Innovative Solutions for the Kadomtsev – Petviashvili Equation via the New Iterative Method

. This research paper presents a new iterative method (NIM) for obtaining the solution to the potential Kadomtsev – Petviashvili (PKP) equation. NIM is a promising approach to solving complex mathematical problems, and its effectiveness and ef ﬁ ciency are highlighted through its application to the PKP equation. The results obtained through the use of NIM are compared to the exact solutions of the PKP equation, and it is found that the NIM approach provides results that are in close agreement with the exact solutions. This demonstrates the utility and accuracy of NIM and makes it a valuable tool for solving similar mathematical problems in the future. Furthermore, the lack of discretization in the NIM approach makes it a more convenient method for solving the PKP equation compared to traditional approaches that require discretization. Overall, the ﬁ ndings of this research paper suggest that NIM is a highly effective and convenient method for obtaining approximate analytical solutions to complex mathematical problems, such as the ð 2 þ 1 Þ -dimensional PKP equation.


Introduction
Nonlinear phenomena are of paramount importance in numerous scientific and engineering disciplines.Despite their widespread significance, the models of real-world problems that incorporate nonlinearity are often challenging to solve, either theoretically or numerically.This has led to a growing interest in developing better and more efficient methods for determining solutions to nonlinear models, whether they be approximate or exact, analytical or numerical.
In recent years, significant progress has been made in this direction, with researchers devoting a great deal of attention to the search for new and improved methods for solving nonlinear problems.This has led to a wealth of studies and publications that explore various approaches to solving nonlinear models (such as those by [1][2][3][4][5][6][7][8][9][10]).These efforts are aimed at improving our understanding of nonlinearity and finding ways to effectively model and solve real-world problems that incorporate nonlinear dynamics.
Despite the progress that has been made, much work remains to be done.Nonlinear models can still be difficult to solve, and the search for more effective and efficient methods for determining their solutions continues to be a focus of research and development efforts in various scientific and engineering disciplines.Nevertheless, the progress that has been made in recent years highlights the promise of continued progress in the future as researchers continue to work toward finding better and more efficient methods for solving nonlinear problems.
The ð2 þ 1Þ-dimensional potential Kadomtsev-Petviashvili (PKP) equation is an important partial differential equation in the field of nonlinear wave phenomena.It is a generalization of the (2 + 1)-dimensional KP equation and describes the propagation of nonlinear waves in a two-dimensional space with an additional degree of freedom.
The importance of PKP lies in its ability to model a wide range of physical phenomena, such as plasma physics, fluid dynamics, and nonlinear optics.For example, PKP has been used to describe the interaction of Langmuir waves and ionacoustic waves in plasma physics, the propagation of waves in shallow water, and the dynamics of optical pulses in nonlinear media.
Moreover, PKP possesses several interesting mathematical properties, such as soliton solutions, integrability, and infinite-dimensional symmetries.These properties make PKP an attractive subject of study in the field of mathematical physics, where it has been investigated extensively using a variety of techniques, including inverse scattering transform, Darboux transformation, and the Hirota bilinear method.
Overall, the PKP equation plays an important role in both the physical and mathematical sciences and continues to be an active area of research [11][12][13].
The PKP equation is a partial differential equation that has applications in various fields of physics, including fluid dynamics, plasma physics, and nonlinear optics.Here are some examples of how PKP equations are used in the real world.
Fluid Dynamics: The PKP equation is used to describe the dynamics of fluids in various contexts, including ocean currents, atmospheric circulation, and turbulence.For example, the PKP equation has been used to study the behavior of waves in shallow water and to model the flow of viscous fluids in channels and pipes.
Plasma Physics: The PKP equation is also used in the study of plasma physics, particularly in the context of magnetohydrodynamics (MHDs).MHD is a branch of plasma physics that studies the behavior of ionized gases in the presence of magnetic fields.The PKP equation has been used to describe the dynamics of MHD waves and to study the stability of plasma configurations.
Nonlinear Optics: The PKP equation has applications in the field of nonlinear optics, which studies the behavior of light in nonlinear media.Nonlinear optics is used in a variety of applications, including telecommunications, laser technology, and medical imaging.The PKP equation has been used to model the propagation of ultrashort pulses of light in optical fibers and to study the dynamics of solitons in nonlinear media [14][15][16].
In 2001, Senthilvelan [17] solved the potential PKP equation by applying the homogenous balance method (HBM).Later, Li and Zhang [18] made improvements to the steps of the HBM and applied the method to the study of the PKP equation.Through their research, they were able to obtain various exact solutions to the equation, including soliton, multisoliton, and rational-type solutions.
In an independent investigation, Li and Zhang [19] successfully derived novel soliton-like solutions for the PKP equation through the application of symbolic computation.Additionally, Kaya and El-Sayed [20] employed the Adomian decomposition method (ADM) to ascertain the solution of the PKP equation.Simultaneously, Batiha and Batiha [21] utilized the variational iteration method (VIM) to uncover solutions for the ð2 þ 1Þ -dimensional PKP equation.These diverse approaches underscore the range of methodologies employed in addressing the PKP equation, showcasing the potential of different methods in resolving intricate nonlinear problems.
The new iterative method (NIM), presented by Daftardar-Gejji and Jafari [22] in 2006, emerged as a potent mathematical tool for tackling both linear and nonlinear functional equations.Demonstrating its efficacy, NIM has been successfully applied to solve a diverse array of nonlinear equations, encompassing integral equations, algebraic equations, as well as ordinary or partial differential equations of both fractional and integer order.
NIM stands out due to its simplicity in implementation and comprehensibility, rendering it accessible to a wide spectrum of researchers and practitioners.In contrast to wellestablished methods like the ADM [23], the homotopy perturbation method [24], and the VIM [25], NIM has demonstrated superior performance and increased efficiency.This has contributed to its popularity among individuals addressing intricate nonlinear problems [26].
In this study, our aim is to use the NIM to obtain a numerical solution for the ð2 þ 1Þ-dimensional PKP Equation, which is given in the following form: The initial conditions are already known and will be used in our analysis.In order to validate the effectiveness of NIM, we will compare the results obtained through its application with the exact solution of the PKP equation.

Overview of the NIM
This section provides an outline of the NIM numerical method, detailing its approach as follows [27][28][29][30]: In the presented equation, f represents a given function, while L and N denote linear and nonlinear operators, respectively.The solution to Equation ( 2) is expressed as follows: Suppose we have the following: Then, we get the following: Therefore, NðyÞ can be decomposed as follows: The recurrence relation is written in the following form: By knowing that L is linear we will have the following: So, So, the solution will be given as follows: 3. Convergence Analysis of NIM Theorem 1.For any n and given real constants L>0 and ku i jj ≤ M < 1 e for i ¼ 1; 2; …, if the nonlinear operator N is C ð1Þ in the neighborhood of u 0 and kN ðnÞ ðu 0 Þjj ≤ L, then the series ∑ 1 n¼0 H n is absolutely convergent, and kH n jj ≤ LM n e n−1 ðe − 1Þ for n ¼ 1; 2; …. Proof.

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Since demonstrating the boundedness of u i for all i poses challenges, a more practical result is provided in the following theorem.This theorem specifies conditions on N ðkÞ ðu 0 Þ that are sufficient to ensure the convergence of the series.
Theorem 2. If the series ∑ 1 n¼0 H n converges absolutely, given that N is C ð1Þ and kN ðnÞ ðu 0 Þjj ≤ M ≤ e −1 for all n.

Examination of the PKP Equation with Numerical Applications
This paper focuses on the (2 + 1)-dimensional PKP equation, defined as follows: The initial conditions for the PKP equation are given as follows: Mathematical Problems in Engineering The exact solution to the PKP equation has been determined and was first published by Senthilvelan [17].The traveling wave solution is expressed as follows [20]: To solve Equation ( 18) by NIM, we integrate Equation ( 18) twice with respect to y from 0 to y and use Equation (19) to get the following: . By using Equation ( 8), we get the following: Thus, 1 illustrates the comparison between the 1-iterate of NIM and the exact solution given by Equation (20).
Table 1 presents a comparison between the first iteration of NIM and the exact solution given by Equation (20).
To validate the accuracy of NIM, we showcase the solution of the PKP Equation ( 18) through ADM, as previously demonstrated in [20].In Table 2, we present a comprehensive comparison between the ADM solution [20] and the exact solution provided by Equation (20).This analysis allows us to assess the precision and reliability of NIM in solving complex mathematical problems.
We will now use the NIM to find an approximate analytical solution to the PKP equation as follows: with the initial conditions are as follows: The exact solution to Equation ( 25) can be found in [17] and is given by the following: To solve Equation ( 25) by NIM, we integrate Equation ( 25) and use Equation (26) to get the following: 4 Mathematical Problems in Engineering By using Equation ( 8), we get the following: Thus,   2 presents the comparison between the 1-iteration of NIM and the exact solution defined by Equation (27).
Table 3 displays a contrast between the initial iteration of NIM and the exact solution defined by Equation (27).
Table 4 showcases a comparison between the VIM [21] and the exact solution provided by Equation ( 27).

Conclusions
The development of the NIM presents a significant breakthrough in the field of differential equations.Unlike traditional numerical methods, NIM does not rely on linearization or make any limiting assumptions, making it highly versatile and applicable to a wide range of linear and nonlinear differential equations.The successful application of NIM in deriving numerical solutions for the (2 + 1)-dimensional PKP equation demonstrates its remarkable efficacy and ease-of-use.Furthermore, the comparison with the exact solution showcases the accuracy and precision of NIM.The potential applications of NIM are vast, as it offers a powerful and efficient tool for uncovering analytical and numerical solutions for a diverse range of differential equations in various fields such as physics, engineering, and mathematics.Overall, NIM has opened up new possibilities for researchers and professionals in the field of differential equations, making complex problems more accessible and solvable.