), with respect to $\eta$ and zero-integration constant, becomes Applying the balancing principle to the terms of Eq. (3) between ${\psi }^{\prime \prime }$ and ${\psi }^4$ leads to $k=1$.

3. Generalized Trial Equation Scheme

Handling the investigated model through the generalized trial equation scheme⁵² involves the following steps, as mentioned earlier:

Step 1. In this step, we have

where ${𝒮}$ is a polynomial of F and its partial derivatives.

Step 2. Utilizing the traveling-wave transformation

where ${\theta }_1$ and ${\theta }_2$ are the nonzero arbitrary values, allows to reduce Eq. (4) into an ODE of $F=F(\xi )$ with the following form :Step 3. The generated solution of (4) is given as where Using Eqs. (7) and (8), the following derivatives are obtained : where $Q_1(\mathrm{\Omega })$ and $Q_2(\mathrm{\Omega })$ are polynomials. Appending these equations into Eq. (4) results in an equation of the polynomial $\mathrm{\Phi }(\mathrm{\Omega })$ of $\mathrm{\Omega }$, By utilizing the balancing principle on (11), the relations of $a,b$ and c are discovered. We can take some values of $a,b$ and c.

Step 4. Each coefficient of polynomial $\mathrm{\Lambda }(\mathrm{\Gamma })$ is substituted with zero to derive an algebraic system as follows :

By solving the above system (12), the parameters namely ${\varphi }_0,{\varphi }_1,\dots ,{\varphi }_b$, ${\phi }_0,{\phi }_1,\dots ,{\phi }_c$ and ${\omega }_0,{\omega }_1,\dots ,{\omega }_a$ are obtained.

Step 5. In this step, the elementary form of the integral by reduction of Eq. (8) is reached as follows :

where ${\xi }_0$ is a free constant.

4. Application of GTES

In this section, we consider Eq. (1). Then, by balancing the terms $F{F}^{\prime \prime }$ or $F^{\prime 2}$ with $F^4$ of Eq. (3), and by determining the values of $a,b$ and c, we can get

For diverse values of the parameters of $a,b$ and c, we obtain two cases as discussed below.

4.1. Case I: $a=1,b=4$ and $c=0$

If we consider $a=1,b=4$ and $c=0$ in Eqs. (7) and (8), then we obtain

where ${\varphi }_4\ne 0$ and ${\rho }_0\ne 0$. Solving the algebraic equation (12) yields different results.

The different sets of categories of solutions are discussed in the following subsections.

4.1.1. Set I

Here we have

Substituting these results into Eqs. (8) and (15) results in Integrating (18), the following solutions to Eq. (1) are attained:

First solution. Let

then the relation (19) results in then the relation (22) results in Moreover, the parameters $r_1,r_2$ and $r_3$ are the roots of the polynomial equation Putting the solutions (20) and (23) into (15), the following traveling-wave solutions of Eq. (1), respectively, are attained :

4.1.2. Set II

Here we have

Substituting these results into Eqs. (8) and (27), we can get Integrating  (28), the below solutions are obtained:

First solution. Let

Therefore, the exact solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,B_1,p_1$ and $p_2$ can be selected as free constants.

Second solution. Let

Hence, the solution is given as where ${\theta }_1,{\theta }_2,{\theta }_4,B_1,p_1$ and $p_2$ can be selected as free constants.

Third solution. Let

Therefore, the solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,A_0,B_1,p_1$ and $p_2$ can be selected as free constants.

Fourth solution. Let

where Therefore, the solution will be

Remark 1. If the modulus $\sigma \to 1$, then the solution can be reduced to the solitary wave solution

where $r_3=r_4$.

Remark 2. If the modulus $\sigma \to 0$, then the solution can be reduced to the solitary wave solution

where $r_2=r_3$.

Figure 1 depicts the impact analysis of singular-soliton solution where the plots of u are given for the following values for Eq. (46) :

The behavior of general singular solitons received from the mentioned technique is investigated, which is presented in Fig. 1. From this figure, it is ostensible that the singular solitons exhibit a stable propagation for the thin-film equation.

Also, the effect on kink–singular-soliton solution u is analyzed and plotted in Fig. 2 with the following values for Eq. (48) :

The dynamics of general kink–singular solitons received from the mentioned technique is investigated, which is presented in Fig. 2. From the figure, it is ostensible that the solitons exhibit a stable propagation in both components of thin-film ferroelectric material equation.

Also, the effect on kink–singular-soliton solution u is analyzed and plotted in Fig. 3 with the following values for Eq. (50) :

The behavior of general two solitons received from the mentioned technique is investigated, which is presented in Fig. 3. From the figure, it is obvious that the solitons exhibit a stable propagation for the thin-film ferroelectric material equation.

Also, the effect on kink–singular-soliton solution u is analyzed and plotted in Fig. 4 with the following values for Eq. (52) :

The behavior of general bright solitons received from the mentioned scheme is investigated, which is presented in Fig. 4. From the figure, it is ostensible that the solitons exhibit a stable propagation for the thin-film ferroelectric material equation.

Also, the effect on periodic wave solution u is analyzed and plotted in Fig. 5 with the following values for Eq. (54) :

The behavior of general periodic wave received from the mentioned technique is investigated, which is presented in Fig. 5. From the figure, it is ostensible that the periodic structure exhibits a stable propagation for the thin-film ferroelectric material equation.

4.1.3. Set III

Here we have

Substituting these results into Eqs. (8) and (55), we get Integrating  (56), the below solutions are obtained:

First solution. Let

Therefore, the exact solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,B_1,p_1$ and $p_2$ can be selected as free constants.

Second solution. Let

Hence, the solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,B_1,p_1$ and $p_2$ can be selected as free constants.

Third solution. Let

Therefore, the solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,A_0,B_1,p_1$ and $p_2$ can be selected as free constants.

Fourth solution. Let

where Therefore, the solution will be

Remark 1. If the modulus $\sigma \to 1$, then the solution can be reduced to the solitary wave solution

where $r_3=r_4$.

Remark 2. If the modulus $\sigma \to 0$, then the solution can be reduced to the solitary wave solution

where $r_2=r_3$.

Figure 6 depicts the impact on singular-soliton solution u for the following values for Eq. (74) :

The behavior of general singular solitons received from the mentioned technique is investigated, which is presented in Fig. 6. From the figure, it is ostensible that the singular solitons exhibit a stable propagation for the thin-film equation.

Also, the effect on kink–singular-soliton solution u is analyzed and plotted in Fig. 7 with the following values for Eq. (76) :

The dynamics of general kink–singular solitons received from the mentioned technique is investigated, which is presented in Fig. 7. From the figure, it is ostensible that the solitons exhibit a stable propagation in both components of thin-film ferroelectric material equation.

Figure 8 depicts the impact on kink–singular-soliton solution u for the following values for Eq. (78) :

The behavior of general two solitons received from the mentioned technique is investigated, which is presented in Fig. 8. From the figure, it is obvious that the solitons exhibit a stable propagation for the thin-film ferroelectric material equation.

Also, the effect on bright soliton solution u is analyzed and plotted in Fig. 9 for the following values for Eq. (80) :

The behavior of general bright solitons received from the mentioned scheme is investigated, which is presented in Fig. 9. From the figure, it is ostensible that the solitons exhibit a stable propagation for the thin-film ferroelectric material equation.

Also, the effect on periodic wave solution u is analyzed and plotted in Fig. 10 for the following values for Eq. (82) :

The behavior of general periodic wave received from the mentioned technique is investigated, which is presented in Fig. 10. From the figure, it is ostensible that the periodic structure exhibits a stable propagation for the thin-film ferroelectric material equation.

4.1.4. Set IV

Here we have

Substituting these results into Eqs. (8) and (83), we get

Integrating (84), the below solutions are obtained:

First solution. Let

Therefore, the exact solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,B_1,p_1$ and $p_2$ can be selected as free constants.

Second solution. Let

Hence, the solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,B_1,p_1$ and $p_2$ can be selected as free constants.

Third solution. Let

Therefore, the solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,A_0,B_1,p_1$ and $p_2$ can be selected as free constants.

Fourth solution. Let

where Therefore, the solution will be

Remark 1. If the modulus $\sigma \to 1$, then the solution can be reduced to the solitary wave solution

where $r_3=r_4$.

Remark 2. If the modulus $\sigma \to 0$, then the solution can be reduced to the solitary wave solution

where $r_2=r_3$.

4.1.5. Set V

Here we have

Substituting these results into Eqs. (8) and (101), we get Integrating (102), the below solutions are achieved:

First solution. Let

Therefore, the exact solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,B_1,p_1$ and $p_2$ can be selected as free constants.

Second solution. Let

Therefore, the solution will be where ${\theta }_1,{\theta }_2,{\theta }_4,B_1,p_1$ and $p_2$ can be selected as free constants.

Third solution. Let

Therefore, the solution is given by where ${\theta }_1,{\theta }_2,{\theta }_4,A_0,B_1,p_1$ and $p_2$ can be selected as free constants.

Fourth solution. Let

where Therefore, the solution will be

Remark 1. If the modulus $\sigma \to 1$, then the solution can be reduced to the solitary wave solution

where $r_3=r_4$.

Remark 2. If the modulus $\sigma \to 0$, then the solution can be reduced to the solitary wave solution

where $r_2=r_3$.

4.2. Case II: $a=2,b=5$ and $c=1$

If we take $a=2,b=5$ and $c=1$ for Eqs. (7) and (8), then we obtain

where ${\varphi }_5\ne 0$ and ${\rho }_1\ne 0$. Solving the algebraic equation (12) yields different results.

The different sets of categories of solutions are discussed in the following subsections.

4.2.1. Set I

Here we have

Substituting these results into Eqs. (8) and (121), we get When ${\mathrm{\Omega }}^5+\frac{{\varphi }_4}{{\varphi }_5}{\mathrm{\Omega }}^4+\frac{{\varphi }_3}{{\varphi }_5}{\mathrm{\Omega }}^3+\frac{{\varphi }_2}{{\varphi }_5}{\mathrm{\Omega }}^2+\frac{{\varphi }_1}{{\varphi }_5}\mathrm{\Omega }+\frac{{\varphi }_0}{{\varphi }_5}={(\mathrm{\Omega }-r_1)}^5$, then we have or Therefore, the solution will be

4.2.2. Set II

Here we have

Substituting these results into Eqs. (8) and (126), we get Therefore, the exact solution will be

4.2.3. Set III

Here we have

Substituting these results into Eqs. (8) and (126), we get Hence, the exact solution will be

4.2.4. Set IV

Here we have

Substituting these results into Eqs. (8) and (132), we get Therefore, the exact solution will be

4.2.5. Set V

Here we have

Substituting these results into Eqs. (8) and (135), we get Therefore, the exact solution will be

4.3. Results

By selecting the appropriate values for the parameters, we were able to generate the desired types of solutions that indicate wave discrepancy. 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. 1–10. The results from the simulations show that through the inherent properties of auxiliary parameters for the adjustment and control of region and rate of convergence of approximate series solutions, generalized trial equation scheme and generalized G-expansion method have proven to be very efficient and capable techniques for handling the nonlinear engineering problems for 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 want to undertake on the plasma physics and crystal lattice theory. The traveling-wave transformation was used along with the extended sinh-Gordon equation expansion technique and the Riccati equation method to construct the soliton and other wave solutions of the thin-film ferroelectric material equation.²⁷ In addition, dark soliton and cnoidal waves that have not been observed in ferroelectrics were obtained, and a bright soliton has been found for TFFME.³¹ The modified simple equation method and Riccati–Bernoulli sub-ODE method were used in TFFME to investigate the waves propagating through ferroelectric materials.³² Moreover, the Paul–Painlevé approach was used to achieve the optical soliton solutions of the thin-film ferroelectric material equation.³³ According to the existing literature and obtained results, we understand there are plenty of solutions including more types of obtained solutions.

5. Conclusion

To conclude, the solitary wave solutions by utilizing the generalized trial equation scheme were analytically constructed, and we also noticed that the equation was nonintegrable. The impact of wave motion in plasma on the physical parameters including speed and amplitudes of solitary waves was focused. Subsequently, several exact explicit solutions to this problem have been investigated utilizing these improvements and symbolic computation. In particular, four forms of function solutions, including soliton, kink–soliton, singular-soliton and periodic wave solutions, were studied. Plenty of exact solutions for the two cases were found. Via graphical illustrations by using the considered methods, the dynamical behavior of results was investigated. Many other nonlinear evolution equations, including coupled ones, can be solved using this technique. Along with the scientific derivation for the analytical findings, the outcomes were graphically displayed to help identify the dynamical aspects easily. Theoretical insights reported here may be considered helpful for future experimental studies. Moreover, various important remarks about the physical meanings of solutions were presented. From these results, it may be seen that MEJM and EEM are power tools to solve such nonlinear partial models arising in applied and engineering sciences. In the future, we can further study their soliton solutions, rogue wave solutions, solitary waves, symmetry, etc. We utilized the Maple program package to carry out the computations and confirm every outcome.

Conflict of Interest

Authors declare they have no competing interests regarding the publication of this paper.

Acknowledgments

The researchers would like to acknowledge the Deanship of Scientific Research, Taif University, for funding this work.

Also, this work was supported by the Jilin Province Science and Technology Association, “The Fourth Batch of Young Talent Support Project”, Grant No. QT202014.