Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

Abstract

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps.

1. Introduction

Bifurcated processes are central to dynamics in applied mathematics and theoretical physics. Finding exact numerical solutions for functions of separated variables is a key part of soliton theory. In particular, traveling wave solutions for nonlinear equations in mathematical physics are very important. In recent years, the search for exact solutions to nonlinear partial differential equations (PDEs) has become a focus. Using specific software, such as MATLAB V2020, Maple V18, and Mathematica V31.1 or any above version makes the difference, as these softwares help carry out complicated algebraic manipulations. Obtaining the exact solutions to these equations is always required since they represent real-life problems in many engineering disciplines, including chemistry, biology, mechanics, and applied physics. Engineers and physicists have developed several efficient approaches to grasp the mechanisms of physical models and comprehend the fundamental issues that arise in underlying problems. Besides Hirota’s bilinear method and the inverse scattering transform [1,2], there are other more effective ways to find the traveling wave explicit solitary solutions of nonlinear equations. These include the Painleve expansions method [3], the homogeneous balance method [4,5], the F-expansion method [6], and the Jacobi elliptic function method [7]. Only recently has the tanh-function method been used to find exact solutions for non-linear differential equations [8,9]. We emphasize that mathematicians have several powerful techniques to find new traveling wave solutions, among which are the Khater methods [10,11], $G^{\prime }/G$-expansion method [12], Kudryashov method [13], exp-function method [14], Sin–Gordon method [15], and Sardar sub-equation method [16]. These approaches are pivotal for extending the research and application of solitons on non-linear fractional differential equations.

The objective of this study is to elaborate and demonstrate the effectiveness of the Ricatti–Bernoulli sub-ode approach using a new nonlinear partial differential equation (PDE). This is relevant to nanosciences, including the transmission line model for nano-ionic currents along microtubules (MTs), which play a role in cellular signaling. The MTs are fibrous cytoplasmic biopolymers in nanotubes that enable cell movement, cell communication and transportation of substances and data inside neuronal extensions. Researchers have also suggested that MTs participate in higher neuronal functions such as memory and consciousness. Nevertheless, how MTs process and operate electrical signals is a mystery. For the first time, researchers formulated a concept for ionic waves along MTs based on the polyelectrolyte nature of cylindrical biopolymers. It is like an electric element, with capacitive, resistive, and negative incrementally resistive properties for each tubulin dimer protein [17]. Nanopores (NPs) between the dimers in the MT wall are given extra attention because they appear to function as ionic channels [18,19]. These stated NPs might help to describe the behavior of MTs as biomolecular transistors, which amplify electrical information in neurons. An analysis of their physical properties provides the foundation for the derivation of the following equation, which may define the ionic currents along MTs. These details are elaborated fully in [18], including the principles upon which they are based and the mathematics that underlie the process. This derivation describes in detail the mechanisms through which biopolymers contribute to the signaling events within cells, including the polyelectrolyte character of MTs, structural aspects of these proteins, and the behavior of nanoionic currents

$$r_2{c}_0{l}^2{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{{\partial }^{2}F}{\partial x^2}}\right)+l^2{\displaystyle \frac{{\partial }^{2}F}{\partial x^2}}+2r_1{c}_0\gamma F{\displaystyle \frac{\partial F}{\partial t}}-r_1{c}_0{\displaystyle \frac{\partial F}{\partial t}}.$$

(1)

This study defines the transverse resistance $r_1={10}^9\Omega$ and the longitudinal resistance $r_2=7\times {10}^6\Omega$ as elementary ring (ER) components. In a given MT, the parameter $\gamma$ represents the nonlinearity of an ER capacitor. Here, setting the inductance $l=8\times {10}^{-9}m$ and the total maximal capacitance ($c_0$) of the ER is $1.8\times {10}^{-15}F$. Sekulic et al. [20] studied the MT equation formulated in terms of the nonlinear RLC transmission line. To gain solitary wave solutions for the problem, they used the method called METF, the extended tanh-function, which was modified to analyze the behavior of ionic currents in microtubules successfully.

Nonlinear PDEs have recently gained significant attention due to their applications in science and engineering, particularly wave processes and biology. In the present work, we added to this field by applying symbolic computations and the Riccati–Bernoulli sub-ODE method with the Bäcklund transformation [21] to obtain new and multiple traveling wave solutions of nonlinear PDEs. This is crucial for nanosciences, biophysics, and applications directly related to the transmission line models of microtubules, which help transport ions in living cells. We specifically construct this nonlinear equation to solve fundamental nanoscience problems, particularly those involving cell signaling and the primary processes in neuron functioning. Thus, by presenting solutions in terms of 3D graphics and contour plots using Mathematica software, we can reproduce previously obtained solitary wave solutions employing a modern approach and present additional novel findings. One could argue that the generality of the presented method makes it more versatile in applying further nonlinear equations in physics, thereby supporting its use in tackling significant issues in complex systems. This work is timely and important because it not only contributes to the field of nonlinear dynamics but also provides opportunities for further investigation in the areas of biophysical modeling and cognitive science. Furthermore, several optional PDEs represent a problem to solve using the standard approaches. A preliminary method described in refs. [22,23,24] is known to resolve high-order algebraic solutions effectively and is utilized to discover standalone wave solutions. Indeed, these PDEs are transformed into algebraic equations using the Bäcklund transformation and the Riccati–Bernoulli equation. Consequently, we can apply this technique effectively to solve various issues in mathematical physics and evaluate its effectiveness to some extent. This methodology ensures finite solutions for the derived equations and gives efficient results for the investigated equations. As we know, it is a relative of PDEs, thus indicating its ability to produce infinitely many solutions, where the Bäcklund transformation is introduced as an infinite chain of solutions.

The present work employs the proposed methodology to solve nonlinear PDEs related to nanobiosciences as transmission lines and to model MTs as nonlinear RLCs. This illustrates the efficiency and utility of the proposed approach to target multifaceted biological organizations. By focusing on the nanoionic current flow, the qualitatively enhanced outcomes provide better insights into real-life physical difficulties and indicate the effective application of the introduced methods in numerous scientific disciplines. These results also provide further evidence of this approach’s applicability in solving other nonlinear equations in a broad range of applications in physics and biology.

2. Methodology

The use of the Riccati–Bernoulli sub-ODE method is appropriate in solving very complex nonlinear partial differential equations (PDEs), particularly by simplifying them into simple and solvable ordinary differential equations (ODEs). This progression lessens the dimensionality of the problem by introducing traveling wave transformations. To illustrate the fundamental concept of the Ricatti–Bernoulli sub-ode method, we consider a given PDE with two variables, expressed as follows:

$$G\left(F,{\displaystyle \frac{\partial F}{\partial x}},{\displaystyle \frac{\partial F}{\partial t}},{\displaystyle \frac{{\partial }^{2}F}{\partial x^2}},\dots \right)=0.$$

(2)

Making use of the traveling wave transformation $\zeta =x\pm ct$, one can transform the partial derivatives in the above PDE into $\zeta$-derivatives, so the PDE can be expressed as an ODE. This prepares the ground for utilizing the Riccati–Bernoulli sub-ODE method, which systematically derives closed form solutions.

$$H\left(f,{\displaystyle \frac{df}{d\zeta }},{\displaystyle \frac{d^{2}f}{d{\zeta }^2}},{\displaystyle \frac{d^{3}f}{d{\zeta }^3}},\dots \right)=0.$$

(3)

Subsequently, we use the following series expansion as a solution to Equation (3):

$$f(x,t)=f\left(\zeta \right)=\sum _{i=-n}^n{z}_i{R\left(\xi \right)}^i,$$

(4)

with

$$R\left(\xi \right)={\displaystyle \frac{-X{k}_2+k_1\varphi \left(\xi \right)}{k_1+k_2\varphi \left(\xi \right)}}.$$

Here, we give a solution as the series expansion, in which $R\left(\xi \right)$ is inferred from the Bäcklund transformation, while $\varphi \left(\xi \right)$ satisfies the Riccati equation, ${\displaystyle \frac{d\varphi }{d\xi }}=X+\varphi {\left(\xi \right)}^2$. It offers a systematic way to balance the nonlinear terms in the ODE and also remarkably reduces the higher-order derivatives. The Bäcklund transformation is essential in bringing in the term $R\left(\xi \right)$ in Equation (4), which requires that $\varphi \left(\xi \right)$ satisfies the Riccati equation, a nonlinear ODE representing the way the newly transformed wave solutions would change. Herein lies the crucial bridge between the Bäcklund transformation and the Riccati–Bernoulli method. That is, while the Bäcklund transformation will yield new variables that will simplify the form or structure of the solution, it is the Riccati equation that describes, or dictates, how these variables would evolve. The constants (X), ($k_1$), and ($k_2$) are fixed parameters with $k_2\ne 0$. Thus, the positive integer (n) can easily be balanced by equating the coefficient of the highest-order derivative term with the coefficient of the nonlinear terms in Equation (3). We substitute the series expansion, Equation (4), into the transformed ODE, Equation (2), to obtain an algebraic system of equations. The system arises naturally from the interaction of the Bäcklund transformation that represents $R\left(\xi \right)$ and the Riccati equation for $\varphi \left(\xi \right)$, together with the given solution. The algebraic system leads to the constants ($z_i$) and wave velocity (c), which in turn give exact traveling wave solutions. Fortunately, the Riccati equation admits several types of solutions [25]:

$$\begin{array}{c}\varphi \left(\xi \right)=\left\{\begin{array}{cc}-\sqrt{-X}tanh\left(\sqrt{-X}\xi \right),\hfill & \mathrm{as}X<0,\hfill \\ -\sqrt{-X}coth\left(\sqrt{-X}\xi \right),\hfill & \mathrm{as}X<0,\hfill \end{array}\right.\hfill \\ \varphi \left(\xi \right)=-{\displaystyle \frac{1}{\xi }},\mathrm{as}X=0,\hfill \\ \varphi \left(\xi \right)=\left\{\begin{array}{cc}\sqrt{X}tan\left(\sqrt{X}\xi \right),\hfill & \mathrm{as}X>0,\hfill \\ -\sqrt{X}cot\left(\sqrt{X}\xi \right),\hfill & \mathrm{as}X>0.\hfill \end{array}\right.\hfill \end{array}$$

(5)

3. Execution of the Problem

Next, we apply the method in the previous section to a new nonlinear PDE that is very important in nanosciences. This is the transmission line model, which shows nano-ionic currents moving along MTs and is a part of how cells communicate. To solve Equation (1), applying the proposed method, we use the traveling wave transformation

$$\chi ={\displaystyle \frac{x}{l}}-{\displaystyle \frac{\omega t}{T}},$$

where ($\omega$) is the dimensionless velocity of the wave. The characteristic time for charging the ER capacitor is set as $T=r_1{c}_0=1.32\times {10}^{-6}$. Thus, Equation (1) is converted into a set of ODEs:

$${\displaystyle \frac{d^{2}f}{d{\chi }^2}}-{\displaystyle \frac{\kappa }{\omega }}{\displaystyle \frac{dF}{d\chi }}+{\displaystyle \frac{\nu }{2}}\left(f^2\right)-\mu f=0,$$

(6)

where $\kappa ={\displaystyle \frac{T}{r_2{c}_0}}$, $\nu ={\displaystyle \frac{2r_1\gamma }{r_2}}$, $\gamma ={\displaystyle \frac{r_1}{r_2}}$, and $T=r_1{c}_0=1.32\times {10}^{-6}$. Next, we compare the leading high-order derivative term with the leading high-order nonlinear term. By equating the order of (f) from the second equation with ($f^2$) in Equation (6), we obtain $n=2$. Inserting Equation (4) into Equation (6), we obtain a system of algebraic equations. By solving the resulting system of algebraic equations using Maple, we can identify several distinct cases, as follows:

$$\begin{array}{c}{\varphi }^0:\nu {z_{-2}}^2{k_2}^8\omega +12z_{-2}{k_2}^8\omega X^2=0,\hfill \\ {\varphi }^1:-4z_{-1}{k_2}^8\omega X^3-4\kappa z_{-2}{k_2}^8{X}^2-2\nu z_{-2}{k_2}^8{z}_{-1}\omega X=0,\hfill \\ {\varphi }^2:2\kappa z_{-1}{k_2}^8{X}^3+16z_{-2}{k_2}^8{X}^3\omega -2\mu z_{-2}{k_2}^8\omega X^2+\nu {z_{-1}}^2{k_2}^8\omega X^2+2\nu z_{-2}{k_2}^8{z}_0\omega X^2=0,\hfill \\ {\varphi }^3:-2\nu z_{-1}{k_2}^8{z}_0\omega X^3-4z_{-1}{k_2}^8{X}^4\omega -4\kappa z_{-2}{k_2}^8{X}^3+2\mu z_{-1}{k_2}^8\omega X^3-2\nu z_{-2}{k_2}^8{z}_1{X}^3\omega =0,\hfill \\ {\varphi }^4:\nu {z_0}^2\omega {k_2}^8{X}^4+2\nu z_{-2}{k_2}^8{z}_2{X}^4\omega -2\mu z_0\omega {k_2}^8{X}^4+2\kappa z_{-1}{k_2}^8{X}^4+2\nu z_{-1}{k_2}^8{z}_1{X}^4\omega +\hfill \\ 4z_{-2}{k_2}^8{X}^4\omega +4z_2{k_2}^8{X}^6\omega -2\kappa z_1{k_2}^8{X}^5=0,\hfill \\ {\varphi }^5:-4z_1{k_2}^8{X}^6\omega +4\kappa z_2{k_2}^8{X}^6-2\nu z_0{z}_1{X}^5{k_2}^8\omega -2\nu z_{-1}{k_2}^8{z}_2{X}^5\omega +2\mu z_1{X}^5{k_2}^8\omega =0,\hfill \\ {\varphi }^6:2\nu z_0{z}_2{X}^6{k_2}^8\omega -2\kappa z_1{k_2}^8{X}^6+16z_2{k_2}^8{X}^7\omega -2\mu z_2{X}^6{k_2}^8\omega +\nu {z_1}^2{X}^6{k_2}^8\omega =0,\hfill \\ {\varphi }^7:4\kappa z_2{k_2}^8{X}^7-4z_1{k_2}^8{X}^7\omega -2\nu z_1{X}^7{k_2}^8{z}_2\omega =0,\hfill \\ {\varphi }^8:12z_2{k_2}^8{X}^8\omega +\nu {z_2}^2{X}^8{k_2}^8\omega =0.\hfill \end{array}$$

(7)

Case 1:

$$\begin{array}{cc}\hfill z_0& ={\displaystyle \frac{1}{2}}{\displaystyle \frac{\mu }{\nu }},z_1=0,z_{-1}=-{\displaystyle \frac{1}{2}}\mu {\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}},\hfill \\ \hfill z_2& =0,z_{-2}=-{\displaystyle \frac{1}{48}}{\displaystyle \frac{{\mu }^2}{\nu }},X={\displaystyle \frac{1}{24}}\mu ,\omega ={\displaystyle \frac{1}{5}}\sqrt{-6{\mu }^{-1}}\kappa .\hfill \end{array}$$

(8)

Case 2:

$$\begin{array}{cc}\hfill z_0=& {\displaystyle \frac{3}{4}}{\displaystyle \frac{\mu }{\nu }},z_1=12{\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}},z_{-1}=-{\displaystyle \frac{1}{8}}\mu {\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}},\hfill \\ \hfill z_2& =-12{\nu }^{-1},z_{-2}=-{\displaystyle \frac{1}{768}}{\displaystyle \frac{{\mu }^2}{\nu }},X={\displaystyle \frac{1}{96}}\mu ,\omega ={\displaystyle \frac{1}{5}}\sqrt{-6{\mu }^{-1}}\kappa .\hfill \end{array}$$

(9)

This study employs the Riccati–Bernoulli sub-ODE technique to simplify the solutions of nonlinear partial differential equations (PDEs) by transforming them into ordinary differential equations (ODEs). This method’s strength is seen in the fact that it considers nonlinear terms analytically and thus offers multiple traveling wave solutions. However, due to the complexity of some of the nonlinear PDEs, better solutions call for further improvements, which is where the Bäcklund transformation comes into play. The Bäcklund transformation can be accomplished as a bridge between the different solutions of the PDE through the transformation of one solution into another more easily soluble solution. In this work, the transformation alters the forms of nonlinear terms in the PDE, enabling the use of the Riccati–Bernoulli technique to solve them. In fact, one can say that the Bäcklund transformation reduces the complexity of the process of combining nonlinear wave solutions by directly relating different solutions of the same equation.

Thus, the Bäcklund transformation and the Riccati–Bernoulli method function synergistically. The Riccati–Bernoulli method yields the exact solutions of the reduced ODE, while on the other hand, the Bäcklund transformation relates the solutions and further analyzes the steeper, uncharted solution profiles. This combined approach plays a crucial role in identifying the wave solutions for the given nonlinear PDE equation, which is crucial for modeling nanoionic currents and cell signaling. It provides both existing analytical solutions and new solutions for the wave solutions while also effectively addressing the nonlinearity of the system.

Solution family 1: Considering case 1 for $X<0$, we obtain the precise traveling wave solution for Equation (1) in the following form:

$$f_1(x,t)=-{\displaystyle \frac{1}{48}}{\displaystyle \frac{{\mu }^2{W}_1^2}{\nu W_2^2}}-{\displaystyle \frac{1}{2}}\mu W_1{\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}}{W}_2^{-1}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mu }{\nu }},$$

(10)

with

$$\begin{array}{cc}\hfill W_1& =k_1-{\displaystyle \frac{1}{12}}{k}_2\sqrt{-6\mu }tanh\left({\displaystyle \frac{1}{12}}\sqrt{-6\mu }\chi \right),\hfill \\ \hfill W_2& =-{\displaystyle \frac{1}{24}}\mu k_2-{\displaystyle \frac{1}{12}}{k}_1\sqrt{-6\mu }tanh\left({\displaystyle \frac{1}{12}}\sqrt{-6\mu }\chi \right),\hfill \end{array}$$

or

$$f_2(x,t)=-{\displaystyle \frac{1}{48}}{\displaystyle \frac{{\mu }^2{W}_3^2}{\nu W_4^2}}-{\displaystyle \frac{1}{2}}\mu W_3{\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}}{W}_4^{-1}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mu }{\nu }},$$

(11)

with

$$\begin{array}{cc}\hfill W_3& =k_1-{\displaystyle \frac{1}{12}}{k}_2\sqrt{-6\mu }coth\left({\displaystyle \frac{1}{12}}\sqrt{-6\mu }\chi \right),\hfill \\ \hfill W_4& =-{\displaystyle \frac{1}{24}}\mu k_2-{\displaystyle \frac{1}{12}}{k}_1\sqrt{-6\mu }coth\left({\displaystyle \frac{1}{12}}\sqrt{-6\mu }\chi \right).\hfill \end{array}$$

Solution family 2: Considering case 1 for $X>0$, we obtain the precise traveling wave solution for Equation (1) in the following form:

$$f_3(x,t)=-{\displaystyle \frac{1}{48}}{\displaystyle \frac{{\mu }^2{S}_1^2}{\nu S_2^2}}-{\displaystyle \frac{1}{2}}\mu S_1{\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}}{S}_2^{-1}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mu }{\nu }},$$

(12)

with

$$\begin{array}{cc}\hfill S_1& =k_1+{\displaystyle \frac{1}{12}}{k}_2\sqrt{6}\sqrt{\mu }tan\left({\displaystyle \frac{1}{12}}\sqrt{6}\sqrt{\mu }\chi \right),\hfill \\ \hfill S_2& =-{\displaystyle \frac{1}{24}}\mu k_2+{\displaystyle \frac{1}{12}}{k}_1\sqrt{6}\sqrt{\mu }tan\left({\displaystyle \frac{1}{12}}\sqrt{6}\sqrt{\mu }\chi \right),\hfill \end{array}$$

or

$$f_4(x,t)=-{\displaystyle \frac{1}{48}}{\displaystyle \frac{{\mu }^2{S}_3^2}{\nu S_4^2}}-{\displaystyle \frac{1}{2}}\mu S_3{\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}}{S}_4^{-1}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mu }{\nu }},$$

(13)

with

$$\begin{array}{cc}\hfill S_3& =k_1-{\displaystyle \frac{1}{12}}{k}_2\sqrt{6}\sqrt{\mu }cot\left({\displaystyle \frac{1}{12}}\sqrt{6}\sqrt{\mu }\chi \right),\hfill \\ \hfill S_4& =-{\displaystyle \frac{1}{24}}\mu k_2-{\displaystyle \frac{1}{12}}{k}_1\sqrt{6}\sqrt{\mu }cot\left({\displaystyle \frac{1}{12}}\sqrt{6}\sqrt{\mu }\chi \right),\hfill \end{array}$$

Solution family 3: Considering case 1 for $X=0$, we obtain the precise traveling wave solution for Equation (1) in the following form:

$$f_5(x,t)=-{\displaystyle \frac{1}{48}}{\mu }^2{S}_5^2{\nu }^{-1}{S}_6^{-2}-{\displaystyle \frac{1}{2}}\mu S_5{\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}}{S}_6^{-1}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mu }{\nu }},$$

(14)

with

$$\begin{array}{cc}\hfill S_5& =k_1-{\displaystyle \frac{k_2}{\chi }},\hfill \\ \hfill S_6& =-{\displaystyle \frac{1}{24}}\mu k_2-{\displaystyle \frac{k_1}{\chi }},\hfill \end{array}$$

Here, $\chi ={\displaystyle \frac{x}{l}}+{\displaystyle \frac{1}{5}}\sqrt{-6{\mu }^{-1}}\kappa t{T}^{-1}.$

Solution family 4: Considering case 2 for $X<0$, we obtain the precise traveling wave solution for Equation (1) in the following form:

$$f_6(x,t)={\displaystyle \frac{z_{-2}{R}_1^2}{R_2^2}}+{\displaystyle \frac{z_{-1}{R}_1}{R_2}}+z_0+{\displaystyle \frac{z_1{R}_2}{R_1}}-12{\displaystyle \frac{R_2^2}{\nu R_1^2}},$$

(15)

with

$$\begin{array}{cc}\hfill R_1& =k_1-k_2\sqrt{-X}tanh\left(\sqrt{-X}\chi \right),\hfill \\ \hfill R_2& =-X{k}_2-k_1\sqrt{-X}tanh\left(\sqrt{-X}\chi \right),\hfill \end{array}$$

or

$$f_7(x,t)={\displaystyle \frac{z_{-2}{R}_3^2}{R_4^2}}+{\displaystyle \frac{z_{-1}{R}_3}{R_4}}+z_0+{\displaystyle \frac{z_1{R}_4}{R_3}}-12{\displaystyle \frac{R_4^2}{\nu R_3^2}},$$

(16)

with

$$\begin{array}{cc}\hfill R_3& =k_1-k_2\sqrt{-X}coth\left(\sqrt{-X}\chi \right),\hfill \\ \hfill R_4& =-X{k}_2-k_1\sqrt{-X}coth\left(\sqrt{-X}\chi \right).\hfill \end{array}$$

Solution family 5: Considering case 2 for $X>0$, we obtain the precise traveling wave solution for Equation (1) in the following form:

$$f_8(x,t)={\displaystyle \frac{z_{-2}{G}_1^2}{G_2^2}}+{\displaystyle \frac{z_{-1}{G}_1}{G_2}}+z_0+{\displaystyle \frac{z_1{G}_2}{G_1}}-12{\displaystyle \frac{G_2^2}{\nu G_1^2}},$$

(17)

with

$$\begin{array}{cc}\hfill G_1& =k_1+k_2\sqrt{X}tan\left(\sqrt{X}\chi \right),\hfill \\ \hfill G_2& =-X{k}_2+k_1\sqrt{X}tan\left(\sqrt{X}\chi \right),\hfill \end{array}$$

or

$$f_9(x,t)={\displaystyle \frac{z_{-2}{G}_3^2}{G_4^2}}+{\displaystyle \frac{z_{-1}{G}_3}{G_4}}+z_0+{\displaystyle \frac{z_1\left(-X{k}_2-k_1\sqrt{X}cot\left(\sqrt{X}\chi \right)\right)}{G_3}}-12{\displaystyle \frac{G_4^2}{\nu G_3^2}},$$

(18)

with

$$\begin{array}{cc}\hfill G_3& =k_1-k_2\sqrt{X}cot\left(\sqrt{X}\chi \right),\hfill \\ \hfill G_4& =-X{k}_2-k_1\sqrt{X}cot\left(\sqrt{X}\chi \right).\hfill \end{array}$$

Solution family 6: Considering case 2 for $X=0$, we obtain the precise traveling wave solution for Equation (1) in the following form:

$$f_{10}(x,t)=z_{-2}{G}_5^2{G}_6^{-2}+z_{-1}{G}_5{G}_6^{-1}+z_0+z_1{G}_6{G}_5^{-1}-12G_6^2{\nu }^{-1}{G}_5^{-2},$$

(19)

with

$$\begin{array}{cc}\hfill G_5& =k_1-{\displaystyle \frac{k_2}{\chi }},\hfill \\ \hfill G_6& =-X{k}_2-{\displaystyle \frac{k_1}{\chi }}.\hfill \end{array}$$

Here, $\chi ={\displaystyle \frac{x}{l}}+{\displaystyle \frac{1}{5}}\sqrt{-6{\mu }^{-1}}\kappa t{T}^{-1},z_1=12{\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}},z_{-1}=-{\displaystyle \frac{1}{8}}\mu {\nu }^{-1}{\displaystyle \frac{1}{\sqrt{-6{\mu }^{-1}}}},z_{-2}=-{\displaystyle \frac{1}{768}}{\displaystyle \frac{{\mu }^2}{\nu }},z_0={\displaystyle \frac{3}{4}}{\displaystyle \frac{\mu }{\nu }}$ and $X={\displaystyle \frac{1}{96}}\mu ,$.

4. Results and Discussion

In this paper, we describe a methodology for nonlinear PDEs that arise in nanosciences and biophysics, with a focus on microtubule transmission line models for nanoionic currents. These PDEs are then transformed into ODEs via the Bäcklund transformation, which helps to reduce the complexity of the problem and makes it easier to solve. We find the coefficients for the series solution using the Riccati–Bernoulli sub-ODE approach, enhancing our understanding of the system’s integrated dynamics. The Riccati–Bernoulli sub-ODE method provides a detailed framework for solving the equation system, revealing three distinct families of solutions. There are three classes of functions, namely the hyperbolic, rational, and trigonometric functions. Despite the differences in the form and structure of these solutions, each of them has specific analytical potential to effectively reveal the system’s features. This framework is especially useful in capturing such patterns as kink solitons, which are important in theoretical applications including solitons in particle physics and dislocations in solids. By identifying these solution families, the method increases the opportunities for understanding the various behaviors of complex systems, underscoring the significance of the results for contemporary biophysics and nanoscience. In our numerical analysis of the obtained solutions, the following values for all related parameters are considered [18]: $k_1=0.1$, $k_2=0.3$, $\kappa ={\displaystyle \frac{T}{r_2{c}_0}}$, $\nu ={\displaystyle \frac{2r_1\gamma }{r_2}}$, $\gamma ={\displaystyle \frac{r_1}{r_2}}$, $T=r_1{c}_0=1.32\times {10}^{-6}$, and $\gamma =-1$. We demonstrated the relationship between the kink and anti-kink dynamics by analyzing the distinct cases derived from the algebraic system. In particular, we presented the development of kink-blow dynamics by examining the solutions obtained for various parameter values and spatial distributions. The solution families, derived from Case 1 and Case 2, reflect different physical conditions, allowing for a more detailed exploration of the system stability, propagation characteristics, and interaction behaviors. Each family provides a distinct representation of how the model behaves under specific parameter configurations.

Figure 1 presents the development of anti-kink solutions through their evolution in the context of microtubules modeled by nonlinear RLC transmission lines in nanobiosciences. We presented the link between the obtained solutions and the dynamics of kinks through the details in the behavior of kink-blow dynamics with different parameters and spatial configurations. The solution provides more evidence of stability and other propagation aspects of the system. The changes in the values of the parameters may affect the interactions in such a way that the physical conditions driving the variation in wave dynamics can be interpreted, hence providing the possibility to describe the interaction processes at the nanoscale in detail.

Figure 2 shows the kink plot of these solutions compared to the nonlinear wave solutions previously presented. Both plots demonstrate the change in space under different parameters and distributions, with contours indicating regions where specific parameters, such as amplitude and phase, are the same. These pictures have given us essential details about how stable microtubules are, how they move, and how they interact at the nanoscale. This is critical for understanding and manipulating microtubules in biological settings.

Figure 3 indicates the dynamics of localized disturbances and nonlinear waves (kink waves) in microtubules formulated as nonlinear RLC transmission lines in nanoscience. This plot concerns how kink profiles change with respect to different parameters, and the distribution of solitons as compared to the anti-symmetric patterns of anti-kink profiles. The contour lines on the plot show areas with constant parameters such as amplitude and phase. This shows how complicated the stability, propagation, and interaction mechanisms are at the nanoscale level for microtubules, which are essential for biological functioning.

The anti-kink plot in Figure 4 shows how the anti-kink modes in microtubules move, as seen through the nonlinear RLC transmission lines used in nanosciences. This plot demonstrates a distinct interference of these solutions under the parameters and spatial distributions, as well as their symmetrical mode against the localized perturbations that could be observed in the kink solutions. This knowledge is essential for explaining the complexity of biological processes that involve microtubules in cells.

Figure 5 shows the kink plot describing the velocities of local disturbances and nonlinear wave dynamics. From the kink plot, it is well evinced that kink solutions present neat velocity shooting profiles for changes in parameters and density distributions. This contrasts with the almost symmetrical speed profile observed in anti-kink solutions. Actually, in the dissimilarity of the velocity modes, interesting information about the very complex biomechanical processes and transport phenomena realized by microtubules within cells can be found. Understanding such variability is crucial for understanding how microtubules function in various dynamic cellular contexts.

To prove the effectiveness of the method used in this study over some methods in analyzing the problem under study, we made a simple comparison between the derived and derived solutions using the modified extended tanh-function method (METFM), as shown in

Table 1. Comparison of the current approach with the alternative approach, the modified extended tanh-function method (METFM) [20].

Case I: $X<0$	Present method	$f(x,t)=\frac{z_{-2}{R}_1^2}{R_2^2}+\frac{z_{-1}{R}_1}{R_2}+z_0+\frac{z_1{R}_2}{R_1}-12\frac{R_2^2}{\nu R_1^2}.$
Case I: $b<0$	METFM	$u(x,t)={\displaystyle \frac{3}{4}}{\displaystyle \frac{\gamma }{\beta }}-a_1\sqrt{{\displaystyle \frac{1}{24}}}\gamma tanh\left({\theta }_1\right)-{\displaystyle \frac{\gamma }{4\beta }}{tanh}^2\left({\theta }_1\right)$
Case II: $X>0$	Present method	$f(x,t)={\displaystyle \frac{z_{-2}{G}_1^2}{G_2^2}}+{\displaystyle \frac{z_{-1}{G}_1}{G_2}}+z_0+{\displaystyle \frac{z_1{G}_2}{G_1}}-12{\displaystyle \frac{G_2^2}{\nu G_1^2}}.$
Case II: $b>0$	METFM	$u(x,t)={\displaystyle \frac{1}{4}}{\displaystyle \frac{\gamma }{\beta }}+a_1\sqrt{{\displaystyle \frac{1}{24}}}\gamma tan\left({\theta }_1\right)-{\displaystyle \frac{\gamma }{4\beta }}{tan}^2\left({\theta }_1\right)$
Case III: $X=0$	Present method	$f(x,t)=z_{-2}{G}_5^2{G}_6^{-2}+z_{-1}{G}_5{G}_6^{-1}+z_0+z_1{G}_6{G}_5^{-1}$$-12G_6^2{\nu }^{-1}{G}_5^{-2}$
Case III: $b=0$	METFM	No solutions at $b=0$.

. It is clear that there are some cases in which METFM does not provide solutions, while the technique under study does, confirming the effectiveness of the method used in this study.

5. Conclusions

• The Riccati–Bernoulli sub-ODE method has been successfully used to obtain solutions to the nonlinear PDE, the nonlinear RLC transmission line model of microtubules (MTs).
• This approach produced new traveling wave solutions of substantial importance in the nascent areas of nanotechnology, biosciences, and biophysics.
• The Riccati–Bernoulli sub-ODE method’s calculated kink solutions match well with-known solutions from other methodologies, demonstrating the method’s efficacy.
• The Riccati–Bernoulli sub-ODE method seems to be an effective, brief, and elegant representation for analyzing wide classes of nonlinear evolution equations in physics.
• We have established the applicability and robustness of the proposed method by demonstrating its ability to yield further solutions.

6. Future Work

Another direction that could be explored in the future would be a continuation of the studies of the usage of the Riccati–Bernoulli sub-ODE method and Bäcklund transformation on more complicated nonlinear phenomena and processes in nanosciences and other branches of science, such as analyzing different nonlinear structures in plasma physics. Future work can be expanded to examine various types of physical PDEs that describe different nonlinear structures in plasma and fluids or to include stochastic perturbations to encapsulate the noise associated with nanoionic currents.

Moreover, the methods introduced in this study can be extended to investigate wave propagation in biological structures other than microtubules, such as in protein complexes or neuronal signal transduction pathways. More experimentally verifiable scenarios involving the calculated solutions may also be another line of research interest, along with real-time integration of the solutions into nanotechnology and biophysical research. These could not only extend the current study to a wider range but also promote more developments in theoretical modeling and application. However, it is crucial to understand that the Riccati–Bernoulli sub-ODE method and Bäcklund transformation offer reliable solutions to the studied nonlinear PDEs, though there are certain shortcomings. When applied to highly nonlinear systems or systems with complex boundary conditions, the method may encounter problems. The use of symbolic computation also becomes problematic when the method runs into computational complexity issues. In addition, this approach could potentially not completely solve high-dimensional cases or variability. It is important to note this study’s limitations in order to set boundaries and expectations for its applicability and further research directions.