Uncovering New Wave Profiles in Boussinesq-Type KdV Systems Through Symbolic and Semi-Analytical Methods

Abstract

We study here the Boussinesq-type Korteweg–de Vries (KdV) equation, a nonlinear partial differential equation, for describing the wave propagation of long, nonlinear, and dispersive waves in shallow water and other physical scenarios. In order to obtain novel families of wave solutions, we apply two efficient analytical techniques: the Modified Extended tanh (ME-tanh) method and the Modified Residual Power Series Method (mRPSM). These methods are used for the very first time in this equation to produce both exact and high-order approximate solutions with rich wave behaviors including soliton formation and energy localization. The ME-tanh method produces a rich class of closed-form soliton solutions via systematic simplification of the PDE into simple ordinary differential forms that are readily solved, while the mRPSM produces fast-convergent approximate solutions via a power series representation by iteration. The accuracy and validity of the results are validated using symbolic computation programs such as Maple and Mathematica. The study not only enriches the current solution set of the Boussinesq-type KdV equation but also demonstrates the efficiency of hybrid analytical techniques in uncovering sophisticated wave patterns in multimensional spaces. Our findings find application in coastal hydrodynamics, nonlinear optics, geophysics, and the theory of elasticity, where accurate modeling of wave evolution is significant.

1. Introduction

Nonlinear partial differential equations (NLPDE) have long been fundamental in describing wave phenomena in various branches of physics and engineering. One of the most prominent among them is the Korteweg–de Vries (KdV) equation, first derived by Diederik Korteweg and Gustav de Vries in 1895 to model long surface gravity waves in shallow water [1]. Since its inception, the KdV equation has become a cornerstone in soliton theory and nonlinear wave analysis.

A notable extension of the classical KdV equation is the Boussinesq-type KdV equation, which incorporates higher-order dispersion and nonlinearity to model bidirectional wave propagation in shallow water, elastic rods, and other nonlinear dispersive media. This equation better captures the dynamics of long waves in weakly nonlinear regimes where both nonlinearity and dispersion significantly influence wave shape and speed [2].

Given the complexity of such nonlinear equations, a wide range of analytical and numerical methods have been developed to obtain their solutions. Classical techniques include the Hirota’s bilinear method [3], the exp-function methods [4,5], the sub-equation method [6,7], the residual power series method [8,9,10], the iterative shehu transform method [11], the tanh method [12,13], the $(G^{\prime }/G)$ and ${(G^{\prime }/G)}^2$-expansion methods [14,15,16,17], hybrid residual power series methods [18,19], and the ME-tanh method [20].

In recent years, hybrid and improved analytical techniques have gained traction for their ability to provide accurate and tractable solutions. Among them, the Modified Extended tanh (ME-tanh) method extends the classical tanh approach by incorporating additional degrees of freedom in the ansatz, making it suitable for constructing a wider variety of wave solutions. On the other hand, the mRPSM builds upon power series theory by recursively generating solution components, offering a flexible semi-analytical framework that bridges analytical and numerical results.

The methods employed in this work each have distinct strengths and limitations that are useful to keep in mind when applying them to nonlinear PDEs. The ME-tanh method is a strong analytical tool for constructing explicit closed-form soliton solutions. Its major strength lies in its simplicity and systematic reduction of a PDE to an ODE through an ansatz, which then generates exact localized waveforms such as solitary waves and periodic structures. However, this approach requires certain balancing conditions between nonlinear and dispersive terms and, therefore, may not capture all solution types in more complex systems. In contrast, the mRPSM is a highly flexible semi-analytical technique that systematically produces higher-order approximations that quickly converge to exact solutions. From this point of view, it controls accuracy at a reasonable computational cost. So, it is useful especially in cases where closed-form solutions cannot be realistically attained. Its drawback rests with the recursive derivation and truncation process. Accuracy depends on how many terms can be retained practically.

By combining these two approaches, we balance the ability of the ME-tanh method to provide clear analytical insight with the strength of the mRPSM to confirm accuracy and extend the analysis into regimes where exact solutions are difficult. This dual application not only enriches the solution space of the Boussinesq-type KdV equation but also provides a framework that can be adapted to other nonlinear PDEs encountered in physics and engineering.

The present work focuses on applying ME-tanh and mRPSM to the Boussinesq type KdV equation, which has recently attracted increasing attention from researchers due to its rich solution structure. Our aim is to obtain both analytical and approximate numerical solutions for the equation using these two approaches. This is due to the need to discover new families of solutions and to verify the capabilities of modern hybrid methods. The equation we work on is as follows [21]:

$${\chi }_{tt}+{\chi }_{xx}+\gamma {\chi }_{xxxx}+\lambda {\left({\chi }^2\right)}_{xx}+\theta {\chi }_{xy}+d{\chi }_{xz}+f{\chi }_{xt}=0.$$

(1)

It coincides with the classical Boussinesq equation when the mixed-derivative couplings are switched off and there is no transverse variation, i.e., set

$$a=b=c=0\mathrm{and}{\chi }_y={\psi }_z=0,$$

which yields

$${\chi }_{tt}+{\chi }_{xx}+\alpha {\chi }_{xxxx}+\beta {\left({\chi }^2\right)}_{xx}=0,$$

up to standard sign/scaling conventions. The KdV equation is then obtained from this Boussinesq form under the usual unidirectional, weakly nonlinear, long-wavelength asymptotic reduction; no additional parameter choices beyond $a=b=c=0$ are required—KdV arises via this scaling of the equation.

In real-life applications, this type of equation is particularly valuable for modeling shallow water waves in environments like coastal regions, rivers, and canals, where wave behavior is influenced by both nonlinearity and dispersion. The model is capable of describing wave phenomena such as solitons localized wave packets that maintain their shape over long distances due to a balance between nonlinear steepening and dispersive spreading. These solitons are not only observed in water waves but also arise in many other physical systems.

Beyond hydrodynamics, Boussinesq-type KdV equations have significant applications in geophysics, particularly in modeling tsunami waves or internal waves in oceans and the atmosphere. The equations are also used in elasticity theory to describe nonlinear stress waves in solid media, making them useful for studying seismic wave propagation in the Earth’s crust or engineered materials. In the field of nonlinear optics, similar models describe the transmission of light pulses through optical fibers, capturing the soliton-like behavior of these pulses under certain conditions.

From a mathematical perspective, these equations are rich in structure and often exhibit integrability properties or allow for exact solutions via techniques like the inverse scattering transform, Lie symmetry analysis, or perturbation methods. The presence of nonlinear terms, such as ${\left({\Phi }^2\right)}_{xx}$, combined with higher-order dispersive terms like ${\Phi }_{xxxx}$, allows the equation to represent a wide range of complex wave behaviors. Additional mixed derivatives like ${\Phi }_{xy},{\Phi }_{xz},$ and ${\Phi }_{xt}$ enable the modeling of wave interactions in multiple spatial dimensions and directions, reflecting more realistic physical systems. Wazwaz and Kaur [22] used the Hirota’s bilinear method to derive multi-soliton (MS) solutions of the Boussinesq type equation, emphasizing its integrability and complex soliton interactions.

To the best of our knowledge, the ME-tanh and mRPSM methods have not yet been applied to this equation. In this work, we solve the Boussinesq-type KdV equation for the first time using these approaches. The modified extended tanh method generalizes the classical tanh framework by introducing greater flexibility in the ansatz, which enables the construction of new closed-form soliton solutions. Complementarily, the mRPSM offers fast-convergent semi-analytical approximations with controllable error quantification, making it effective for modeling complex nonlinear dynamics. This combined strategy couples exact and approximate solutions, significantly enriching the known solution space and providing insight beyond earlier studies that primarily relied on Hirota’s bilinear or exp-function methods.

Of interest in our study is the fact that the exact soliton trajectories obtained via the ME-tanh method and the fast-converging approximations derived through the mRPSM have never previously appeared in the context of Boussinesq-type KdV theory. Earlier works focused mainly on multisoliton or rational solutions obtained through bilinear or exp-function techniques, whereas our results introduce entirely new waveform families. These include localized solitary structures, energy-confined profiles, and systematically verified approximations, thereby delivering both novel analytical solutions and rigorously converging iterative schemes as original contributions of this work.

In this study, Section 2 includes the definitions of the methods we use and theorems. Section 3 includes the solutions obtained using the ME-tanh method and their graphical representations. Section 4 includes the interpretation of the graphics obtained from the exact solution. Section 5 includes the numerical solutions obtained by mRPSM and their tabular representations. Section 6 includes the physical interpretation of the numerical results, and finally, Section 7 includes the conclusions.

2. Solution Techniques

2.1. ME-Tanh Method

The fundamental principle of the modified extended tanh method [20] is illustrated through the analysis of a general partial differential equation (PDE) of the form:

$$\mathcal{P}(\chi ,{\chi }_x,{\chi }_y,{\chi }_z,{\chi }_t,{\chi }_{xx},{\chi }_{xy},{\chi }_{xz},{\chi }_{xt},\dots )=0,$$

(2)

where $\mathcal{P}$ denotes a polynomial function that includes nonlinear terms involving partial derivatives of the function $\chi =\chi (x,y,z,t)$.

To reduce the PDE to an ordinary differential equation (ODE), the following transformation is employed:

$$\chi (x,y,z,t)=\chi \left(\xi \right),\xi =hx+py+rz+gt,$$

(3)

where $h,p,r,$ and g are constants. Substituting the transformation (3) into Equation (2) yields an ODE of the form:

$$\mathcal{H}\left(\chi ,{\chi }^{\prime },{\chi }^{″},\dots \right)=0.$$

(4)

Let us consider that the solution to Equation (4) takes the following form:

$$\chi =u_0+\sum _{r=1}^N\left(u_r\Lambda {\left(\xi \right)}^r\Lambda \left(\xi \right))+y_r\Lambda {\left(\xi \right)}^{-r}\left(\xi \right)\right),r\in \{1,2,\dots ,N\},$$

(5)

where $u_r$ and $y_r$ are constants that must be found, and $u_N\ne 0$, $y_N\ne 0$ with $u_N,y_N$ being numbers that are not both zero simultaneously, and $\Lambda \left(\xi \right)$ meets the Riccati equation as follows:

$${\Lambda }^{\prime }\left(\xi \right)=\delta +{\Lambda \left(\xi \right)}^2.$$

(6)

Here, $\delta$ is a parameter that will be found out afterward. Numerous solutions to Equation (6) are possible, as may be shown in

Table 1. Forms of $\Lambda \left(\xi \right)$ for different signs of $\delta$.

If $\mathit{\delta }<0$	If $\mathit{\delta }>0$
${\Lambda }_1=-\sqrt{-\delta }tanh\left(\sqrt{-\delta }(\xi +{\xi }_0)\right)$	${\Lambda }_{11}=\sqrt{\delta }tan\left(\sqrt{\delta }(\xi +{\xi }_0)\right)$
${\Lambda }_2=-\sqrt{-\delta }coth\left(\sqrt{-\delta }(\xi +{\xi }_0)\right)$	${\Lambda }_{12}=-\sqrt{\delta }cot\left(\sqrt{\delta }(\xi +{\xi }_0)\right)$
${\Lambda }_3=-\sqrt{-\delta }\left[tanh\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)+i\sec \mathrm{h}\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)\right]$	${\Lambda }_{13}=\sqrt{\delta }\left[tan\left(2\sqrt{\delta }(\xi +{\xi }_0)\right)+sec\left(2\sqrt{\delta }(\xi +{\xi }_0)\right)\right]$
${\Lambda }_4=-\sqrt{-\delta }\left[coth\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)+\csc \mathrm{h}\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)\right]$	${\Lambda }_{14}={\displaystyle \frac{1}{2}}\sqrt{\delta }\left[tan\left({\displaystyle \frac{\sqrt{\delta }(\xi +{\xi }_0)}{2}}\right)-cot\left({\displaystyle \frac{\sqrt{\delta }(\xi +{\xi }_0)}{2}}\right)\right]$
${\Lambda }_5=-{\displaystyle \frac{1}{2}}\sqrt{-\delta }\left[tanh\left({\displaystyle \frac{\sqrt{-\delta }(\xi +{\xi }_0)}{2}}\right)+coth\left({\displaystyle \frac{\sqrt{-\delta }(\xi +{\xi }_0)}{2}}\right)\right]$	${\Lambda }_{15}=-{\displaystyle \frac{\sqrt{\delta }\left[1-tan\left(\sqrt{\delta }(\xi +{\xi }_0)\right)\right]}{1+tan\left(\sqrt{\delta }(\xi +{\xi }_0)\right)}}$
${\Lambda }_6={\displaystyle \frac{\delta -\sqrt{-\delta }tanh\left(\sqrt{-\delta }(\xi +{\xi }_0)\right)}{1+\sqrt{-\delta }tanh\left(\sqrt{-\delta }(\xi +{\xi }_0)\right)}}$	${\Lambda }_{16}={\displaystyle \frac{\sqrt{\delta }\left[4-5cos\left(2\sqrt{\delta }(\xi +{\xi }_0)\right)\right]}{3+5sin\left(2\sqrt{\delta }(\xi +{\xi }_0)\right)}}$
${\Lambda }_7=-{\displaystyle \frac{\sqrt{-\delta }\left[5-4cosh\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)\right]}{3+4sinh\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)}}$	${\Lambda }_{17}=-{\displaystyle \frac{2\sqrt{\delta }}{tan\left(\frac{\sqrt{\delta }(\xi +{\xi }_0)}{2}\right)-cot\left(\frac{\sqrt{\delta }(\xi +{\xi }_0)}{2}\right)}}$
${\Lambda }_8={\displaystyle \frac{\sqrt{-\delta (a^2+b^2)}-a\sqrt{-\delta }cosh\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)}{asinh\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)+b}}$	${\Lambda }_{18}=i\sqrt{\delta }\left[1-{\displaystyle \frac{2a}{a+cos\left(2\sqrt{\delta }(\xi +{\xi }_0)\right)-isin\left(2\sqrt{\delta }(\xi +{\xi }_0)\right)}}\right]$
${\Lambda }_9=-{\displaystyle \frac{2\sqrt{-\delta }}{tanh\left(\frac{\sqrt{-\delta }(\xi +{\xi }_0)}{2}\right)+coth\left(\frac{\sqrt{-\delta }(\xi +{\xi }_0)}{2}\right)}}$	If $\delta =0$
${\Lambda }_{10}=\sqrt{-\delta }\left[1-{\displaystyle \frac{2a}{a+cosh\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)-sinh\left(2\sqrt{-\delta }(\xi +{\xi }_0)\right)}}\right]$	${\Lambda }_{19}={\displaystyle \frac{1}{\xi +{\xi }_0}}$

.

The highest-order derivative and the largest nonlinear variable are balanced to provide the positive integer N in Equation (5).

By replacing Equation (5), its derivative, and Equation (6) into Equation (4), as well as collecting all the terms of the same power ${\Lambda }^r$, $(r=1,2,\dots ,N)$ and equating them to zero, one can use a symbolic computation tool to determine the values of $u_N$ and $y_N$. By entering these values and the solutions into Equation (5), one can obtain the exact solutions to Equation (2).

2.2. The mRPSM

We discuss several definitions and theorems pertaining to RPSM that are utilized in this study in this part.

Definition 1.

The power series is given by

$$\sum _{m=0}^{\infty }{a}_m{\left(x-x_0\right)}^m=a_0+a_1\left(x-x_0\right)+a_2{\left(x-x_0\right)}^2+\dots ,$$

(7)

where $x\le x_0,$ is called power series (PS) about $x=x_0$.

Theorem 1.

Assume the function h has a PS at $x=x_0$ as

$$h\left(x\right)=\sum _{k=0}^{\infty }{a}_k{\left(x-x_0\right)}^k,x_0\le x\le x_0+R,$$

(8)

where R is the radius of convergence. If $h^{\left(k\right)}\left(x\right)\in C\left(x_0,x_0+R\right),$$k=0,1,2,\dots ,$ then $a_k={\displaystyle \frac{h^{\left(k\right)}\left(x_0\right)}{k!}},$ and $h^{\left(k\right)}\left(x\right)$ is the kth derivative of the h function.

Definition 2.

A series

$$\sum _{k=0}^{\infty }{h}_m\left(t\right){\left(x-x_0\right)}^k$$

(9)

is called the multiple power series (MPS) about $x=x_0$.

Theorem 2.

Assume $v(z,x)$ has MPS at $x=x_0$

$$v(z,x)=\sum _{k=0}^{\infty }{h}_k\left(z\right){\left(x-x_0\right)}^k,z\in J,x_0\le x\le x_0+R.$$

(10)

If $v_x^{\left(k\right)}v\left(z,x\right)\in C\left(J\times \left(x_0,x_0+R\right)\right),$ for $k=0,1,2,\dots ,$ then $h_k\left(z\right)={\displaystyle \frac{v_x^{\left(k\right)}v\left(z,x_0\right)}{k!}}$.

The RPS method that suggested the problem’s resolution is now provided by

$$u\left(x,t\right)=\sum _{n=0}^{\infty }{c}_n\left(x\right){\displaystyle \frac{t^n}{n!}}.$$

(11)

To obtain the approximate values of the above series Equation (11), we consider its jth truncated series $u_j\left(x,t\right)$ which has the form

$$u_j\left(x,t\right)=\sum _{n=0}^k{c}_n\left(x\right){\displaystyle \frac{t^n}{n!}}.$$

(12)

Since $u\left(x,0\right)=c_0\left(x\right)=A\left(x\right)$ and $u_t\left(x,0\right)=c_1\left(x\right)=B\left(x\right),$ we rewrite Equation (12) as

$$\begin{array}{ccc}\hfill u_j\left(x,t\right)& =& u_1\left(x,t\right)+\sum _{n=2}^j{c}_n\left(x\right){\displaystyle \frac{t^n}{n!}},k=2,3,\dots \hfill \\ & =& A\left(x\right)+B\left(x\right)t+\sum _{n=2}^j{c}_n\left(x\right){\displaystyle \frac{t^n}{n!}},\hfill \end{array}$$

(13)

where $u_1(x,t)=A\left(x\right)+B\left(x\right)t$ is considered to be the first RPS approximate solution of $u(x,t)$.

To clarify the mRPSM steps, let $F\left[u\right]$ denote the operator in Equation (1). For the truncation $u_j$ in Equation (13), define the residual ${\mathrm{Res}}_j(x,y,z,t)=F\left[u_j(x,y,z,t)\right]$ and expand it in t about 0 as ${\mathrm{Res}}_j={\sum }_{k\ge 0}\frac{t^k}{k!}{R}_{j,k}(x,y,z)$, with $R_{j,k}=\left({\partial }_t^k{\mathrm{Res}}_j\right){|}_{t=0}$. Because Equation (1) is second order in time, the new coefficient $c_j$ first appears in $R_{j,j-2}$; hence, at each step we compute $c_j$ from the residual condition Equation (13) using only A, B, and the previously obtained $c_2,\dots ,c_{j-1}$. Iterating $j=2,3,\dots$ yields $c_2,c_3,\dots$ and the k-th approximation $u_k=A+tB+{\sum }_{n=2}^k\frac{t^n}{n!}{c}_n$.

Finding the explicit form of the RPS coefficients for the Boussinesq-type KdV equation is the focus of the following section.

3. Solutions of the Model

The ME Tanh-Function Method Solutions

Letting the transformations

$$\chi (x,y,z,t)=\chi \left(\xi \right),\xi =hx+py+rz+gt,$$

(14)

and by utilizing Equation (2) and integrating twice while setting the integration constants to zero, the aforementioned equation reduces to an ODE as

$$\gamma h^4{\chi }^{″}+\left(h(dr+fg+p\theta )+h^2+g^2\right)\chi +\lambda h^2{\chi }^2.$$

(15)

The balancing rule for Equation (15) using ${\chi }^{″}$ with ${\chi }^2$ yields $N=2$. Therefore, Equation (5) becomes

$$\chi =u_0+u_1\Lambda \left(\xi \right)+u_2\Lambda {\left(\xi \right)}^2+{\displaystyle \frac{y_1}{\Lambda \left(\xi \right)}}+{\displaystyle \frac{y_2}{\Lambda {\left(\xi \right)}^2}}.$$

(16)

When combined with Equation (6), the algebraic equation system that follows is created.

$$\begin{array}{c}\Lambda {\left(\xi \right)}^4:6\gamma h^4{u}_2+h^2\lambda u_2^2=0,\\ \Lambda {\left(\xi \right)}^3:2\gamma h^4{u}_1+2h^2\lambda u_1{u}_2=0,\\ \Lambda {\left(\xi \right)}^2:8\delta \gamma h^4{u}_2+2h^2\lambda u_0{u}_2+h^2\lambda u_1^2+dhr{u}_2+fgh{u}_2+hp\theta u_2+g^2{u}_2+h^2{u}_2=0,\\ \Lambda {\left(\xi \right)}^1:2\delta \gamma h^4{u}_1+2h^2\lambda u_0{u}_1+2h^2\lambda u_2{y}_1+dhr{u}_1+fgh{u}_1+hp\theta u_1+g^2{u}_1+h^2{u}_1=0,\\ \Lambda {\left(\xi \right)}^0:2{\delta }^2\gamma h^4{u}_2+2\gamma h^4{y}_2+h^2\lambda u_0^2+2h^2\lambda u_1{y}_1+2h^2\lambda u_2{y}_2+dhr{u}_0+fgh{u}_0+hp\theta u_0+g^2{u}_0+h^2{u}_0=0,\\ \Lambda {\left(\xi \right)}^{-1}:2\delta \gamma h^4{y}_1+2h^2\lambda u_0{y}_1+2h^2\lambda u_1{y}_2+dhr{y}_1+fgh{y}_1+hp\theta y_1+g^2{y}_1+h^2{y}_1=0,\\ \Lambda {\left(\xi \right)}^{-2}:8\delta \gamma h^4{y}_2+2h^2\lambda u_0{y}_2+h^2\lambda y_1^2+dhr{y}_2+fgh{y}_2+hp\theta y_2+g^2{y}_2+h^2{y}_2=0,\\ \Lambda {\left(\xi \right)}^{-3}:2{\delta }^2\gamma h^4{y}_1+2h^2\lambda y_1{y}_{2}0,\\ \Lambda {\left(\xi \right)}^{-4}:6{\delta }^2\gamma h^4{y}_2+h^2\lambda y_2^2=0.\end{array}$$

For $u_0$, $u_1$, $u_2$, $y_1$, and $y_2$, we have two cases and two sets of solutions here.

Case 1.

$$\left\{u_0=-{\displaystyle \frac{12\gamma h^2\delta }{\lambda }},u_1=0,u_2=-{\displaystyle \frac{6\gamma h^2}{\lambda }},y_1=0,y_2=-{\displaystyle \frac{6\gamma h^2{\delta }^2}{\lambda }},\theta ={\displaystyle \frac{-dhr-fgh-g^2+16\gamma h^4\delta -h^2}{hp}}\right\}.$$

(17)

Set 1.

For $\delta <0$,

$$\begin{array}{c}\hfill {\chi }_1={\displaystyle \frac{6\delta \gamma h^2{\left(tanh\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-1\right)}^2{\left(tanh\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+1\right)}^2}{\lambda tanh{\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_2={\displaystyle \frac{6\delta \gamma h^2{\left(coth\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-1\right)}^2{\left(coth\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+1\right)}^2}{\lambda coth{\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_3={\displaystyle \frac{6\gamma \delta {\left(\mathrm{I}\mathrm{sech}\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+tanh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+1\right)}^2{\left(\mathrm{I}\mathrm{sech}\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+tanh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-1\right)}^2{h}^2}{\lambda {\left(tanh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+\mathrm{I}\mathrm{sech}\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_4={\displaystyle \frac{6\delta \gamma h^2{\left(coth\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+\mathrm{csch}\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+1\right)}^2{\left(coth\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+\mathrm{csch}\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-1\right)}^2}{\lambda {\left(coth\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+\mathrm{csch}\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_5={\displaystyle \frac{3\delta \gamma h^2{\left(tanh\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)+coth\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)+2\right)}^2{\left(tanh\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)+coth\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)-2\right)}^2}{2\lambda {\left(tanh\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)+coth\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_6=-{\displaystyle \frac{6\gamma h^2{\delta }^2{\left(\delta +1\right)}^2{\left(tanh\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-1\right)}^2{\left(tanh\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+1\right)}^2}{\lambda {\left(1+\sqrt{-\delta }tanh\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2{\left(\sqrt{-\delta }tanh\left(\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-\delta \right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_7={\displaystyle \frac{384\delta \gamma h^2{\left(2cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-1+2sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2{\left(cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-2-sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}{\lambda {\left(3+4sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2{\left(-5+4cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}},\end{array}$$

$${\chi }_8=-{\displaystyle \frac{6\gamma h^{2}a\left(\begin{array}{c}cosh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^4{a}^3{\delta }^2-4{\left(-\delta \right)}^{\frac{3}{2}}\sqrt{-\delta \left(a^2+b^2\right)}cosh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^3{a}^2\\ -2cosh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^{2}sinh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^2{a}^3{\delta }^2+sinh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^4{a}^3{\delta }^2\\ -4cosh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^{2}sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)a^{2}b{\delta }^2\\ -4\sqrt{-\delta }\sqrt{-\delta \left(a^2+b^2\right)}cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)sinh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^2{a}^2\delta \\ +4sinh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^3{a}^{2}b{\delta }^2+6cosh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^2{a}^3{\delta }^2+4cosh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^{2}a{b}^2{\delta }^2\\ -8\sqrt{-\delta }\sqrt{-\delta \left(a^2+b^2\right)}cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)ab\delta \\ -2sinh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^2{a}^3{\delta }^2+4sinh{\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)}^{2}a{b}^2{\delta }^2\\ -4\sqrt{-\delta }\sqrt{-\delta \left(a^2+b^2\right)}cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)b^2\delta -4sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)a^{2}b{\delta }^2\\ -4\sqrt{-\delta }{\left(-\delta \left(a^2+b^2\right)\right)}^{\frac{3}{2}}cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+a^3{\delta }^2\end{array}\right)}{\lambda {\left(asinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)+b\right)}^2{\left(a\sqrt{-\delta }cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-\sqrt{-\delta \left(a^2+b^2\right)}\right)}^2}},$$

$$\begin{array}{c}\hfill {\chi }_9={\displaystyle \frac{3\delta \gamma h^2{\left(tanh\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)+coth\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)+2\right)}^2{\left(tanh\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)+coth\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)-2\right)}^2}{2\lambda {\left(tanh\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)+coth\left(\frac{\sqrt{-\delta }\left(\xi +{\xi }_0\right)}{2}\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{10}={\displaystyle \frac{96\delta \gamma h^2{a}^2{\left(cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}{\lambda {\left(a+cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2{\left(-a+cosh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)-sinh\left(2\sqrt{-\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}},\end{array}$$

$$\mathrm{If}\delta >0$$

$$\begin{array}{c}\hfill {\chi }_{11}=-{\displaystyle \frac{6\delta \gamma h^2{\left(tan{\left(\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2+1\right)}^2}{\lambda tan{\left(\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{12}=-{\displaystyle \frac{6\delta \gamma h^2{\left(cot{\left(\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2+1\right)}^2}{\lambda cot{\left(\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{13}=-{\displaystyle \frac{6\delta \gamma h^2{\left(tan{\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2+2tan\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)sec\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)+sec{\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2+1\right)}^2}{\lambda {\left(tan\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)+sec\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{14}=-{\displaystyle \frac{3\delta \gamma h^2{\left(tan{\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)}^2-2tan\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)cot\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)+cot{\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)}^2+4\right)}^2}{2\lambda {\left(tan\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)-cot\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{15}=-{\displaystyle \frac{24\delta \gamma h^2{\left(tan{\left(\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2+1\right)}^2}{\lambda {\left(1+tan\left(\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2{\left(-1+tan\left(\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{16}=-{\displaystyle \frac{150\delta \gamma h^2{\left(5cos{\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2+5sin{\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)}^2-8cos\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)+6sin\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)+5\right)}^2}{\lambda {\left(3+5sin\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2{\left(-4+5cos\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{17}=-{\displaystyle \frac{3\delta \gamma h^2{\left(tan{\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)}^2-2tan\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)cot\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)+cot{\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)}^2+4\right)}^2}{2\lambda {\left(tan\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)-cot\left(\frac{\sqrt{\delta }\left(\xi +{\xi }_0\right)}{2}\right)\right)}^2}},\end{array}$$

$$\begin{array}{c}\hfill {\chi }_{18}=-{\displaystyle \frac{96\gamma \delta {\left(\mathrm{I}cos\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)+sin\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)\right)}^2{a}^2{h}^2}{\lambda {\left(sin\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)+\mathrm{I}cos\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)+\mathrm{I}a\right)}^2{\left(sin\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)+\mathrm{I}cos\left(2\sqrt{\delta }\left(\xi +{\xi }_0\right)\right)-\mathrm{I}a\right)}^2}}.\end{array}$$

If $\delta =0$

$$\begin{array}{c}\hfill {\chi }_{19}=-{\displaystyle \frac{6\gamma h^2}{\lambda {\left(\xi +{\xi }_0\right)}^2}}.\end{array}$$

4. Physical Interpretation of the Figures

Figure 1, Figure 2 and Figure 3 are plotted using the values $z=1$, $\lambda =2$, $\delta =-0.4$, $g=1$, $h=1$, $x=1$, $\gamma =-1.5$, $p=2$, $y=1$, and $r=0.3$. The parameter values used in these figures were selected primarily for two reasons: (i) to satisfy the mathematical constraints required by the ME-tanh method (such as the balancing conditions ensuring soliton-type solutions), and (ii) to produce clear illustrative plots that demonstrate the soliton structures, localization, and stability properties of the obtained solutions. While the chosen values are illustrative rather than tied to a single physical system, they remain within ranges that are consistent with physically realistic regimes in shallow water wave dynamics, elasticity, and nonlinear optics, where nonlinearity ($\lambda$) and dispersion ($\gamma$) interact to sustain solitary waves. Thus, the selections serve both as mathematically valid test cases and as representative examples of physically relevant parameter regimes. The graphical results presented in Figure 1, Figure 2 and Figure 3 provide a more qualitative understanding of the wave dynamics represented by the analytical solution ${\chi }_3(x,y,z,t)$ solved using the ME-tanh method. They consist of 3D surface plots, 2D projections, and contour plots of the real, imaginary, and absolute square parts of the solution, representing the space-time dynamics of its complete visualization.

Figure 1 shows the real part of the solution ${\chi }_3(x,y,z,t)$ at various time levels. The 3D and 2D plots show a smooth, localized wave profile that is coherent in time, characteristic of solitonic behavior. The height and shape of the wave profile evolve with time, showing nonlinear interactions governed by the interplay between dispersion and nonlinearity. Physically, the persistence and localized nature of the wavefront enhance the soliton-like behavior of the solution. Such structures are essential in shallow water wave modeling, where solitary waves travel long distances without form loss. The real part is the leading observable wave amplitude in many physical systems, such as fluid surface elevations or electric field envelopes in optical fibers.

Figure 2 plots the imaginary part of ${\chi }_3(x,y,z,t)$. The amplitude of this component is negligible, having values near machine precision, as may be observed in both surface and contour plots. This result indicates that the solution is very nearly real-valued, as should be expected for the majority of physical waveforms calculated from real-valued generalized ansatz functions. On physical grounds, the small imaginary part means that the energy and momentum of the model are carried effectively by the real part, in support of the interpretation of a purely propagating solitary wave without complex phase interactions. This is particularly significant in hydrodynamics and other conservative wave systems.

Figure 3 is a graph of the squared amplitude $|{\chi }_3{(x,y,z,t)|}^2$, which provides a measure of the wave’s energy density or intensity. The graphs show a highly peaked distribution that evolves smoothly in time, indicative of the stability and localization of the wave structure. In physical systems, this quantity typically corresponds to conserved or measurable quantities such as energy flux, probability density, or wave power, depending on the context. This steep localization here indicates a highly confined soliton that would be suitable for everyday applications in optical communications, quantum mechanics, or geophysical wave modeling.

The findings collectively confirm the ME-tanh method to be furnishing solutions with significant solitonic features, satisfactory localization, and temporal stability. The consistency between visual and numerical findings points to the validity of the method for modeling actual nonlinear wave phenomena in multidimensional settings. The graphical findings vindicate the usability of the solution derived in physical contexts where solitary wave dynamics need to be accurately modeled.

Beyond the illustrative parameter set ($\delta =-0.4$, $\gamma =-1.5$), it is important to note that varying these parameters systematically alters the dispersion–nonlinearity balance and hence the observed soliton features. For instance, decreasing $\delta$ enhances dispersive effects and broadens the soliton profile, whereas more negative values of $\gamma$ reinforce localization and improve temporal stability. This shows that parameter variation is crucial in the control of soliton persistence, energy containment, and robustness. Moreover, the imaginary part of ${\chi }_3$ remains at the order of ${10}^{-15}$, so the solution is effectively real-valued and only the real part contributes to measurable physical quantities such as wave amplitude and energy density.

5. Numerical Solutions by mRPSM

First, we assume that the first iteration solution is

$$u_1(x,y,z,t)=A(x,y,z)+B(x,y,z)t.$$

(18)

The coefficient $c_2(x,y,z)$ in the expansion Equation (12) is then found by substituting the second RPS approximation solution

$$u_2(x,y,z,t)=A(x,y,z)+tB(x,y,z)+{\displaystyle \frac{1}{2}}{t}^2{c}_2(x,y,z),$$

(19)

into ${\mathrm{Res}}_2(x,y,z,t)$ as

$$\begin{array}{ccc}\hfill {\mathrm{Res}}_2(x,y,z,t)& =& d\left(A_{xz}+t{B}_{xz}+{\displaystyle \frac{1}{2}}{t}^2{\left(\left(c_2\right)\right)}_{xz}\right)+\gamma \left(A_{xxxx}+t{B}_{xxxx}+{\displaystyle \frac{1}{2}}{t}^2{\left(\left(c_2\right)\right)}_{xxxx}\right)\hfill \\ & & +\theta \left(A_{xy}+t{B}_{xy}+{\displaystyle \frac{1}{2}}{t}^2{\left(\left(c_2\right)\right)}_{xy}\right)+2\lambda {\left(A_x+t{B}_x+{\displaystyle \frac{1}{2}}{t}^2{\left(\left(c_2\right)\right)}_x\right)}^2\hfill \\ & & +2\lambda \left(A+tB+{\displaystyle \frac{1}{2}}{t}^2\left(c_2\right)\right)\left(A_{xx}+t{B}_{xx}+{\displaystyle \frac{1}{2}}{t}^2{\left(c_2\right)}_{xx}\right)+A_{xx}\hfill \\ & & +f\left(B_x+t{\left(c_2\right)}_x\right)+t{B}_{xx}+{\displaystyle \frac{1}{2}}{t}^2{\left(c_2\right)}_{xx}+c_2.\hfill \end{array}$$

(20)

Then, we solve ${\mathrm{Res}}_2(x,y,z,0)=0$ and obtain

$$c_2(x,y,z)=-d{A}_{xz}-\gamma A_{xxxx}-\theta A_{xy}-2\lambda (A_x^2+A{A}_{xx})-A_{xx}-f{B}_x.$$

(21)

Thus, we attain the 2nd mRPSM solution as,

$$u_2(x,y,z,t)=A+tB+{\displaystyle \frac{1}{2}}{t}^2\left(-d{A}_{xz}-\gamma A_{\left(xxxx\right)}-\theta A_{xy}-2\lambda (A_x^2-A{A}_{xx})-A_{xx}-f{B}_x\right).$$

(22)

To find $c_3(x,y,z)$, we write

$$u_3(x,y,z,t)=A(x,y)+B(x,y)t+f_3(x,y){\displaystyle \frac{1}{2}}{t}^2+f_3(x,y){\displaystyle \frac{1}{6}}{t}^3$$

(23)

in ${\mathrm{Res}}_3(x,y,z,t)$ and obtain

$$\begin{array}{cc}\hfill {\mathrm{Res}}_3(x,y,z,t)& =d\left(A_{xz}+t{B}_{xz}+{\displaystyle \frac{1}{2}}{t}^2{\left(c_2\right)}_{xz}+{\displaystyle \frac{1}{6}}{t}^3{\left(c_3\right)}_{xz}\right)+\gamma \left(A_{xxxx}+t{B}_{xxxx}+{\displaystyle \frac{1}{2}}{t}^2{\left(c_2\right)}_{xxxx}+{\displaystyle \frac{1}{6}}{t}^3{\left(c_3\right)}_{xxxx}\right)\hfill \\ \hfill & +\theta \left(A_{xy}+t{B}_{xy}+{\displaystyle \frac{1}{2}}{t}^2{\left(c_2\right)}_{xy}+{\displaystyle \frac{1}{6}}{t}^3{\left(c_3\right)}_{xy}\right)+\lambda (2{\left(A_x+t{B}_x+{\displaystyle \frac{1}{2}}{t}^2{\left(c_2\right)}_x+{\displaystyle \frac{1}{6}}{t}^3{\left(c_3\right)}_x\right)}^2\hfill \\ \hfill & +2\left(A+tB+{\displaystyle \frac{1}{2}}{t}^2{c}_2+{\displaystyle \frac{1}{6}}{t}^3{c}_3\right)\left(A_{xx}+t{B}_{xx}+{\displaystyle \frac{1}{2}}{t}^2{\left(c_2\right)}_{xx}+{\displaystyle \frac{1}{6}}{t}^3{\left(c_3\right)}_{xx}\right))\hfill \\ \hfill & +A_{xx}+t{B}_{xx}+{\displaystyle \frac{1}{2}}{t}^2{\left(c_2\right)}_{xx}+{\displaystyle \frac{1}{6}}{t}^3{\left(c_3\right)}_{xx}\hfill \\ \hfill & +f\left(B_x+t{\left(c_2\right)}_x+{\displaystyle \frac{1}{2}}{t}^2{\left(c_3\right)}_x\right)+t\left(c_3\right)+c_2.\hfill \end{array}$$

(24)

Then, we solve ${\left({\mathrm{Res}}_3\right)}_t(x,y,z,0)=0$ to get

$$c_3(x,y,z)=-4\lambda A_x{B}_x-2B\lambda A_{xx}-2A\lambda B_{xx}-d{B}_{xz}-B_{xx}-\gamma B_{xxxx}-\theta B_{xy}-f{\left(c_2\right)}_x.$$

(25)

Thus, we obtain the 3rd mRPSM solution as

$$\begin{array}{ccc}\hfill u_3(x,y,z,t)& =& A+Bt+{\displaystyle \frac{t^2}{2}}\left(c_2\right)-{\displaystyle \frac{t^3}{6}}4\lambda A_x{B}_x-2{\displaystyle \frac{t^3}{6}}B\lambda A_{xx}-2{\displaystyle \frac{t^3}{6}}A\lambda B_{xx}\hfill \\ & & -{\displaystyle \frac{t^3}{6}}d{B}_{xz}-{\displaystyle \frac{t^3}{6}}{B}_{xx}-{\displaystyle \frac{t^3}{6}}\gamma B_{xxxx}-\theta B_{xy}-{\displaystyle \frac{t^3}{6}}f{\left(c_2\right)}_x.\hfill \end{array}$$

(26)

Following the similar procedure, the rest of the unknown functions are calculated as

$$\begin{array}{ccc}\hfill c_4(x,y,z)& =& -4\lambda A_x{\left(c_2\right)}_x-2\left(c_2\right)\lambda A_{xx}-2A\lambda {\left(c_2\right)}_{xx}-4\lambda {B^2}_x-4B\lambda B_{xx}\hfill \\ & & -d{\left(c_2\right)}_{xz}-f{\left(c_3\right)}_x-{\left(c_2\right)}_{xx}-\gamma {\left(c_2\right)}_{xxxx}-\theta {\left(c_2\right)}_{xy},\hfill \end{array}$$

(27)

and

$$\begin{array}{ccc}\hfill c_5(x,y,z)& =& -4\lambda A_x{\left(c_3\right)}_x-2\lambda A_{xx}{c}_3-2\lambda A{\left(c_3\right)}_{xx}-12\lambda B_x{\left(c_2\right)}_x-6\lambda B_{xx}{c}_2\hfill \\ & & -6\lambda B{\left(c_2\right)}_{xx}-d{\left(c_3\right)}_{xz}-f{\left(c_4\right)}_x-\gamma {\left(c_3\right)}_{xxxx}-\theta {\left(c_3\right)}_{xy}-{\left(c_3\right)}_{xx}.\hfill \end{array}$$

(28)

Thus, other iteration results are obtained as follows:

$$\begin{array}{cc}\hfill u_4(x,y,z,t)& =A+tB+{\displaystyle \frac{1}{2}}{t}^2\left(c_2\right)+{\displaystyle \frac{1}{6}}{t}^3\left(c_3\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{24}}{t}^4\left(-4\lambda B_x^2-4\lambda A_x{\left(c_2\right)}_x-f{\left(c_3\right)}_x-d{\left(c_2\right)}_{xz}-\theta {\left(c_2\right)}_{xy}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{24}}{t}^4\left(-2\lambda \left(c_2\right)A_{xx}-4\lambda B{B}_{xx}-{\left(c_2\right)}_{xx}-2\lambda A{\left(c_2\right)}_{xx}-\gamma {\left(c_2\right)}_{xxxx}\right),\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill u_5(x,y,z,t)& =A+tB+{\displaystyle \frac{1}{2}}{t}^2\left(c_2\right)+{\displaystyle \frac{1}{6}}{t}^3{c}_3+{\displaystyle \frac{1}{24}}{t}^4{c}_4\hfill \\ \hfill & +{\displaystyle \frac{1}{120}}{t}^5\left(-12\lambda B_x{\left(c_2\right)}_x-4\lambda A_x{\left(c_3\right)}_x-f{\left(c_4\right)}_x-d{\left(c_3\right)}_{xz}-\theta {\left(c_3\right)}_{xy}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{120}}{t}^5\left(-2\lambda c_3{A}_{xx}-6\lambda c_2{B}_{xx}-6\lambda B{\left(c_2\right)}_{xx}-{\left(c_3\right)}_{xx}-2\lambda A{\left(c_3\right)}_{xx}-\gamma {\left(c_3\right)}_{xxxx}\right).\hfill \end{array}$$

(30)

The modified residual power series method has provided reliable approximate analytical solutions that align well with the exact solutions derived from the ME-tanh method. With these solutions in hand, we now turn to a detailed discussion of their physical implications. In the next section, we explore the physical interpretation of the numerical results, analyzing the behavior of the solutions in different contexts and their relevance to real-world phenomena.

6. Physical Interpretation of the Numerical Results

Table 2. Comparison of approximate solutions $u_3$, $u_4$, and $u_5$ with the exact solution at various t levels.

t	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	$\mathit{ex}$	$|{\mathit{u}}_3-\mathit{ex}|$	$|{\mathit{u}}_4-\mathit{ex}|$	$|{\mathit{u}}_5-\mathit{ex}|$
0.0	0.010452	0.0104529	0.0104529	0.010452	$0.0000$	$0.0000$	$0.0000$
0.1	0.010968	0.010968	0.010968	0.010968	$6.4719\times {10}^{-9}$	$1.2861\times {10}^{-10}$	$2.4940\times {10}^{-12}$
0.2	0.0115125	0.0115126	0.0115126	0.011512	$1.0569\times {10}^{-7}$	$4.1986\times {10}^{-9}$	$1.6276\times {10}^{-10}$
0.3	0.0120885	0.012089	0.012089	0.012089	$5.4635\times {10}^{-7}$	$3.2538\times {10}^{-8}$	$1.8913\times {10}^{-9}$
0.4	0.0126976	0.0126992	0.0126994	0.012699	$1.7639\times {10}^{-6}$	$1.3999\times {10}^{-7}$	$1.0845\times {10}^{-8}$
0.5	0.0133418	0.0133458	0.013346	0.013346	$4.4009\times {10}^{-6}$	$4.3636\times {10}^{-7}$	$4.2235\times {10}^{-8}$
0.6	0.0140229	0.0140311	0.014032	0.014032	$9.3304\times {10}^{-6}$	$1.1095\times {10}^{-6}$	$1.2881\times {10}^{-7}$
0.7	0.0147427	0.0147579	0.014760	0.014760	0.0000176819	$2.4516\times {10}^{-6}$	$3.3189\times {10}^{-7}$
0.8	0.0155031	0.0155291	0.015533	0.015534	0.0000308708	$4.8887\times {10}^{-6}$	$7.5599\times {10}^{-7}$
0.9	0.016306	0.0163476	0.016355	0.016356	0.0000506331	$9.0148\times {10}^{-6}$	$1.5675\times {10}^{-6}$
1.0	0.0171531	0.0172166	0.017229	0.017232	0.0000790632	0.0000156302	$3.0182\times {10}^{-6}$

and

Table 3. Comparison of approximate solutions $u_3$, $u_4$, and $u_5$ with the exact solution at various x levels.

t	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	$\mathit{ex}$	$|{\mathit{u}}_3-\mathit{ex}|$	$|{\mathit{u}}_4-\mathit{ex}|$	$|{\mathit{u}}_5-\mathit{ex}|$
0.0	0.006875	0.006875	0.006875	0.006875	$2.5776\times {10}^{-9}$	$4.3712\times {10}^{-11}$	$7.2615\times {10}^{-13}$
0.1	0.007195	0.007195	0.007195	0.007195	$2.8083\times {10}^{-9}$	$4.8342\times {10}^{-11}$	$8.1481\times {10}^{-13}$
0.2	0.007531	0.007531	0.007531	0.007531	$3.0637\times {10}^{-9}$	$5.3542\times {10}^{-11}$	$9.1583\times {10}^{-13}$
0.3	0.007886	0.007886	0.007886	0.007886	$3.3467\times {10}^{-9}$	$5.9392\times {10}^{-11}$	$1.0312\times {10}^{-12}$
0.4	0.008259	0.008259	0.008259	0.008259	$3.6609\times {10}^{-9}$	$6.5985\times {10}^{-11}$	$1.1631\times {10}^{-12}$
0.5	0.008652	0.008652	0.008652	0.008652	$4.0103\times {10}^{-9}$	$7.3429\times {10}^{-11}$	$1.3143\times {10}^{-12}$
0.6	0.009067	0.009067	0.009067	0.009067	$4.3995\times {10}^{-9}$	$8.1848\times {10}^{-11}$	$1.4879\times {10}^{-12}$
0.7	0.009504	0.009504	0.009504	0.009504	$4.8337\times {10}^{-9}$	$9.1390\times {10}^{-11}$	$1.6878\times {10}^{-12}$
0.8	0.009965	0.009965	0.009965	0.009965	$5.3190\times {10}^{-9}$	$1.0223\times {10}^{-10}$	$1.9184\times {10}^{-12}$
0.9	0.010453	0.010453	0.010453	0.010453	$5.8624\times {10}^{-9}$	$1.1456\times {10}^{-10}$	$2.1850\times {10}^{-12}$
1.0	0.010968	0.010968	0.010968	0.010968	$6.4719\times {10}^{-9}$	$1.2861\times {10}^{-10}$	$2.4940\times {10}^{-12}$

show the quantitative data concerning the temporal and spatial behavior of the approximate solutions resulting from the application of the mRPSM to the Boussinesq-type KdV equation. These tables enable us to analyze both the accuracy and physical accuracy of the iterative solutions $u_3(x,y,t)$, $u_4(x,y,t)$, and $u_5(x,y,t)$ in relation to an exact reference solution.

Table 2 focuses on the variation of the approximate solutions with time at a specific location. The most significant observation here is that the numerical solution becomes closer to the exact solution with the increase in approximation order. Exactly, the discrepancy between the approximate and the exact solution drops very strongly from $u_3$ to $u_5$. For $t=1.0$, the absolute error decreases from $7.90\times {10}^{-5}$ for $u_3$ to $3.01\times {10}^{-6}$ for $u_5$. This convergence pattern indicates the strong temporal stability and computational efficacy of the mRPSM algorithm in capturing the time-evolution of nonlinear wave forms. From a physical perspective, this temporal precision is needed in order to effectively simulate wave propagation processes such as shallow water wave dynamics, whose wave properties such as phase velocity, amplitude modulation, and dispersive spreading are time-dependent sensitivities. The high fidelity of $u_5$ ensures these wave properties are resolved and predicted correctly even at large time steps.

Table 3 focuses on spatial variation along the x-axis at a particular instant. Here again, the mRPSM shows excellent convergence with rising order. For example, at $x=1.0$, absolute error decreases from $6.47\times {10}^{-9}$ for $u_3$ to just $2.49\times {10}^{-12}$ for $u_5$. This precision shows the robustness of the technique to solve spatial features, for instance, wave steeping and neighborhood structures like solitons. This spatial accuracy is essential in physical systems whose behavior is modeled by nonlinear dispersive wave equations, where small errors in spatial resolution can lead to large errors in predicting wave interaction, soliton collisions, or energy localization. The capability of the method to handle fine spatial gradients also makes it applicable in other domains like plasma physics and optical fiber communication, where waveforms can contain sudden changes and complicated profiles.

Numerical solutions confirm the assertion that the mRPSM is not only a computational approximation method but also an exactly physically consistent and reliable method for solving nonlinear PDEs like the Boussinesq-type KdV equation. It preserves the balance of dispersion and nonlinearity, which is crucial when modeling real-world processes such as tsunami evolution, elastic wave propagation in solids, and pulse dynamics in nonlinear optical media. In addition, the systematic convergence behavior in both space and time domains implies that mRPSM successfully models the intrinsic physical laws driving wave motion. The iterative nature of its algorithm provides flexibility for adaptable control of precision and is, thus, suitable for precision applications over large domains and timescales.

7. Error Analysis

This section presents graphical comparisons of the absolute errors related to consecutive mRPSM approximations, and so provides a clear image of the method’s convergence in both temporal and spatial dimensions to validate the numerical findings.

The error plots presented for $u_3$, $u_4$, and $u_5$ versus the exact solution offer valuable insight into the accuracy and convergence behavior of the mRPSM over both spatial and time domains. In Figure 4, we notice that the absolute error for every higher approximation decreases by a considerable factor with higher values of time, with $u_5$ registering errors from ${10}^{-6}$ to ${10}^{-12}$, reflecting its higher fidelity even when the solution evolves dynamically. This confirms that higher-order approximations in the series not only increase accuracy but also continue to be stable with time. Likewise, the error-vs-space plot shows systematic decrease in the absolute value of the error with spatial location, which suggests that the scheme accurately captures the spatial structure of the wave solution. The utilization of logarithmic scales for both plots highlights the exponential decrease of error from $u_3$ to $u_5$ and emphasizes the quick convergence of mRPSM. Physically, this means that the method can come close to approximating complicated, multidimensional wave phenomena described by Boussinesq-type KdV systems and hence a meaningful and useful tool for characterizing both short- and long-time dynamics of nonlinear dispersive waves to a good degree of accuracy.

8. Conclusions

In this study, we considered the Boussinesq-type Korteweg–de Vries (KdV) equation modeling the propagation of long nonlinear dispersive waves in various physical media. The equation is convenient to capture both nonlinear and dispersive effects more accurately than the standard KdV equation. For obtaining analytical and approximate solutions of this equation for the first time, we utilized two effective analytical methods: the ME-tanh technique and the mRPSM. While ME-tanh offers closed-form analytical solutions, explicitly showcasing wave structures and soliton profiles, mRPSM provides iterative approximate solutions for modeling complex nonlinear activity. Beyond the methodological contribution, the results carry important physical implications. Soliton solutions obtained by ME-tanh will be interesting to localized waves that remain stable and relevant to water waves, optical fibers, and elasticity, where the persistence of a solitary pulse is of paramount importance. In support of this, the convergence analysis of mRPSM shows that it would equally capture the temporal and spatial evolution, making it suitable for the modeling of tsunamis, propagation of stresses in solids, and optical pulse transmission. Unlike preceding methods such as Hirota’s bilinear method that are largely limited to generating multi-soliton solutions, the present ME-tanh with the mRPSM is more versatile and effective since it combines exact analysis with reliable approximations. Our findings, therefore, provide further solutions of the Boussinesq-type KdV equation and should also serve as testimony to the advantage that hybrid analytical methods offer in deepening physical insight and improving the accuracy of physical predictions in nonlinear dispersive wave dynamics.