Advanced computational techniques for solving the modified KdV-KP equation and modeling nonlinear waves

Abstract

This research employs recent and precise computational techniques to identify new and accurate solitary wave solutions of the modified Korteweg-de Vries-Kadomtsev-Petviashvili (KdV-KP) equation. The KdV-KP equation is a nonlinear partial differential equation that characterizes the evolution of waves in diverse physical systems, including nonlinear optics, fluid dynamics, and plasma physics. The equation generalizes the KdV and KP equations, which are fundamental models in their respective fields. Physically, the modified KdV-KP equation describes wave propagation where the effects of nonlinearity and dispersion can produce intriguing and complicated wave phenomena such as wave turbulence and solitary waves. The equation has numerous real-world applications, including modeling shallow water waves such as tsunamis and river waves in fluid dynamics, describing the behavior of plasma waves in fusion reactors and astrophysical plasmas, and studying the propagation of light in nonlinear media such as optical fibers in nonlinear optics. One of the most significant applications of the equation is in the study of solitons, which are self-sustaining solitary waves that preserve their shape and velocity even after colliding with other solitons. Solitons are used in various applications such as optical communications, where they transmit information over long distances without distortion, and in fluid dynamics, where they model long-lasting waves in the ocean. The study’s importance lies in its impact on the utilization of the modified KdV-KP equation in diverse fields of physics, including fluid dynamics and plasma physics. The effectiveness of the proposed techniques is demonstrated by comparing them with other computational methods, indicating their superiority.

Appendix

Here, we explain the highlights of the above–employed analytical (Khater II) method and approximate (variational iteration) method.

1.1 Khater II method (Khater 2023a, 2022a, b, 2023b, c, 2022c, 2023d, e; Khater et al. 2023; Khater 2023f)

This section gives the Khater II method’s headlines where it is first time to be used as follows:

Assume the following form for the equation of nonlinear evolution:

$$\begin{aligned} {\mathcal {G}}({\mathcal {U}},\,{\mathcal {U}}_{x},\,{\mathcal {U}}_{t},\,{\mathcal {U}}_{x,\,t},\ldots )=0, \end{aligned}$$

(A1)

where ${\mathcal {G}}={\mathcal {G}}(x,t)$ is a polynomial of ${\mathcal {U}}(x,t)$ and its partial derivatives wherein the highest order derivatives and nonlinear terms are concerned. The main steps of the employed method are as follows

Step 1. The traveling wave transformation

$$\begin{aligned} {\mathcal {U}}(x,\, t)={\mathcal {V}}(\xi ), \quad \xi =x+c\,t, \end{aligned}$$

(A2)

converting Equation (A1) into the following ODE

$$\begin{aligned} {\mathcal {C}}\left( {\mathcal {U}},\, {\mathcal {U}}^{\prime }, {\mathcal {U}}^{\prime \prime },\, \ldots \right) =0, \end{aligned}$$

(A3)

where ${\mathcal {C}}$ is a polynomial in ${\mathcal {U}}(\xi )$ and its total derivatives, wherein $\varphi ^{\prime }(\xi )=\frac{d \varphi }{d \xi }$.

Step 2. We suppose the solution of (A3) is of the form

$$\begin{aligned} {\mathcal {V}}(\xi )=\sum \limits _{i=1}^n \left( a_i \, f(\xi )^i+b_i \, \phi (\xi )\, f(\xi )^{i-1}\right) +a_0, \end{aligned}$$

(A4)

where $a_{i},\, b_{i}\,(i=0,1,2,3, \ldots ,n)$ are arbitrary constants to be determined, such that $a_{n} \ne 0$, and $\phi (\xi ), \, f(\xi )$ satisfies the following equation

$$\begin{aligned} f'(\xi )&= -\delta -f(\xi )^2, \end{aligned}$$

(A5)

$$\begin{aligned} \phi '(\xi )&= -f(\xi ) \, \phi (\xi ), \end{aligned}$$

(A6)

where $\delta$ is arbitrary constant. While $f(\xi )$ and $\phi (\xi )$ have the following form

$$\begin{aligned} f(\xi )&= -\sqrt{\delta } \tan \left( \sqrt{\delta } \xi \right) , \end{aligned}$$

(A7)

$$\begin{aligned} \phi (\xi )&= \sec \left( \sqrt{\delta } \xi \right) . \end{aligned}$$

(A8)

Step 3. We determine the positive integer n come out in the suggested general solution by considering the homogeneous balance between the highest order derivatives and the highest order nonlinear terms occurring in (A3) as following

$$\begin{aligned} D\left[ \frac{d^{q} {\mathcal {U}}}{d\xi ^{q}}\right] =n+q,\ \ \ D\left[ {\mathcal {U}}^{p}\left( \frac{d^{q} {\mathcal {U}}}{d\xi ^{q}}\right) ^{s}\right] =np+s(n+q). \end{aligned}$$

(A9)

Step 4. We compute all the required derivatives ${\mathcal {V}}^{\prime },\, {\mathcal {V}}^{\prime \prime },\, \ldots$, and substitute (A4) and the derivatives into (A3) and then we account for the function $\phi (\xi ), \, f(\xi )$. As a result of this substitution, we obtain a polynomial of $\phi (\xi ), \, f(\xi )$ and its derivatives. In this polynomial, we equate all the coefficients to zero. This procedure yields a system of equations whichever can be solved to find $a_k$ and $\phi (\xi ), \, f(\xi )$.

1.2 $\mathcal{V}\mathcal{I}$ method (Momani and Abuasad 2006; Tatari and Dehghan 2007; Liu and Gurram 2009)

Here is a general nonlinear differential equation, which illustrates the basic concept of He’s variational iteration method:

$$\begin{aligned} {\mathcal {L}}\, {\mathcal {U}}(x,t)+{\mathcal {N}}\, {\mathcal {U}}(x,t)=g(x,t), \end{aligned}$$

(A7)

where ${\mathcal {L}},\, {\mathcal {N}},\, g(x,t)$ are respectively describe a linear operator, a nonlinear operator, and a known analytical function. According to the variational method, a correction functional can be constructed as follows:

$$\begin{aligned} {\mathcal {U}}_{\rho +1}(x,t)={\mathcal {U}}_{\rho }(x,t)+\int _{0}^{t} \varsigma \, \left( L {\mathcal {U}}_{\rho }(x,\, \xi )+N \tilde{{\mathcal {U}}}_{\rho }(x,\, \xi )-g(x,\, \xi )\right) d \xi . \end{aligned}$$

(A8)

In this case, $\varsigma$ denotes a general Lagrange multiplier that can be identified optimally using variational theory, and $\tilde{{\mathcal {U}}}_{\rho }$ represents a restricted variation, i.e. $\delta \, \tilde{{\mathcal {U}}}_{\rho }=0$. The stationary conditions

$$\begin{aligned} \left\{ \begin{array}{c} 1+\varsigma =0, \\ \\ \varsigma ^{\prime }=0.~~~~~ \end{array} \right. \end{aligned}$$

(A9)

This in turn gives

$$\begin{aligned} \varsigma =-1. \end{aligned}$$

(A10)

Using He’s variational iteration technique to the investigated model for investigating the model’s approximate solution, gets the following values of the analytical, numerical, and absolute difference between these two values.