Extended -expansion method for solving the coupled KdV equations with two arbitrary constants and its application to MEMS system

The coupled Korteweg-de Vries (cKdV) equations with two arbitrary constants hold significant importance in the field of micro-electro-mechanical systems (MEMS). These equations describe the behavior of nonlinear waves in MEMS devices. In MEMS applications, the cKdV equations can be used to analyze the dynamics of microstructures such as cantilevers, membranes, and resonators. By solving these equations, researchers can predict the behavior of MEMS devices under different operating conditions. In this paper, the $(\frac{G^{\prime }}{G})$-expansion method is extended to seek more general travelling solutions of the cKdV equations with two arbitrary constants. The two arbitrary constants offer flexibility in modeling different physical phenomena and boundary conditions. As a result, many new and more general exact travelling wave solutions are obtained including soliton solutions, hyperbolic function solutions, trigonometric function solutions and rational solutions. They help in understanding the complex interactions between mechanical and electrical properties. Additionally, the study of these equations provides insights into the nonlinear behavior of MEMS systems, which is crucial for improving their performance and reliability. Overall, the cKdV equations with two arbitrary constants play a vital role in advancing the design and understanding of MEMS applications.

1 Introduction

In the fields of physics and other disciplines, numerous phenomena are often described by nonlinear evolution equations (NLEEs). To gain a deep understanding of the physical mechanisms behind natural phenomena represented by the NLEEs, it is crucial to study their exact solutions. Many methods have been developed to obtain exact solutions for NLEES, such as the inverse scattering transform [1], the Darboux transformation [2], Bäcklund transformation [3], the Hirota method [4], the Wronskian technique [5], homogeneous balance method [6, 7], truncated Painlev$\acute{e}$ expansion method [8, 9], symmetry method [10], F-expansion method [11, 12], the generalized auxiliary equation method [13]. Among the numerous types of NLEEs, the cKdV equations hold a special place. The cKdV equations are widely used to model the interaction of multiple waves in different physical scenarios. For instance, in the field of micro-electro-mechanical systems (MEMS) [14], they can describe the behavior of nonlinear waves in MEMS devices. The dynamics of MEMS frequently exhibit nonlinear characteristics arising from large deformations, material nonlinearity, or electrostatic coupling effects. Such nonlinear behaviors are typically modeled using NLEEs. Soliton is stable, localized waves with inherent waveform preservation. This property proves advantageous for enhancing signal transmission efficiency in MEMS resonators and communication components. However, solving cKdV equations is a challenging task due to their inherent nonlinearity and complexity.

In recent years, the generalized $(\frac{G^{\prime }}{G})$-expansion method [15–17] have emerged as a promising and powerful technique for obtaining exact solutions of NLEEs. This method offers several advantages over traditional methods. It is more flexible and can be applied to a broader class of equations. By using the generalized $(\frac{G^{\prime }}{G})$-expansion method, we can obtain a variety of solutions, including solitary wave solutions, periodic wave solutions, rational function solutions, and more. These solutions can provide a more comprehensive understanding of the behavior of the physical systems described by the equations.

In this paper, we aim to extend the generalized $(\frac{G^{\prime }}{G})$-expansion method. Subsequently, we apply the extended $(\frac{G^{\prime }}{G})$-expansion method to solve the cKdV equations with two arbitrary constants. Our objective is to obtain numerous novel and more general travelling wave solutions, which can provide valuable insights into the wave-wave interactions described by the cKdV equations. Additionally, we will explore the application of these solutions to the MEMS field.

2 Introduction of the extended $(\frac{G^{\prime }}{G})$-expansion method

For a given NLEEs with variable $x=(t,x_1,x_2,\dots ,x_m)$ and $u(x)$

$P\left(u,u_t,u_{x_1},u_{x_2},\dots ,u_{x_m},u_{tt},u_{x_1{x}_1},u_{x_{1}t},u_{x_2{x}_2},\dots \right)=0,(2.1)$

through the application of the travelling wave transformation $u(x)=u(\xi ),\xi =k(x_1+l_1{x}_2+l_2{x}_3+\cdots +l_{m-1}{x}_m+Vt)$(where $k,V$ and $l_i(i=\mathrm{1,2},\dots ,m-1)$ are all constants.), Equation 2.1 can be reduced to an ordinary differential equation (ODE):

$Q\left(u,u^{\prime },u^{″},\cdots \right)=0,(2.2)$

we work towards getting its solutions in a more general form:

$u={\displaystyle \sum _{i=-n}^n}{a}_i{\left(\frac{G^{\prime }}{G}\right)}^i,(2.3)$

in which $G=G(\xi )$ complies with the ODE

$G^{″}+\lambda G^{\prime }+\mu G=0.(2.4)$

Since Equation 2.3 contains $2n$ arbitrary constants, the solutions derived from the extended $(\frac{G^{\prime }}{G})$-expansion method are more general in scope compared to those obtained through traditional approaches. To optimize the utilization of the extended $(\frac{G^{\prime }}{G})$-expansion method, we list its main steps as follows:

Step 1. Determine the integer value of $n$. Substitute Equation 2.3 along with Equation 2.4 into Equation 2.2. By balancing the highest-order derivative term with the nonlinear terms in Equation 2.2, we are able to derive the algebraic equation related to $n$.

Step 2. Derive an algebraic equation system. Substituting Equation 2.3 and Equation 2.4 into Equation 2.2 with the $n$ value from step 1. Collecting the coefficients of ${(\frac{G^{\prime }}{G})}^p(p=0,\pm 1,\pm 2,\cdots )$, then setting each coefficient to zero, we can get a set of over-determined algebraic equations for $a_i(i=0,\pm 1,\pm 2,\dots ,\pm n),k,l_i(i=\mathrm{1,2},\dots ,m-1),V,\mu$ and $\lambda$.

Step 3. Solve the algebraic equation system. Employ Maple to solve the algebraic equation system and obtain the explicit expressions for $a_i(i=0,\pm 1,\pm 2,\dots ,\pm n),k,l_i(i=\mathrm{1,2},\dots ,m-1),V,\mu$ and $\lambda$.

Step 4. Get the exact solutions. By substituting the outcomes from the previous steps, we are able to obtain a series of travelling solutions of Equation 2.2 which rely on the fundamental solution $G$ of Equation 2.4.

3 Solutions of the cKdV equations

In this part, we intend to utilize our method to acquire new and more general exact travelling solutions for the cKdV equations [18].

$u_t=a\left(u_{\mathit{xxx}}+6u{u}_x\right)+2bv{v}_x,(3.1)$

$v_t=-v_{\mathit{xxx}}-3u{v}_x.(3.2)$

Suppose that

$\begin{array}{ll}\hfill & \hfill u=u\left(x,t\right)=u\left(\xi \right),\\ \hfill & \hfill v=v\left(x,t\right)=v\left(\xi \right),\\ \hfill & \hfill \xi =x-Vt,\end{array}(3.3)$

then, upon inserting Equation 3.3 into Equations 3.1, 3.2 separately, we get.

$-V{u}^{\prime }-6au{u}^{\prime }-a{u}^{‴}-2bv{v}^{\prime }=0,(3.4)$

$-V{v}^{\prime }+3u{v}^{\prime }+v^{‴}=0,(3.5)$

by integrating Equation 3.4 with respect to $\xi$ for one time, we get

$-Vu-3a{u}^2-a{u}^{″}-b{v}^2+C=0,(3.6)$

Based on step 1, we find that $n=2$ for $u$ and $v$. We postulate that Equations 3.5, 3.6 possess the following formal solutions

$u={\displaystyle \sum _{i=-2}^2}{c}_i{\left(\frac{G^{\prime }}{G}\right)}^i,(3.7)$

$v={\displaystyle \sum _{i=-2}^2}{d}_i{\left(\frac{G^{\prime }}{G}\right)}^i,(3.8)$

where $c_i$ and $d_i(i=0,\pm 1,\pm 2)$ are all constants to be determined.

Upon substituting Equation 3.7 and Equation 3.8 along with Equation 2.4 into Equations 3.5, 3.6, the following results are achieved.

Case 1:

$c_1=c_2=d_1=d_2=d_{-2}=0,c_{-2}=-2{\mu }^2,c_{-1}=-2\lambda \mu ,d_0=\frac{\lambda d_{-1}}{2\mu },$

$c_0=-\frac{16a{\mu }^3+2a{\mu }^2{\lambda }^2-b{d}_{-1}^2+2{\mu }^2{\lambda }^2+4{\mu }^3}{6{\mu }^2\left(1+2a\right)},$

$V=\frac{-8a{\mu }^3+2a{\mu }^2{\lambda }^2+b{d}_{-1}^2}{2{\mu }^2\left(1+2a\right)},$

$\begin{array}{ll}\hfill C=& \left(8a^2{\mu }^2{\lambda }^{2}b{d}_{-1}^2-32a^2{\mu }^{3}b{d}_{-1}^2+8a{\mu }^2{\lambda }^{2}b{d}_{-1}^2+b^2{d}_{-1}^4-32a{\mu }^{3}b{d}_{-1}^2\right.\hfill \\ \hfill & +64a^2{\mu }^6-32a^2{\mu }^5{\lambda }^2+b{d}_{-1}^2{\mu }^2{\lambda }^2-4b{d}_{-1}^2{\mu }^3+64a^3{\mu }^6+4a^2{\mu }^4{\lambda }^4\hfill \\ \hfill & \left.+a{b}^2{d}_{-1}^4+4a^3{\mu }^4{\lambda }^4-32a^3{\mu }^5{\lambda }^2\right)/12{\mu }^4{\left(1+2a\right)}^2,\hfill \end{array}(3.9)$

where λ, μ and d₋₁ are arbitrary constants.

Case 2:

$c_1=c_2=d_1=d_2=0,c_{-1}=-4\lambda \mu ,c_{-2}=-4{\mu }^2,$

$d_{-1}=\pm 2\lambda \mu \sqrt{\frac{-6a}{b}},d_{-2}=\pm 2{\mu }^2\sqrt{\frac{-6a}{b}},$

$c_0=\frac{\mp d_{0}b\sqrt{\frac{-6a}{b}}+\left(1+a\right)\left({\lambda }^2+8\mu \right)}{1+2a},$

$V=\frac{a{\lambda }^2+8a\mu \pm d_{0}b\sqrt{\frac{-6a}{b}}}{1+2a},$

$\begin{array}{ll}\hfill & \hfill C=\left(\mp \left(2a^2{\lambda }^2+16a\mu +{\lambda }^2+8\mu +16a^2\mu \right)d_{0}b\sqrt{\frac{-6a}{b}}-2a{\lambda }^2{d}_{0}b\right.\\ \hfill & \hfill +a^3{\lambda }^4-24a{\mu }^2-32a^3{\lambda }^2\mu -32a^3{\mu }^2+3b{d}_0^2+a^2{\lambda }^4-32a^2{\mu }^2\\ \hfill & \hfill \left.-32a^2{\lambda }^2\mu -12a\mu {\lambda }^2+6ab{d}_0^2+6a^{2}b{d}_0^2\right)/3{\left(1+2a\right)}^2,\end{array}(3.10)$

where λ, μ and d₀ are arbitrary constants.

Case 3:

$c_{-2}=c_{-1}=d_{-2}=d_{-1}=d_2=0,c_1=-2\lambda ,c_2=-2,V={\lambda }^2+2\mu +3c_0,$

$d_1=\pm \sqrt{\frac{2a{\lambda }^2+12a{c}_0+16a\mu +6c_0+2{\lambda }^2+4\mu }{b}},$

$d_0=\pm \frac{\lambda }{2}\sqrt{\frac{2a{\lambda }^2+12a{c}_0+16a\mu +6c_0+2{\lambda }^2+4\mu }{b}},$

$C=\frac{a{\lambda }^4}{2}+3a{c}_0{\lambda }^2+2a\mu {\lambda }^2+3a{c}_0^2-4a{\mu }^2+\frac{{\lambda }^4}{2}+{\lambda }^2\mu +\frac{5c_0{\lambda }^2}{2}+2c_0\mu +3c_0^2,$

where λ, μ and c₀ are arbitrary constants.

Case 4:

$d_{-2}=d_2=0,c_2=-2,c_1=-2\lambda ,$

$d_{-1}=d_1\mu ,c_{-1}=-2\lambda \mu ,c_{-2}=-2{\mu }^2,$

$V=\frac{2a{\lambda }^2+16a\mu +b{d}_1^2}{2\left(1+2a\right)},$

$c_0=-\frac{2a{\lambda }^2+16a\mu +2{\lambda }^2+16\mu -b{d}_1^2}{6\left(1+2a\right)},$

$d_0=\frac{\lambda \left(24a\mu -b{d}_1^2\right)}{2b{d}_1},$

$\begin{array}{ll}\hfill & \hfill C=\left(1728a^2{\lambda }^2{\mu }^2+8b^2{d}_1^4\mu +b^2{d}_1^4{\lambda }^2+448b{d}_1^2{a}^2{\lambda }^2\mu +96b{d}_1^{2}a{\lambda }^2\mu \right.\\ \hfill & \hfill +6912a^4{\lambda }^2{\mu }^2+64b^2{d}_1^{4}a\mu +8d_1^{4}a{\lambda }^2{b}^2+6912a^3{\lambda }^2{\mu }^2+b^3{d}_1^6\\ \hfill & \hfill +4b{d}_1^2{a}^2{\lambda }^4+1024b{d}_1^2{a}^2{\mu }^2+192b{d}_1^{2}a{\mu }^2+4a^{3}b{d}_1^2{\lambda }^4+a{b}^3{d}_1^6\\ \hfill & \hfill +1024a^{3}b{d}_1^2{\mu }^2+8a^2{b}^2{d}_1^4{\lambda }^2+64a^2{b}^2{d}_1^4\mu \\ \hfill & \hfill \left.+448a^{3}b{d}_1^2{\lambda }^2\mu \right)/12b{d}_1^2{\left(1+2a\right)}^2,\end{array}(3.11)$

where λ, μ and d₁ are arbitrary constants.

Case 5:

$c_{-1}=c_{-2}=d_{-1}=d_{-2}=0,c_1=-4\lambda ,c_2=-4,$

$d_1=\pm 2\lambda \sqrt{\frac{-6a}{b}},d_2=\pm 2\sqrt{\frac{-6a}{b}},$

$V={\lambda }^2+8\mu +3c_0,$

$d_0=\pm \frac{{\lambda }^2+a{\lambda }^2+8a\mu +6a{c}_0+8\mu +3c_0}{\sqrt{-6ab}},$

$\begin{array}{ll}\hfill C=& -\left(128a{\mu }^2+9c_0^2+{\lambda }^4+96a{c}_0\mu +6c_0{\lambda }^2+a^2{\lambda }^4\right.\hfill \\ \hfill & +18a^2{c}_0^2+112a^2{\mu }^2+2a{\lambda }^4+12a{c}_0{\lambda }^2\hfill \\ \hfill & +40a^2\mu {\lambda }^2+12a{c}_0{\lambda }^2+96a^2{c}_0\mu +64{\mu }^2\hfill \\ \hfill & \left.+48c_0\mu +16{\lambda }^2\mu +18a{c}_0^2+32a{\lambda }^2\mu \right)/6a,\hfill \end{array}(3.12)$

where λ, μ and c₀ are arbitrary constants.

Substituting Equations 3.9–3.12 into Equations 3.7, 3.8 respectively, we have five kinds of formal solutions of Equations 3.1, 3.2:

$u_1=-2{\mu }^2{\left(\frac{G^{\prime }}{G}\right)}^{-2}-2\lambda \mu {\left(\frac{G^{\prime }}{G}\right)}^{-1}-\frac{16a{\mu }^3+2a{\mu }^2{\lambda }^2-b{d}_{-1}^2+2{\mu }^2{\lambda }^2+4{\mu }^3}{6{\mu }^2\left(1+2a\right)},(3.13)$

$v_1=d_{-1}\frac{G}{G^{\prime }}+\frac{d_{-1}\lambda }{2\mu },(3.14)$

where $\xi =x-\frac{-8a{\mu }^3+2a{\mu }^2{\lambda }^2+b{d}_{-1}^2}{2{\mu }^2(1+2a)}t$.

$u_2=-4{\mu }^2{\left(\frac{G^{\prime }}{G}\right)}^{-2}-4\lambda \mu {\left(\frac{G^{\prime }}{G}\right)}^{-1}+\frac{a{\lambda }^2\mp d_{0}b\sqrt{\frac{-6a}{b}}+{\lambda }^2+8\mu +8a\mu }{1+2a},(3.15)$

$v_2=\pm 2{\mu }^2\sqrt{\frac{-6a}{b}}{\left(\frac{G^{\prime }}{G}\right)}^{-2}\pm 2\lambda \mu \sqrt{\frac{-6a}{b}}{\left(\frac{G^{\prime }}{G}\right)}^{-1}+d_0,(3.16)$

where $\xi =x-\frac{a{\lambda }^2+8a\mu \pm d_{0}b\sqrt{\frac{-6a}{b}}}{1+2a}t$.

$u_3=-2{\left(\frac{G^{\prime }}{G}\right)}^2-2\lambda \frac{G^{\prime }}{G}+c_0,(3.17)$

$v_3=\pm \sqrt{\frac{2a{\lambda }^2+12a{c}_0+16a\mu +6c_0+2{\lambda }^2+4\mu }{b}}\left(\frac{G^{\prime }}{G}+\frac{\lambda }{2}\right).(3.18)$

where $\xi =x-({\lambda }^2+2\mu +2c_0)t$.

$\begin{array}{ll}\hfill & \hfill u_4=-2{\mu }^2{\left(\frac{G^{\prime }}{G}\right)}^{-2}-2\lambda \mu {\left(\frac{G^{\prime }}{G}\right)}^{-1}-2\lambda \frac{G^{\prime }}{G}-2{\left(\frac{G^{\prime }}{G}\right)}^2\\ \hfill & \hfill -\frac{2a{\lambda }^2+16a\mu +2{\lambda }^2+16\mu -b{d}_1^2}{6+12a},\end{array}(3.19)$

$v_4=d_1\mu {\left(\frac{G^{\prime }}{G}\right)}^{-1}+d_1\frac{G^{\prime }}{G}-\frac{\lambda \left(24a\mu -b{d}_1^2\right)}{2b{d}_1},(3.20)$

where $\xi =x-\frac{2a{\lambda }^2+16a\mu +b{d}_1^2}{2(1+2a)}t$.

$u_5=-4{\left(\frac{G^{\prime }}{G}\right)}^2-4\lambda \frac{G^{\prime }}{G}+c_0,(3.21)$

$v_5=\pm 2\sqrt{\frac{-6a}{b}}{\left(\frac{G^{\prime }}{G}\right)}^2\pm 2\lambda \sqrt{\frac{-6a}{b}}\frac{G^{\prime }}{G}\pm \frac{{\lambda }^2+a{\lambda }^2+8a\mu +6a{c}_0+8\mu +3c_0}{\sqrt{-6ab}},(3.22)$

where $\xi =x-({\lambda }^2+8\mu +3c_0)t$.

Then, by substituting the solutions of Equation 2.4 into Equations 3.13, 3.14, we derive three types travelling solutions of the cKdV equations as follows:

When ${\lambda }^2-4\mu >0$,

$\begin{array}{l}\hfill u_{1_1}=\frac{-2{\mu }^2}{{\left(-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}{C_{1}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}\right)}^2}\\ \hfill -\frac{2\lambda \mu }{-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}{C_{1}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}}\\ \hfill -\frac{16a{\mu }^3+2a{\mu }^2{\lambda }^2-b{d}_{-1}^2+2{\mu }^2{\lambda }^2+4{\mu }^3}{6{\mu }^2\left(1+2a\right)},\end{array}(3.23)$

$v_{1_1}=\frac{d_{-1}}{-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }}+\frac{d_{-1}\lambda }{2\mu }.(3.24)$

When ${\lambda }^2-4\mu <0$, we obtain

$\begin{array}{l}\hfill u_{1_2}=\frac{-2{\mu }^2}{{\left(-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }\right)}^2}\\ \hfill -\frac{2\lambda \mu }{-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }}\\ \hfill -\frac{16a{\mu }^3+2a{\mu }^2{\lambda }^2-b{d}_{-1}^2+2{\mu }^2{\lambda }^2+4{\mu }^3}{6{\mu }^2\left(1+2a\right)},\end{array}(3.25)$

$v_{1_2}=\frac{d_{-1}}{-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }}+\frac{d_{-1}\lambda }{2\mu }.(3.26)$

When ${\lambda }^2-4\mu =0$,

$\begin{array}{ll}\hfill u_{1_3}=& \frac{-2{\mu }^2}{{\left(-\frac{\lambda }{2}+\frac{C_2}{C_1+C_2\xi }\right)}^2}-\frac{2\lambda \mu }{-\frac{\lambda }{2}+\frac{C_2}{C_1+C_2\xi }}\hfill \\ \hfill & -\frac{16a{\mu }^3+2a{\mu }^2{\lambda }^2-b{d}_{-1}^2+2{\mu }^2{\lambda }^2+4{\mu }^3}{6{\mu }^2\left(1+2a\right)},\hfill \end{array}(3.27)$

$v_{1_3}=\frac{d_{-1}}{-\frac{\lambda }{2}+\frac{C_2}{C_1+C_2\xi }}+\frac{d_{-1}\lambda }{2\mu }.(3.28)$

By substituting the solutions of Equation 2.4 into Equations 3.15, 3.16, We possess three types of travelling solutions of the cKdV equations in the following:

When ${\lambda }^2-4\mu >0$,

$\begin{array}{l}\hfill u_{2_1}=\frac{-4{\mu }^2}{{\left(-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}{C_{1}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}\right)}^2}\\ \hfill -\frac{4\lambda \mu }{-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}{C_{1}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}}\\ \hfill +\frac{a{\lambda }^2\mp d_{0}b\sqrt{\frac{-6a}{b}}+{\lambda }^2+8\mu +8a\mu }{1+2a},\end{array}(3.29)$

$\begin{array}{l}\hfill v_{2_1}=\frac{\pm 2{\mu }^2\sqrt{\frac{-6a}{b}}}{{\left(-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}{C_{1}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}\right)}^2}+d_0\\ \hfill \pm \frac{2\lambda \mu \sqrt{\frac{-6a}{b}}}{-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}{C_{1}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}}.\end{array}(3.30)$

When ${\lambda }^2-4\mu <0$,

$\begin{array}{l}\hfill u_{2_2}=\frac{-4{\mu }^2}{{\left(-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }\right)}^2}\\ \hfill -\frac{4\lambda \mu }{-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }}\\ \hfill +\frac{a{\lambda }^2\mp d_{0}b\sqrt{\frac{-6a}{b}}+{\lambda }^2+8\mu +8a\mu }{1+2a},\end{array}(3.31)$

$\begin{array}{l}\hfill v_{2_2}=\frac{\pm 2{\mu }^2\sqrt{\frac{-6a}{b}}}{{\left(-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }\right)}^2}+d_0\\ \hfill \pm \frac{2\lambda \mu \sqrt{\frac{-6a}{b}}}{-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }}.\end{array}(3.32)$

When ${\lambda }^2-4\mu =0$,

$u_{2_3}=\frac{-4{\mu }^2}{{\left(-\frac{\lambda }{2}+\frac{C_2}{C_1+C_2\xi }\right)}^2}-\frac{4\lambda \mu }{-\frac{\lambda }{2}+\frac{C_2}{C_1+C_2\xi }}+\frac{a{\lambda }^2\mp d_{0}b\sqrt{\frac{-6a}{b}}+{\lambda }^2+8\mu +8a\mu }{1+2a},(3.33)$

$v_{2_3}=\frac{\pm 2{\mu }^2\sqrt{\frac{-6a}{b}}}{{\left(-\frac{\lambda }{2}+\frac{C_2}{C_1+C_2\xi }\right)}^2}\pm \frac{2\lambda \mu \sqrt{\frac{-6a}{b}}}{-\frac{\lambda }{2}+\frac{C_2}{C_1+C_2\xi }}+d_0.(3.34)$

Upon substituting the general solutions of Equation 2.4 into Equations 3.17, 3.18, here are three types of travelling solutions of the cKdV equations.

When ${\lambda }^2-4\mu >0$,

$u_{3_1}=\left(2\mu -\frac{{\lambda }^2}{2}\right){\left(\frac{C_{1}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}{C_{1}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}\right)}^2+\frac{{\lambda }^2}{2}+c_0,(3.35)$

$\begin{array}{ll}\hfill & \hfill v_{3_1}=\pm \sqrt{\frac{2\left({\lambda }^2-4\mu \right)\left(a{\lambda }^2+6a{c}_0+8a\mu +3c_0+{\lambda }^2+2\mu \right)}{b}}\\ \hfill & \hfill \times \frac{C_{1}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}{C_{1}sinh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+C_{2}cosh\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}.\end{array}(3.36)$

Taking $c_0=-\frac{a{\lambda }^2+8a\mu +{\lambda }^2+2\mu }{3+6a}$ in Equation 3.36, i.e., $v_{3_1}=0$, then Equations 3.1, 3.2 become the KdV equation

$u_t=a\left(u_{\mathit{xxx}}+6u{u}_x\right),(3.37)$

from Equation 3.35, the solutions of Equation 3.37 can be rewritten as

$\begin{array}{ll}\hfill & \hfill u_{3_1}=\left(2\mu -\frac{{\lambda }^2}{2}\right)\frac{2C_1{e}^{\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}+2C_1{e}^{-\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}+C_2{e}^{\sqrt{{\lambda }^2-4\mu }\xi }+C_2{e}^{-\sqrt{{\lambda }^2-4\mu }\xi }+2C_2}{2C_1{e}^{\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}-2C_1{e}^{-\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}}+C_2{e}^{\sqrt{{\lambda }^2-4\mu }\xi }+C_2{e}^{-\sqrt{{\lambda }^2-4\mu }\xi }+2C_2}\\ \hfill & \hfill +\left(4a{\lambda }^2-4\mu -16a\mu +{\lambda }^2\right)/\left(6+12a\right),\end{array}(3.38)$

if we set $C_1=0$, Equation 3.38 becomes

$u_{3_1}=\frac{4\mu -a{\lambda }^2-{\lambda }^2+4a\mu }{3+6a}-\frac{2C_2\left(4\mu -{\lambda }^2\right)}{C_2{e}^{\sqrt{{\lambda }^2-4\mu }\xi }+C_2{e}^{-\sqrt{{\lambda }^2-4\mu }\xi }+2C_2}.(3.39)$

Comparing our results in Equation 3.39 with other results by Exp-function method in [19], then it can be seen that the forms are similar.

When ${\lambda }^2-4\mu <0$,

$u_{3_2}=\left(\frac{{\lambda }^2}{2}-2\mu \right){\left(\frac{C_{1}cos\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi -C_{2}sin\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sin\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cos\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }\right)}^2+\frac{{\lambda }^2}{2}+c_0,(3.40)$

$\begin{array}{ll}\hfill & \hfill v_{3_2}=\mp \sqrt{\frac{2\left(4\mu -{\lambda }^2\right)\left(a{\lambda }^2+6a{c}_0+8a\mu +3c_0+{\lambda }^2+2\mu \right)}{b}}\\ \hfill & \hfill \times \frac{C_{1}cos\frac{\sqrt{4\mu -{\lambda }^2}\xi }{2}-C_{2}sin\frac{\sqrt{4\mu -{\lambda }^2}\xi }{2}}{C_{1}sin\frac{\sqrt{4\mu -{\lambda }^2-}\xi }{2}+C_{2}cos\frac{\sqrt{4\mu -{\lambda }^2}\xi }{2}}\end{array}.(3.41)$

When ${\lambda }^2-4\mu =0$,

$u_{3_3}=\frac{C_2^2\left(2c_0+\lambda \right){\xi }^2+2C_1{C}_2\left(2c_0+{\lambda }^2\right)\xi +2c_0{C}_1^2-4C_2^2+C_1^2{\lambda }^2}{2{\left(C_1+C_2\xi \right)}^2},(3.42)$

$v_{3_3}=\pm \sqrt{\frac{2a{\lambda }^2+12a{c}_0+16a\mu +6c_0+2{\lambda }^2+4\mu }{b}}\frac{C_2}{C_1+C_2\xi }.(3.43)$

Puting the general solutions of Equation 2.4 into Equations 3.19, 3.20, three types of travelling solutions of the cKdV equations are given in the following:

When ${\lambda }^2-4\mu >0$,

$\begin{array}{ll}\hfill & \hfill u_{4_1}=\frac{-2{\mu }^2}{{\left(-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }\right)}^2}\\ \hfill & \hfill -\frac{2\lambda \mu }{-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }}-\frac{{\lambda }^2-4\mu }{2}\\ \hfill & \hfill \times {\left(\frac{C_{1}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }\right)}^2\\ \hfill & \hfill -\frac{16a\mu -4a{\lambda }^2-{\lambda }^2+16\mu -b{d}_1^2}{6+12a},\end{array}(3.44)$

$\begin{array}{ll}\hfill & \hfill v_{4_1}=\frac{d_1\mu }{-\frac{\lambda }{2}+\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }}\\ \hfill & \hfill +\frac{\sqrt{{\lambda }^2-4\mu }}{2}\frac{C_{1}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }-\frac{12\lambda a\mu }{b{d}_1}.\end{array}(3.45)$

When ${\lambda }^2-4\mu <0$,

$\begin{array}{ll}\hfill & \hfill u_{4_2}=\frac{-2{\mu }^2}{{\left(-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }\right)}^2}\\ \hfill & \hfill -\frac{2\lambda \mu }{-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }}\\ \hfill & \hfill +\frac{{\lambda }^2-4\mu }{2}{\left(\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }\right)}^2\\ \hfill & \hfill -\frac{16a\mu -4a{\lambda }^2-{\lambda }^2+16\mu -b{d}_1^2}{6+12a},\end{array}(3.46)$

$\begin{array}{ll}\hfill & \hfill v_{4_2}=\frac{d_1\mu }{-\frac{\lambda }{2}+\frac{\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }}\\ \hfill & \hfill +\frac{d_1\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }-\frac{12\lambda a\mu }{b{d}_1}.\end{array}(3.47)$

When ${\lambda }^2-4\mu =0$,

$\begin{array}{ll}\hfill & \hfill u_{4_3}=\frac{-2{\mu }^2}{{\left(\frac{-\lambda }{2}+\frac{C_2}{C_1+C_2\xi }\right)}^2}-\frac{2\lambda \mu }{\frac{-\lambda }{2}+\frac{C_2}{C_1+C_2\xi }}-\frac{2C_2^2}{{\left(C_1+C_2\xi \right)}^2}\\ \hfill & \hfill -\frac{16a\mu -4a{\lambda }^2-{\lambda }^2+16\mu -b{d}_1^2}{6+12a},\end{array}(3.48)$

$v_{4_3}=\frac{d_1\mu }{\frac{-\lambda }{2}+\frac{C_2}{C_1+C_2\xi }}+\frac{d_1\sqrt{4\mu -{\lambda }^2}}{2}\frac{C_2}{C_1+C_2\xi }-\frac{12\lambda a\mu }{b{d}_1}.(3.49)$

Substituting the general solutions of Equation 2.4 into Equations 3.21, 3.22, we have three types travelling solutions of the cKdV equations in the following:

When ${\lambda }^2-4\mu >0$,

$u_{5_1}=-\left({\lambda }^2-4\mu \right){\left(\frac{C_{1}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }\right)}^2+{\lambda }^2+c_0,(3.50)$

$\begin{array}{ll}\hfill & \hfill v_{5_1}=\pm \frac{\left({\lambda }^2-4\mu \right)\sqrt{\frac{-6a}{b}}}{2}{\left(\frac{C_{1}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_{1}sinh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_{2}cosh\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }\right)}^2\\ \hfill & \hfill \pm \frac{{\lambda }^2+4a{\lambda }^2+8a\mu +6a{c}_0+8\mu +6c_0}{\sqrt{-6ab}}.\end{array}(3.51)$

If $C_2>0,C_1^2<C_2^2$, then from Equations 3.23, 3.24, we can obtain bell soliton solutions

$u_{5_2}=\left({\lambda }^2-4\mu \right)sec{h}^2\left(\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +{\xi }_0\right)+4\mu +c_0,(3.52)$

$\begin{array}{ll}\hfill & \hfill v_{5_2}=\pm \sqrt{\frac{-6a}{b}}\frac{\left({\lambda }^2-4\mu \right)}{2}sec{h}^2\left(\frac{\sqrt{{\lambda }^2-4\mu }\xi }{2}+{\xi }_0\right)\\ \hfill & \hfill \pm \frac{{\lambda }^2+a{\lambda }^2+20a\mu +6a{c}_0+8\mu +3c_0}{\sqrt{-6ab}},\end{array}(3.53)$

where ${\xi }_0=tan{h}^{-1}\frac{C_1}{C_2}$.

When ${\lambda }^2-4\mu <0$,

$u_{5_3}=\left({\lambda }^2-4\mu \right){\left(\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }\right)}^2+{\lambda }^2+c_0,(3.54)$

$\begin{array}{ll}\hfill & \hfill v_{5_3}=\pm \frac{\left(4\mu -{\lambda }^2\right)\sqrt{\frac{-6a}{b}}}{2}{\left(\frac{C_{1}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi -C_{2}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_{1}sin\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_{2}cos\frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }\right)}^2\\ \hfill & \hfill \pm \frac{{\lambda }^2+4a{\lambda }^2+8a\mu +6a{c}_0+8\mu +3c_0}{\sqrt{-6ab}}.\end{array}(3.55)$

When ${\lambda }^2-4\mu =0$,

$u_{6_1}=\frac{-4C_2^2}{{\left(C_1+C_2\xi \right)}^2}+{\lambda }^2+c_0,(3.56)$

$v_{6_1}=\pm \sqrt{\frac{-6a}{b}}\frac{2C_2^2}{{\left(C_1+C_2\xi \right)}^2}\pm \frac{{\lambda }^2+4a{\lambda }^2+8a\mu +6a{c}_0+8\mu +3c_0}{\sqrt{-6ab}}.(3.57)$

4 Conclusion

In summary, the extended $(\frac{G^{\prime }}{G})$-expansion method has been proposed and applied to construct exact solutions of the cKdV equations. With the aid of Maple, we have obtained many new and more general exact travelling wave solutions, presented as Equations 3.32–3.36 and 3.40–3.57. These solutions span a wide spectrum, including soliton solutions, hyperbolic function solutions, trigonometric function solutions, as well as rational solutions. By applying the solutions obtained from the cKdV equations to MEMS systems, the presence of the two arbitrary constants allows for customization of the model to fit specific experimental data or design requirements. This enables more accurate predictions and optimization of MEMS devices. Additionally, the study of these equations provides insights into the nonlinear behavior of MEMS systems, which is crucial for improving their performance and reliability. Overall, the cKdV equations with two arbitrary constants play a vital role in advancing the design and understanding of MEMS applications. We hope to contribute to the development of more efficient and reliable MEMS devices.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

JZ: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Writing – original draft, Writing – review and editing. FY: Software, Supervision, Validation, Visualization, Writing – review and editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. This work was supported by Natural Science Foundation of LiaoningProvince (2023-MSLH-229, 2024-MSLH-336) and Scientific Research Fund of Liaoning Provincial Education Department (LJ212411632012, LJ222411632015).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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