Solitary wave analysis of the Kadomtsev–Petviashvili model in mathematical physics

Abstract

The (3 + 1)-dimensional Kadomtsev–Petviashvili equation has significant applications that emerge in the study of fluids and multi-component plasmas. The main object of our study is to investigate stable and efficient solitary solutions to the stated wave equations through two different schemes that provide generic solutions like exponential functions, trigonometric functions, hyperbolic functions, etc. The graphical illustrations for different values of the corresponding parameters of the attained solutions are explained in order to expose the intimate structure of the substantial phenomena. In particular, changes in wave position and category with respect to time are deliberated extensively through 2D figures. It has been found that wave velocity depends on some free parameters associated with the methods used to solve the equation. It has been demonstrated that for different values of the free parameters, the obtained solutions of the KP model exhibit dark soliton, peakon shape soliton, dark soliton-type wave, bright soliton, bell-shaped soliton, etc. Moreover, the direction and position of the solitons for the variation of other parameters are measured, from which a simple and comprehensive interpretation of the various aspects of water waves can be obtained. We think the obtained solution could be highly useful for studying nonlinear events in fluid dynamics, wind waves, and wind-generated waves.

Keywords:

• soliton
• the improved Bernoulli subequation function method
• the Kadomtsev–Petviashvili equation
• the new auxiliary equation method
• travelling wave solution

1. Introduction

Mathematical modeling is expected to be a quantitative tool having the high potential to understand the natural behavior of various systems in the field of science and engineering providing more reliable and significant information. Among all, differential equations are the most significant instruments for expressing such models. The majority of those differential equations that demonstrate diverse dynamical characteristics of a wide range of physical systems are the nonlinear partial differential equations (PDE). Nonlinear PDEs are considered very useful tools that aid in understanding the physical phenomena in almost every area of science and engineering, especially, in fluid dynamics, aerodynamics, mathematical physics, quantum physics, signal processing, rheology, image processing, control theory, biology, chemistry, medicine, elasticity, astronomy, environmental sciences, social sciences, finance, and many other fields (Li & Ma, 2017; Seadawy, 2014; Triki, Biswas, Moshokoa, & Belic, 2017).

The Kadomtsev–Petviashvili equation (briefly known as KP equation), a highly nonlinear partial differential equation having spatiotemporal coordinates, is an integrable system in three dimensions that is considered an extension of the Korteweg–de Vries (KdV) equation to two spatial dimensions and was first introduced by Boris Kadomtsev and Vladimir Petviashvili in 1970 (Chakravarty & Kodama, 2009; Kadomtsev & Petviashvili, 1970). The most common nonlinear PDE that models the nonlinear wave theory is the KP equation, which was deduced by various scientists to model different physical phenomena, e.g., Ablowitz and Segur (1979) used this equation to model the surface and internal water waves, and Pelinovsky, Stepanyants, and Kivshar (1995) used in nonlinear optics. The KP equation describes the progression of nonlinear waves with a long wavelength and small amplitude with a weak dependence on the transverse coordinate. The KP equation has been used to describe the behavior of periodic waves in shallow water. For instance, in (Hammack, Mccallister, Scheffner, & Segur, 1995; Hammack, Scheffner, & Segur, 1989; Segur & Finkel, 1985), the authors derived a set of equations that describes the wave profile, velocity, and pressure, and these equations were solved using the perturbation method. The resulting analytical solutions allow for a better understanding of the behavior of shallow water waves, including the effects of wave height, frequency, and water depth. Other schemes, such as the inverse scattering method, the Riemann–Hilbert approach, etc. have also been used to solve the KP equation (Biondini & Pelinovsky, 2008). The KP equation is also useful as a model for verifying new mathematical methods like as in applications of the methods of the dynamical system for water waves (Groves & Sun, 2008). This equation can be used to model the behavior of nonlinear waves, such as rogue waves, which can be hazardous to ships and offshore structures (Xu, Chen, & Dai, 2014). Moreover, the KP equation has also been used to study plasma waves in space, such as those found in the Earth’s inosphere or the solar wind (Deng, Tian, Sun, Zhang, & Hu, 2019). This model used to study the formation of “oceanic eddies”, which are large-scale rotating structures that play an important role in ocean circulation and climate. By modeling the behavior of these eddies using the KP equation, scientist can gain insights into the physical mechanisms that drive their formation and evolution (Haupt, 1988).

Although a lot of successful efforts have already been done for the approximation of nonlinear PDEs using numerical methods, it is still true that many others remain to find solutions to physical systems that can be expressed in closed forms. Because there is no general method to approximate nonlinear PDEs, and in fact, in such circumstances, each problem needs to be investigated as individual work. Consequently, there arise many different methods to investigate analytical solutions for nonlinear PDEs. Therefore, searching for appropriate analytical solutions to nonlinear PDEs has become a vital interest for mathematicians. Scientists have been trying to develop various methods including inverse scattering transformation (Ablowitz & Segur, 1981), the Hirota bilinear method (Geng, Mou, & Dai, 2023; Rizvi et al., 2020), the lie symmetry analysis (Dhiman & Kumar, 2022; Kumar, Rani, & Mann, 2022; Kumar & Dhiman, 2022; Kumar & Rani, 2020, 2021; Malik, Almusawa, Kumar, Wazwaz, & Osman, 2021), the improve Bernoulli subequation function method (Baskonus & Bulut, 2015; Baskonus & Gómez-Aguilar, 2019; Islam & Akbar, 2020; Ismael, Bulut, Baskonus, & Gao, 2021; Kumar, Ilhan, Ciancio, Yel, & Baskonus, 2021; Yel, Sulaiman, & Baskonus, 2020), the new auxiliary equation method (Cheemaa, Seadawy, & Chen, 2019; Fatema, Islam, Arafat, & Akbar, 2022; Seadawy & Cheemaa, 2020), the exp-function method (Djennadi et al., 2021; Nisar et al., 2021), the modified Kudryshov approach (Kumar & Mann, 2022; Seadawy, Kumar, Hosseini, & Samadani, 2018), the square operator method (Bo, Wang, Fang, Wang, & Dai, 2023; Wang, Wang, & Dai, 2022) and many others (Ali, Seadawy, Rizvi, Younis, & Ali, 2020; Arqub et al., 2022; Baskonus & Kayan, 2021; Chen & Ren, 2022; Fang, Mou, Zhang, & Wang, 2021; Fang, Wu, Wang, & Dai, 2021; García Guirao, Baskonus, Kumar, Rawat, & Yel, 2019; Islam, Kundu, Akbar, Gepreel, & Alotaibi, 2021; Ismael, Bulut, Park, & Osman, 2020; Ismael et al., 2022; Nonlaopon, Mann, Kumar, Rezaei, & Abdou, 2022; Park et al., 2020; Rasool, Hussain, Al Sharif, Mahmoud, & Osman, 2023; Seadawy, 2014, 2016; Siddique, Jaradat, Zafar, Mehdi, & Osman, 2021; Wang, Shehzad, Seadawy, Arshad, & Asmat, 2023; Wen et al., 2021) for finding exact solutions of nonlinear PDEs. Two of the above methods, namely, the improved Bernoulli subequation function method (IBSEFM) and the new auxiliary equation method (NAEM) provide interesting results which can be further extended.

The KP equation (Ismael, Sulaiman, Nabi, Mahmoud, & Osman, 2023) has already been investigated by several methods such as the sine-Gordon expansion method (Baskonus, Sulaiman, & Bulut, 2017), the tanh–coth method (Jawad & Adham, 2013), the mapping method (Peng & Krishnan, 2005), the modified Khater method and Jacobi elliptical function method (Yue, Khater, Attia, & Lu, 2020), etc. According to the literature, the stated KP equation has not yet been investigated through the IBSEF method and the NAE method. Since there is no single best method to search for analytical solutions for the KP equation, therefore, in this work we are interested to carry out extensive analytical soliton-like solutions of the Kadomtsev–Petviashvili equation by these two methods. Moreover, depicting three-dimensional graphics, we have discussed the physical characteristics of various types of soliton-like solutions for the different value of free parameters of the attained solutions demonstrated in the 3D plot via Mathematica. The obtained results may enhance understanding of the dynamical characteristics of exact solutions of the KP equation that models highly nonlinear phenomena in the field of science and engineering. These two methods are user friendly, straight forward and informative. Scientists might be interested to use the effect of the free parameter of the obtained solution to investigate the real phenomena in future research.

2. Description of two proposed methods

We will discuss the procedure of the improved Bernoulli subequation function method (IBSEF) and the new auxiliary equation method (NLEE) in this section.

2.1. The IBSEF method

The NLEE accompanying two independent variables $x$ and $t$ is considered to illustrate the IBSEF method of the form (2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) where $R$ is a polynomial of the variable $u=u(x,t).$

Using the wave transformation (2.1.2) $$u\left(x,t\right)=u\left(\eta \right),\mathrm{ }\mathit{\text{ η}}=\alpha x-ct,$$(2.1.2) where $\alpha$ and $c$ represent the wave number and the velocity of the travelling wave, respectively. The NLEE (2.1.1)(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) simplifies to the following nonlinear ordinary differential equation (ODE) due to the influence of the wave transformation (2.1.2)(2.1.2) $$u\left(x,t\right)=u\left(\eta \right),\mathrm{ }\mathit{\text{ η}}=\alpha x-ct,$$(2.1.2) as follows (2.1.3) $$L\left(u,\mathrm{ }\frac{du}{d\eta },\mathrm{ }\frac{d^{2}u}{d{\eta }^2},\mathrm{ }\dots \right)=0,$$(2.1.3) where $L$ is a polynomial of $u(\eta )$ and its derivatives. With the help of the IBSEF, the solution of Eq. (2.1.3)(2.1.3) $$L\left(u,\mathrm{ }\frac{du}{d\eta },\mathrm{ }\frac{d^{2}u}{d{\eta }^2},\mathrm{ }\dots \right)=0,$$(2.1.3) can be written as (2.1.4) $$u\left(\eta \right)=\frac{{\sum }_{i=0}^n{p}_i{H}^i}{{\sum }_{j=0}^m{q}_j{H}^j}=\frac{p_0+p_{1}H+p_2{H}^2+\mathrm{ }\dots +p_n{H}^n}{q_0+q_{1}H+q_2{H}^2+q_3{H}^3+\mathrm{ }\dots +q_m{H}^m},$$(2.1.4) where $H=H(\eta )$ fulfills the improved Bernoulli equation (2.1.5) $$H^{\prime }=\sigma H+\beta H^P,\text{ where }P\in R-\left\{0,1,2\right\},\mathrm{ }\sigma \ne 0,\beta \ne 0.$$(2.1.5)

To calculate the value of $m$ and $n$ of unknown parameters using the balancing principle of the highest order linear term with the top order nonlinear term can be used and the values of $m,n,$ and $P$ are obtained. Substituting solution (2.1.4)(2.1.4) $$u\left(\eta \right)=\frac{{\sum }_{i=0}^n{p}_i{H}^i}{{\sum }_{j=0}^m{q}_j{H}^j}=\frac{p_0+p_{1}H+p_2{H}^2+\mathrm{ }\dots +p_n{H}^n}{q_0+q_{1}H+q_2{H}^2+q_3{H}^3+\mathrm{ }\dots +q_m{H}^m},$$(2.1.4) along with Eq. (2.1.5(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ) in Eq. (2.1.3(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ) provides a polynomial of $H$ is attained as: (2.1.6) $$\Omega \left(H\right)={\sigma }_s{H}^s+ \dots +{\sigma }_{1}H+{\sigma }_0=0.$$(2.1.6)

The coefficients of $\Omega \left(H\right)$ will be equalized to zero, resulting in a set of equations: (2.1.7) $${\sigma }_k=0;\mathrm{ }k=0,\mathrm{ }\dots ,s.$$(2.1.7)

To obtain the values of $p_0,p_1,\dots ,p_n$ and $q_0,q_1,\dots ,q_m,$ we can use Mathematica software to solve the system of algebraic equations. The solutions to the improved Bernoulli’s equation depend on the Bernoulli’s parameter $\sigma$ and $\beta$ as subsequential (2.1.8) $$H\left(\eta \right)={\left[-\frac{\beta }{\sigma }+\frac{\tau }{e^{\sigma \left(P-1\right)\eta \mathrm{ }}}\right]}^{\frac{1}{1-P}},\mathrm{ }\sigma \ne \beta ,$$(2.1.8) (2.1.9) $$H\left(\eta \right)={\left[\frac{\left(\tau -1\right)+\left(\tau +1\right)\tanh \hspace{0.17em}\left(\frac{\sigma \left(1-P\right)\eta }{2}\right)\mathrm{ }}{1-\tanh \hspace{0.17em}\left(\frac{\sigma \left(1-P\right)\eta }{2}\right)\mathrm{ }}\right]}^{\frac{1}{1-P}},\mathrm{ }\sigma =\beta ,\mathrm{ }\tau \in R.$$(2.1.9)

Using the parameters $p_i(i=0, 1, 2, \dots , n),$ $q_j (j=0, 1, 2, \dots , m),$ and the solutions to Eq. (2.1.5(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ) with the wave variable, further generic and some well-known solutions for the definite values of the subjective parameters to the nonlinear evolution equation (2.1.1(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ) can be found in the literature.

2.2. The new auxiliary equation method

Let us choose the nonlinear equation (2.2.1) $$H\left(q,q_t,q_x,q_y,q_z,q_{xx},\dots \dots \dots \right)\mathrm{ }=\mathrm{ }0,$$(2.2.1) where $H$ is a function of $q(x,y,z,t)$ is also a wave function.

To convert Eq. (2.2.1(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ), into an ODE, we have to select a wave variable as (2.2.2) $$q\left(x,y,z,t\right)=Q\left(\mu \right),\mathrm{ }\mu =\alpha x+\beta y+\gamma z-\delta t,$$(2.2.2) and the modified nonlinear equation is (2.2.3) $$M\left(Q,Q^{\prime },Q^{″},Q^{\prime ″},\dots \right)=0,$$(2.2.3) where prime indicates the number of the derivative with respect to $\mu .$

According to the new auxiliary equation method, Eq. (2.2.3(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ) is estimated the travelling wave solution as (2.2.4) $$Q\left(\mu \right)=\sum _{k=0}^L{S}_k{a}^{kg\left(\mu \right)},$$(2.2.4) here $S_0,S_1,S_2,\dots ,S_L$ are unknown parameters to be computed with $S_L\ne 0$ and $g(\mu )$ being the nonlinear equation solution (2.2.5) $$g^{\prime }\left(\mu \right)=\frac{1}{\ln \hspace{0.17em}a }\left\{p{a}^{-g\left(\mu \right)}+q+r{a}^{g\left(\mu \right)}\right\}.$$(2.2.5)

To get $L$ in Eq. (2.2.4(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ), the balancing principle must be used. Replacing the solution (2.2.4)(2.2.4) $$Q\left(\mu \right)=\sum _{k=0}^L{S}_k{a}^{kg\left(\mu \right)},$$(2.2.4) together with (2.2.5)(2.2.5) $$g^{\prime }\left(\mu \right)=\frac{1}{\ln \hspace{0.17em}a }\left\{p{a}^{-g\left(\mu \right)}+q+r{a}^{g\left(\mu \right)}\right\}.$$(2.2.5) then Eq. (2.2.3(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ) provides a polynomial of $a^{kg\left(\mu \right)},$ $\{k=0,1,2,\dots \}.$ Moreover, by setting every coefficient of the resulting polynomials to zero, a system of algebraic equations is obtained.

The values of $S_k,p,q,r$ acquired in the above inspire the solution $g(\mu )$ which is produced in (2.2.5)(2.2.5) $$g^{\prime }\left(\mu \right)=\frac{1}{\ln \hspace{0.17em}a }\left\{p{a}^{-g\left(\mu \right)}+q+r{a}^{g\left(\mu \right)}\right\}.$$(2.2.5) using several instances described in detail (Fatema et al., 2022) in Eq. (2.2.4(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ), providing a sufficient solitary wave solution to Eq. (2.2.1(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ).

3. Application

The (3 + 1)-dimensional KP is of the form: (3.1) $${\left(u_t+6{uu}_x+u_{\mathit{xxx}}\right)}_x-3u_{yy}-3u_{zz}=0.$$(3.1)

Now we will look at wave transformation (3.2) $$u\left(x,y,z,t\right)=u\left(\xi \right),\mathrm{ }\mathrm{ }\xi =kx+ly+mz-ct,$$(3.2) here $k,$ $l,$ $m$ are the wave numbers and $c$ is the wave velocity. Considering Eq. (3.2)(3.2) $$u\left(x,y,z,t\right)=u\left(\xi \right),\mathrm{ }\mathrm{ }\xi =kx+ly+mz-ct,$$(3.2) , the KP equation (3.1)(3.1) $${\left(u_t+6{uu}_x+u_{\mathit{xxx}}\right)}_x-3u_{yy}-3u_{zz}=0.$$(3.1) becomes the undermentioned nonlinear ordinary differential equation as (3.3) $${\left(-c{ku}^{\prime }+6k^{2}u{u}^{\prime }+k^4{u}^{\prime ″}\right)}^{\prime }-3{l^{2}u}^{″}-3{m^{2}u}^{″}=0.$$(3.3)

Twice integrating (3.3)(3.3) $${\left(-c{ku}^{\prime }+6k^{2}u{u}^{\prime }+k^4{u}^{\prime ″}\right)}^{\prime }-3{l^{2}u}^{″}-3{m^{2}u}^{″}=0.$$(3.3) and ignoring the constants, we obtain (3.4) $$-\mathit{cku}+3k^2{u}^2+k^4{u}^{″}-3l^{2}u-3m^{2}u=0.$$(3.4)

3.1. Solitons through the IBSEF method

Using the balancing concept between $u″$ and $u^2$ in Eq. (3.4)(3.4) $$-\mathit{cku}+3k^2{u}^2+k^4{u}^{″}-3l^{2}u-3m^{2}u=0.$$(3.4) relating to the IBSEF approach, we are able to determine the relationship between $m,n,$ and $P:$ $$2P+m=n+2.$$

Choosing $m=1,P=3,$ we get $n=5.$ The trial answer to Eq. (2.1.4(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ) thus becomes (3.1.1) $$u\left(\eta \right)=\frac{a_0+a_{1}H\left(\eta \right)+a_2{H}^2\left(\eta \right)+a_3{H}^3\left(\eta \right)+a_4{H}^4\left(\eta \right)+a_5{H}^5\left(\eta \right)}{b_0+b_{1}H\left(\eta \right)},$$(3.1.1) where $H(\xi )$ is the answer to the more advantageous Bernoulli equation (2.1.5(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ).

Equation (3.4)(3.4) $$-\mathit{cku}+3k^2{u}^2+k^4{u}^{″}-3l^{2}u-3m^{2}u=0.$$(3.4) is changed into a polynomial in $H$ by substituting equations (3.1.1(3.1) $${\left(u_t+6{uu}_x+u_{\mathit{xxx}}\right)}_x-3u_{yy}-3u_{zz}=0.$$(3.1) ) and (2.1.5)(2.1.5) $$H^{\prime }=\sigma H+\beta H^P,\text{ where }P\in R-\left\{0,1,2\right\},\mathrm{ }\sigma \ne 0,\beta \ne 0.$$(2.1.5) . After resolving the system of algebraic equations, we obtain the following set of parameter solutions:

• Set-1: $c\mathrm{ }=\mathrm{ }-\frac{\left(4k^4{\sigma }^2+3l^2+3m^2\right)}{k},$ $a_0\mathrm{ }=\mathrm{ }-\frac{\left(4b_0{k}^2{\sigma }^2\right)}{3},$ $a_1=\mathrm{ }-\frac{\left(4b_1{k}^2{\sigma }^2\right)}{3},$ $a_2=-8b_0\beta k^2\sigma ,$ $a_3=-8b_1\beta k^2\sigma ,$ $a_4=-8b_0{\beta }^2{k}^2,$ $a_5=-8b_1{\beta }^2{\mathrm{k}}^2,$ $b_0=b_0,$ $b_1=b_1$

• Set-2: $c\mathrm{ }=\mathrm{ }-\frac{\left(4k^4{\sigma }^2+3l^2+3m^2\right)}{k},$ $a_0\mathrm{ }=\mathrm{ }0,$ $a_1=\mathrm{ }0,$ $a_2=-8b_0\beta k^2\sigma ,$ $a_3=-8b_1\beta k^2\sigma ,$ $a_4=-8b_0{\beta }^2{k}^2,$ $a_5=-8b_1{\beta }^2{k}^2,$ $b_0=b_0,$ $b_1=b_1$

We found the following soliton solutions when $\sigma \ne \beta :$

• Case 1(a): Nonlinear ordinary differential equation (3.1.1)(3.1.1) $$u\left(\eta \right)=\frac{a_0+a_{1}H\left(\eta \right)+a_2{H}^2\left(\eta \right)+a_3{H}^3\left(\eta \right)+a_4{H}^4\left(\eta \right)+a_5{H}^5\left(\eta \right)}{b_0+b_{1}H\left(\eta \right)},$$(3.1.1) investigates the exponential function solution as a result of using the values provided in set 1 and equation (2.1.8) (3.1.2) $$u_1\left(x,y,z,t\right)=-\frac{4{\sigma }^2{k}^2\left(4\beta \sigma \tau e^{2\sigma \eta }\mathrm{ }+{\sigma }^2{\tau }^2\mathrm{ }+\mathrm{ }{\beta }^2{e}^{4\sigma \eta }\right)}{3{\left(\beta e^{2\sigma \eta }\mathrm{ }-\sigma \tau \right)}^2},$$(3.1.2)

  The hyperbolic form of the modification of the solution function (3.1.2)(3.1.2) $$u_1\left(x,y,z,t\right)=-\frac{4{\sigma }^2{k}^2\left(4\beta \sigma \tau e^{2\sigma \eta }\mathrm{ }+{\sigma }^2{\tau }^2\mathrm{ }+\mathrm{ }{\beta }^2{e}^{4\sigma \eta }\right)}{3{\left(\beta e^{2\sigma \eta }\mathrm{ }-\sigma \tau \right)}^2},$$(3.1.2) is $$u_{1_{10}}\left(x,y,z,t\right)=\frac{-4}{3}{\left(\sigma k\right)}^2\left(1+\frac{6d\sigma \tau }{{\left(\left(\beta -\sigma \tau \right)c\mathit{oshcosh}\mathrm{ }\theta \mathrm{ }+\left(\beta +\sigma \tau \right)s\mathit{inhsinh}\mathrm{ }\theta \right)}^2}\right),$$

where $\theta =\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right).$

Using $\beta =-\sigma \tau$ in $u_{1_{10}}(x,y,z,t)$ we obtain $$u_{1_{11}}\left(x,y,z,t\right)=\frac{-4}{3}{\sigma }^2{k}^2\left(1+\frac{3}{2}\left(\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right)\right)\mathrm{ }\right)$$

If we consider $\beta =\sigma \tau$ then $u_{1_{10}}(x,y,z,t)$ yields $$u_{1_{12}}\left(x,y,z,t\right)=\frac{-4}{3}{\sigma }^2{k}^2\left(1+\frac{3}{2}\left(\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right)\right)\mathrm{ }\right).$$

We obtained the subsequent solutions for $\beta =\sigma :$

• Case 1(b): Equation (3.1.1)(3.1.1) $$u\left(\eta \right)=\frac{a_0+a_{1}H\left(\eta \right)+a_2{H}^2\left(\eta \right)+a_3{H}^3\left(\eta \right)+a_4{H}^4\left(\eta \right)+a_5{H}^5\left(\eta \right)}{b_0+b_{1}H\left(\eta \right)},$$(3.1.1) can be solved for the exponential function in the following form by substituting the parameter values from Set 1 along with Eqs. (2.1.2)(2.1.2) $$u\left(x,t\right)=u\left(\eta \right),\mathrm{ }\mathit{\text{ η}}=\alpha x-ct,$$(2.1.2) and (2.1.9)(2.1.9) $$H\left(\eta \right)={\left[\frac{\left(\tau -1\right)+\left(\tau +1\right)\tanh \hspace{0.17em}\left(\frac{\sigma \left(1-P\right)\eta }{2}\right)\mathrm{ }}{1-\tanh \hspace{0.17em}\left(\frac{\sigma \left(1-P\right)\eta }{2}\right)\mathrm{ }}\right]}^{\frac{1}{1-P}},\mathrm{ }\sigma =\beta ,\mathrm{ }\tau \in R.$$(2.1.9) : $$u_2\left(x,y,z,t\right)=-\frac{4{\sigma }^2{k}^2\left(\left({\tau }^2\text{-}\mathrm{ }4\tau +1\right)t\mathit{anh}\mathrm{ }{\left(\sigma \eta \right)}^2\mathrm{ }+\left(-2{\tau }^2+2\right)\tanh \hspace{0.17em}\left(\sigma \eta \right)+{\tau }^2+4\tau +1\right)}{3{\left(\left(\tau \mathrm{ }+\mathrm{ }1\right)\tanh \hspace{0.17em}\left(\sigma \eta \right)-\left(\tau \text{-}\mathrm{ }1\right)\right)}^2},$$ $u_2(x,y,z,t)$ symbolizes as if we use the real constant $\tau =1,$ $$u_{2_{11}}\left(x,y,z,t\right)=\frac{2}{3}{\left(\sigma k\right)}^2\left(1-3\theta \right),$$

where $\theta =\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right).$

$u_2(x,y,z,t)$ represents as follows if we use the real constant $\tau =-1$ $$u_{2_{12}}\left(x,y,z,t\right)=\frac{2}{3}{\left(\sigma k\right)}^2(1-3(\theta \left)\right),\mathrm{ }$$ where $\theta =\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right).$

• Case 2(a): Using the value of the parameters in Set 2 into solution (3.1.1)(3.1.1) $$u\left(\eta \right)=\frac{a_0+a_{1}H\left(\eta \right)+a_2{H}^2\left(\eta \right)+a_3{H}^3\left(\eta \right)+a_4{H}^4\left(\eta \right)+a_5{H}^5\left(\eta \right)}{b_0+b_{1}H\left(\eta \right)},$$(3.1.1) with the help of equations (2.1.8) we attain the solution for $\sigma \ne \beta$ as follows $$u_3\left(x,y,z,t\right)=-\frac{8{\sigma }^3\tau \beta k^2}{{\left(\beta e^{2\sigma \eta }\mathrm{ }-\sigma \tau \right)}^2}.$$

  The hyperbolic structure form is $$u_{3_{10}}\left(x,y,z,t\right)=\frac{-8{\sigma }^3\tau \beta k^2}{{\left(\left(\beta -\sigma \tau \right)\cosh \hspace{0.17em}\theta +\left(\beta +\sigma \tau \right)\sinh \hspace{0.17em}\theta \right)}^2},$$

where $\theta =\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right).$

By taking $\beta =\sigma \tau$ in $u_{3_{10}}(x,y,z,t)$ yields $$u_{3_{11}}\left(x,y,z,t\right)=-2{\sigma }^2{k}^2\left(\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right)\right)\mathrm{ }$$

Recall $u_{3_{10}}(x,y,z,t)$ by substituting $\beta =-\sigma \tau ,$ we attain $$u_{3_{12}}\left(x,y,z,t\right)=-2{\sigma }^2{k}^2\left(\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right)\right).$$

• Case 2(b): Using equations (2.1.9) and taking into account Set 2 from solution (3.1.1)(3.1.1) $$u\left(\eta \right)=\frac{a_0+a_{1}H\left(\eta \right)+a_2{H}^2\left(\eta \right)+a_3{H}^3\left(\eta \right)+a_4{H}^4\left(\eta \right)+a_5{H}^5\left(\eta \right)}{b_0+b_{1}H\left(\eta \right)},$$(3.1.1) , we get the solution for $\beta =\sigma$ $$u_4\left(x,y,\mathrm{z},t\right)=\frac{8\tau {\sigma }^2{k}^2\left(\theta \mathrm{ }-1\right)}{{\left(\left(\tau +1\right)t\mathit{anh}\mathrm{ }\theta \mathrm{ }-\left(\tau -1\right)\right)}^2},$$

where $\theta =\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right).$

For $\tau =1,$ the solution function $u_4\left(x,y,z,t\right)$ represent as $${u_4}_{11}\left(x,y,z,t\right)=2{\sigma }^2{k}^2(1-\theta ).$$

For $\tau =-1,$ the solution function $u_4\left(x,y,z,t\right)$ represent a kink soliton as $${u_4}_{12}\left(x,y,z,t\right)=2{\sigma }^2{k}^2(1-\theta ),\mathrm{ }$$ where $\theta =\sigma \left(kx+ly+mz+\left(\frac{4k^4{\sigma }^2+3l^2+3m^2}{k}\right)t\right).$

It is crucial to mention that the wave solutions to the Kadomtsev–Petviashvili equation presented are efficient, and inventive, and have not been established in previous studies.

3.2. Investigation of the stated equation through the new auxiliary equation method

We obtain the value of $L=2$ based on the balancing principle between $u″$ and $u^2$ in Eq. (3.4)(3.4) $$-\mathit{cku}+3k^2{u}^2+k^4{u}^{″}-3l^{2}u-3m^{2}u=0.$$(3.4) through the new auxiliary equation method.

Using the NAE method and substituting the value of $L$ into Eq. (2.2.4(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ) we get (3.2.1) $$u\left(\eta \right)=a_0+a_1{a}^{g\left(\eta \right)}+a_2{a}^{2g\left(\eta \right)},$$(3.2.1) where $g(\eta )$ is the root of the nonlinear equation (2.2.5(2.1.1) $$R\left(u,\frac{\partial u}{\partial t},\mathrm{ }\frac{\partial u}{\partial x},\mathrm{ }\frac{{\partial }^{2}u}{\partial x\partial t},\frac{{\partial }^{2}u}{\partial t^2},\mathrm{ }\frac{{\partial }^{2}u}{\partial x^2},\dots \right)=0,$$(2.1.1) ).

Considering the solution (3.2.1)(3.2.1) $$u\left(\eta \right)=a_0+a_1{a}^{g\left(\eta \right)}+a_2{a}^{2g\left(\eta \right)},$$(3.2.1) with help of (2.2.5)(2.2.5) $$g^{\prime }\left(\mu \right)=\frac{1}{\ln \hspace{0.17em}a }\left\{p{a}^{-g\left(\mu \right)}+q+r{a}^{g\left(\mu \right)}\right\}.$$(2.2.5) from Eq. (3.8), we obtain (3.2.2) $$\left(6k^4{r}^2{a}_2+3k^2{a}_2^2\right){\left\{a^{g\left(\eta \right)}\right\}}^4+\left(10k^{4}qr{a}_2+2k^4{r}^2{a}_1+6k^2{a}_1{a}_2\right){\left\{a^{g\left(\eta \right)}\right\}}^3+\left(8k^{4}pr{a}_2+4k^4{q}^2{a}_2+3k^{4}qr{a}_1+6k^2{a}_0{a}_2+3k^2{a}_1^2-ck{a}_2-3l^2{a}_2-3m^2{a}_2\right){\left\{a^{g\left(\eta \right)}\right\}}^2+\left(6k^{4}pq{a}_2+2k^{4}pr{a}_1+k^4{q}^2{a}_1+6k^2{a}_0{a}_1-ck{a}_1-3l^2{a}_1-3m^2{a}_1\right)\left\{a^{g\left(\eta \right)}\right\}+\left(2k^4{p}^2{a}_2+k^{4}pq{a}_1+3k^2{a}_0^2-ck{a}_0\text{-}\mathrm{ }3l^2{a}_0-3m^2{a}_0\right)=0.$$(3.2.2)

From the above Eq. (3.2.2(3.1) $${\left(u_t+6{uu}_x+u_{\mathit{xxx}}\right)}_x-3u_{yy}-3u_{zz}=0.$$(3.1) ) we get a system of algebraic equations by equating the coefficients of all the equal powers of $\left\{a^{g(\eta )}\right\}$ and by solving the obtained system we get the following result:

• Set 1: $c\mathrm{ }=-\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k},\mathrm{ }{A}_0=-2k^{2}pr,\mathrm{ }{A}_1=-2k^{2}qr,\mathrm{ }{A}_2=-2k^2{r}^2$

• Set 2: $c\mathrm{ }=\frac{4k^{4}pr-k^4{q}^2-3l^2-3m^2}{k},\mathrm{ }{A}_0=-\frac{2}{3}{k}^{2}pr-\frac{1}{3}{k}^2{q}^2,A_1=-2k^{2}qr,\mathrm{ }{A}_2=-2k^2{r}^2$

Using the above values of Set 1 along with (2.2.5)(2.2.5) $$g^{\prime }\left(\mu \right)=\frac{1}{\ln \hspace{0.17em}a }\left\{p{a}^{-g\left(\mu \right)}+q+r{a}^{g\left(\mu \right)}\right\}.$$(2.2.5) into Eq. (3.2.1(3.1) $${\left(u_t+6{uu}_x+u_{\mathit{xxx}}\right)}_x-3u_{yy}-3u_{zz}=0.$$(3.1) ), we get the solutions functions of Eq. (3.1)(3.1) $${\left(u_t+6{uu}_x+u_{\mathit{xxx}}\right)}_x-3u_{yy}-3u_{zz}=0.$$(3.1) as follows:

If $q^2\text{-}\mathrm{ }4pr\mathrm{ }<0$ and $r\ne 0,$ $${u_5}_1\left(x,y,z,t\right)=-2k^2\left(pr-\frac{q^2}{4}\right)\left(\frac{\sqrt{\left(4pr-q^2\right)}}{2}\eta \right),$$ or $${u_5}_2\left(x,y,z,t\right)=-2k^2\left(pr-\frac{q^2}{4}\right)\left(\frac{\sqrt{\left(4pr-q^2\right)}}{2}\eta \right)\mathrm{ }.$$

When $q^2\text{-}\mathrm{ }4\mathrm{ }pr>0$ and $r\ne 0,$ $${u_6}_1\left(x,y,z,t\right)=-2k^2\left(pr-\frac{q^2}{4}\right)\left(\frac{\sqrt{\left(q^2-4pr\right)}}{2}\eta \right),$$ or $${u_6}_2\left(x,y,z,t\right)=-2k^2\left(pr-\frac{q^2}{4}\right)\left(\frac{\sqrt{\left(q^2-4pr\right)}}{2}\eta \right)\mathrm{ }.$$

If $q^2+\mathrm{ }4\mathrm{ }{p}^2<0,$ $r\ne 0$ and $r=-p,$ $${u_7}_1\left(x,y,z,t\right)=2k^2\left(p^2+\frac{q^2}{4}\right)\left(\frac{\sqrt{{-4p^2-q}^2}}{2}\eta \right),\mathrm{ }$$ or $${u_7}_2\left(x,y,z,t\right)=2k^2\left(p^2+\frac{q^2}{4}\right)\left(\frac{\sqrt{{-4p^2-q}^2}}{2}\eta \right)\mathrm{ }.$$

When $q^2+\mathrm{ }4\mathrm{ }{p}^2>0,$ $n\ne 0$ and $r=-p,$ $${u_8}_1\left(x,y,z,t\right)=2k^2\left(p^2+\frac{q^2}{4}\right)\left(\frac{\sqrt{\left(q^2+4p^2\right)}}{2}\eta \right),$$ or $${u_8}_2\left(x,y,z,t\right)=2k^2\left(p^2+\frac{q^2}{4}\right)\left(\frac{\sqrt{\left(q^2+4p^2\right)}}{2}\eta \right).$$

If $q^2\text{-}\mathrm{ }4\mathrm{ }{p}^2<0,$ and $r=p,$ $${u_9}_1\left(x,y,z,t\right)=-2k^2\left(p^2-\frac{q^2}{4}\right)\left(\frac{\sqrt{{4p^2-q}^2}}{2}\eta \right),$$ or $${u_9}_2\left(x,y,z,t\right)=-2k^2\left(p^2-\frac{q^2}{4}\right)\left(\frac{\sqrt{{4p^2-q}^2}}{2}\eta \right)$$

When $q^2\text{-}\mathrm{ }4\mathrm{ }{p}^2>0$ and $r=p,$ $${u_{10}}_1\left(x,y,z,t\right)=-2k^2\left(p^2-\frac{q^2}{4}\right)\left(\frac{\sqrt{q^2-4p^2}}{2}\eta \right)$$ or $${u_{10}}_2\left(x,y,z,t\right)=-2k^2\left(p^2-\frac{q^2}{4}\right)\left(\frac{\sqrt{q^2-4p^2}}{2}\eta \right)$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

If $q^2=4pr$ $$u_{11}\left(x,y,z,t\right)=-\frac{2k^2}{{\left(kx+ly+mz+ct\right)}^2}$$ where $c=-\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right).$

When $rp<0,$ $q=0$ and $r\ne 0$ $$u_{{12}_1}\left(x,\mathrm{y},z,t\right)=-2k^{2}pr\left(\sqrt{-rp}\mathrm{ }\eta \right),$$ or $$u_{{12}_2}\left(x,y,z,t\right)=2k^{2}pr\left(\sqrt{-rp}\mathrm{ }\eta \right).$$

If $q=0$ and $p=-r,$ $$u_{13}\left(x,y,z,t\right)=8k^2{r}^2\left(r\eta \right),$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

When $q=r=K,$ $p=0,$ $$u_{14}\left(x,y,z,t\right)=-\frac{2{\left(kK\right)}^2{e}^{K\eta }}{{\left(-1+e^{K\eta }\right)}^2}.$$

If $q=(p+r),$ $$u_{15}\left(x,y,z,t\right)=-\frac{2k^{2}r{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)}^2{e}^{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)\eta }}{{\left(r{e}^{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)\eta }-1\right)}^2}.$$

When $q=-(p+r),$ $$u_{16}\left(x,y,z,t\right)=-\frac{2k^{2}r{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)}^2{e}^{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)\eta }}{{\left(r-e^{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)\eta }\right)}^2}.$$

If $l=0,$ $$u_{17}\left(x,y,z,t\right)=-\frac{2q^2{e}^{q\eta }{k}^{2}r}{{\left(r{e}^{q\eta }-1\right)}^2}.$$

When $n=m=l\ne 0,$ $$u_{18}\left(x,y,z,t\right)=-\frac{3}{2}{k}^2{p}^2\left(\frac{\sqrt{3}p\eta }{2}\right)\mathrm{ }.$$

When $=p,\mathrm{ }q=0,$ $$u_{19}\left(x,y,z,t\right)=-2k^2{p}^2\left(p\eta \right),$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

Applying the values of the parameters of Set 2 along with (2.2.5)(2.2.5) $$g^{\prime }\left(\mu \right)=\frac{1}{\ln \hspace{0.17em}a }\left\{p{a}^{-g\left(\mu \right)}+q+r{a}^{g\left(\mu \right)}\right\}.$$(2.2.5) into Eq. (3.2.1(3.1) $${\left(u_t+6{uu}_x+u_{\mathit{xxx}}\right)}_x-3u_{yy}-3u_{zz}=0.$$(3.1) ), we obtain the solution functions of Eq. (3.1)(3.1) $${\left(u_t+6{uu}_x+u_{\mathit{xxx}}\right)}_x-3u_{yy}-3u_{zz}=0.$$(3.1) as follows:

When $q^2\text{-}\mathrm{ }4pr\mathrm{ }<0$ and $r\ne 0,$ $${u_{20}}_1\left(x,y,z,t\right)=-2k^2\left(pr-\frac{q^2}{4}\right)\left(\left(\frac{\sqrt{\left(4pr-q^2\right)}}{2}\eta \right)\mathrm{ }\right),$$ or $${u_{20}}_2\left(x,y,z,t\right)=-2k^2\left(pr-\frac{q^2}{4}\right)\left(\left(\frac{\sqrt{\left(4pr-q^2\right)}}{2}\eta \right)\mathrm{ }\right).$$

When $q^2\text{-}\mathrm{ }4\mathrm{ }pr>0$ and $r\ne 0,$ $${u_{21}}_1\left(x,y,z,t\right)=-2k^2\left(pr-\frac{q^2}{4}\right)\left(\frac{1}{3}+\left(\frac{\sqrt{\left(q^2-4pr\right)}}{2}\eta \right)\mathrm{ }\right),$$ or $${u_{21}}_2\left(x,y,z,t\right)=-2k^2\left(pr-\frac{q^2}{4}\right)\left(\frac{1}{3}+\left(\frac{\sqrt{\left(q^2-4pr\right)}}{2}\eta \right)\mathrm{ }\right),$$ where $c=kx+ly+mz-\left(\frac{4k^{4}pr-k^4{q}^2-3(l^2+m^2)}{k}\right)t.$

When $q^2+\mathrm{ }4\mathrm{ }{p}^2<0,$ $r\ne 0$ and $r=-p,$ $${u_{22}}_1\left(x,y,z,t\right)=2k^2\left(p^2+\frac{q^2}{4}\right)\left(\frac{1}{3}+\left(\frac{\sqrt{{-4p^2-q}^2}}{2}\eta \right)\mathrm{ }\right),$$ or $${u_{22}}_2\left(x,y,z,t\right)=2k^2\left(p^2+\frac{q^2}{4}\right)\left(\frac{1}{3}+\left(\frac{\sqrt{{-4p^2-q}^2}}{2}\eta \right)\mathrm{ }\right),$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

When $q^2+\mathrm{ }4\mathrm{ }{p}^2>0,$ $n\ne 0$ and $r=-p,$ $${u_{23}}_1\left(x,y,z,t\right)=2k^2\left(p^2+\frac{q^2}{4}\right)\left(\frac{1}{3}-\left(\frac{\sqrt{\left(q^2+4p^2\right)}}{2}\eta \right) \right),$$ or $${u_{23}}_2\left(x,y,z,t\right)=2k^2\left(p^2+\frac{q^2}{4}\right)\left(\frac{1}{3}-\left(\frac{\sqrt{\left(q^2+4{}^2\right)}}{2}\eta \right) \right),$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

When $q^2\text{-}\mathrm{ }4\mathrm{ }{p}^2<0,$ and $r=p,$ $${u_{24}}_1\left(x,y,z,t\right)=-2k^2\left(p^2-\frac{q^2}{4}\right)\left(\frac{1}{3}+\left(\frac{\sqrt{{4p^2-q}^2}}{2}\eta \right) \right),$$ or $${u_{24}}_2\left(x,y,z,t\right)=-2k^2\left(p^2-\frac{q^2}{4}\right)\left(\frac{1}{3}+\left(\frac{\sqrt{{4p^2-q}^2}}{2}\eta \right)\mathrm{ }\right),$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

When $q^2\text{-}\mathrm{ }4\mathrm{ }{p}^2>0\mathrm{ }$ and $r=p,$ $${u_{25}}_1\left(x,y,z,t\right)=-2k^2\left(p^2-\frac{q^2}{4}\right)\left(\frac{1}{3}+\left(\frac{\sqrt{q^2-4p^2}}{2}\eta \right)\mathrm{ }\right),$$ or $${u_{25}}_2\left(x,y,z,t\right)=-2k^2\left(p^2-\frac{q^2}{4}\right)\left(\frac{1}{3}+\left(\frac{\sqrt{q^2-4p^2}}{2}\eta \right)\mathrm{ }\right),$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

When $q^2=4pr$ $$u_{26}\left(x,y,z,t\right)=-\frac{2k^2}{{\left(kx+ly+mz+ct\right)}^2},$$ where $c=-\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right).$

When $p<0,$ $q=0$ and $r\ne 0$ $$u_{{27}_1}\left(x,y,z,t\right)=-2k^{2}pr\left(\frac{1}{3}-\left(\sqrt{-rp}\mathrm{ }\eta \right)\mathrm{ }\right)$$ or $$u_{{27}_2}\left(x,y,z,t\right)=-2k^{2}pr\left(\frac{1}{3}-\left(\sqrt{-rp}\mathrm{ }\eta \right)\mathrm{ }\right),$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

When $q=0$ and $p=-r,$ $$u_{28}\left(x,y,z,t\right)=-2k^2{r}^2\left(\frac{2}{3}+\left(r\eta \right)\mathrm{ }\right).$$

When $q=r=K,$ $p=0$ $$u_{29}\left(x,y,z,t\right)=-2{\left(kK\right)}^2\left(1+\frac{3}{2}\left(\frac{K\eta }{2}\right)\mathrm{ }\right).$$ When $q=(p+r),$ $$u_{30}\left(x,y,z,t\right)=-k^2{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)}^2\left(1+\frac{6r{e}^{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)\eta }}{{\left(r{e}^{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)\eta }-1\right)}^2}\right).$$

When $q=-(p+r),$ $$u_{31}\left(x,y,z,t\right)=-k^2{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)}^2\left(1+\frac{6r{e}^{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)\eta }}{{\left(r-e^{\left(p\mathrm{ }\text{-}\mathrm{ }r\right)\eta }\right)}^2}\right).$$

When $p=0,$ $$u_{32}\left(x,y,z,t\right)=-\frac{q^2{k}^2}{3}\left(1+\frac{{6re}^{q\eta }}{{\left(r{e}^{q\eta }-1\right)}^2}\right).$$

When $n=m=l\ne 0,$ $$u_{33}\left(x,y,z,t\right)=-\frac{3}{2}{k}^2{p}^2\left(\frac{1}{3}+\left(\frac{\sqrt{3}p\eta }{2}\right)\mathrm{ }\right).$$

When $=p,\mathrm{ }q=0,$ $$u_{34}\left(x,y,z,t\right)=\frac{2k^2{p}^2}{3}\left(2-3\left(p\eta \right)\right),$$ where $\eta =kx+ly+mz+\left(\frac{4k^{4}pr-k^4{q}^2+3l^2+3m^2}{k}\right)t.$

From the aforementioned discussion, we have obtained different types of solutions to the stated equation such as hyperbolic function, trigonometric function, exponential function, and rational function.

4. Physical explanation of the travelling wave solutions

This section aims to describe our obtained traveling wave solutions of the Kadomtsev–Petviashvili equation through in the 3D and 2D figures and talk over the natures of those waves for dissimilar values of the free parameters with the aid of software Mathematica.

The obtained solutions via IBSEFM are related to the velocity and wave number of the travelling wave related to the free parameters are $c$ and $k,l,m$ respectively. Here the wave velocity $c$ is purely dependent on the free parameter $k$ which plays a very important role in wave profile. Beginning the values of the free parameters $k,l,m,\sigma$ as $k=10, l=4,m=4,\sigma =0.1,$ then the solution ${u_1}_{11}\left(x,y,z\right)$ represents a dark soliton asserted in Figure 1(a). Moreover, Figure 1(b) and (c) represent the 2D graph of the same solution for the distinct values of $x$ and $t$ respectively. Without changing the values of others parameters but decreasing the value $\sigma =0.01,$ the same solution function portrays peakon shape soliton shown in Figure 2(a). The 2D figures are shown in Figure 2(b) and 2(c) for different values of $x$ and $t$ respectively. Subsequently, by decreasing the value of the parameter $k$ as $k=1$ without changing the other values the same solution function ${u_1}_{11}\left(x,y,t\right)$ portrays the dark soliton-type wave outline in Figure 3(a). Now only increase the value of the Bernoulli parameter $\sigma =1$ and keep the value of other parameters as $k=1, l=4,m=4,$ the soliton shows lump type wave asserted in Figure 3(b), and 3(c) represents the 2D graph of the soliton.

Figure 1. Soliton of ${u_1}_{11}(x,y,z)$ for $k=10,\mathrm{ }l=4,m=4,\sigma =0.1.$

Figure 2. Soliton of ${u_1}_{11}(x,y,z)$ for $k=10, l=4,m=4,\sigma =0.01.$

Figure 3. Soliton of ${u_1}_{11}(x,y,z)$ for (a) $k=1, l=4,m=4,\sigma =0.01$ (b) $k=1, l=4,m=4,\sigma =1.$

Using the values of the free parameters as previous $k=.001,l=4,m=4,\sigma =1,$ the solution ${u_1}_{12}(x,y,t)$ demonstrates bright soliton shown in Figure 4(a) and 4(b) represents the 2D graph for several values of $x.$ when we extended the value of $k=2$ as well as decreased the wave velocity then the solution shows a bell-shaped soliton displayed in Figure 5(a) and for $x=-1, x=0, x=1,$ the same soliton sketched in 2D soliton in Figure 5(b). For $k=1, l=4,m=4,\sigma =1$ the traveling wave solution represents the lump soliton sketched in Figure 6(a) and the 2D graph of the soliton in Figure 6(b). After extending the number of the parameter $k$ in big amounts as $k=200$ the same solution demonstrates the breather-type soliton attached in Figure 7(a) and 7(b) provides the 2D soliton for different values of $x.$

Figure 4. Wave profile of ${u_1}_{12}(x,y,t)$ for $k=0.001, l=4,m=4,\sigma =0.001.$

Figure 5. Wave profile of ${u_1}_{12}(x,y,t)$ for $k=2, l=4,m=4,\sigma =0.01.$

Figure 6. Wave profile of ${u_1}_{12}(x,y,t)$ for $k=1, l=4,m=4,\sigma =1.$

Figure 7. Wave profile of ${u_1}_{12}(x,y,t)$ for $k=200, l=4,m=4,\sigma =0.01.$

Besides, the other obtained results via the IBSEF method and the new auxiliary equation method represent more analogous waves for the different values of the free parameters which are not stated here for sagacity.

5. Comparison of the solutions

Many researchers studied the Kadomtsev–Petviashvili equation by using different methods and achieved atypical analytic solutions. This equation was examined in the early literature through several methods, such as the tanh–coth method (Jawad & Adham, 2013), the mapping method (Peng & Krishnan, 2005), the modified Khater method and Jacobi elliptical function method (Yue et al., 2020) have achieved several solutions for the Kadomtsev–Petviashvili equation. We compare the obtained solutions with those established by Yue et al. (2020) and shown in Table 1.

Table 1. Comparison between Yue et al. (2020) solutions and the obtained solutions obtained in this article.

SL	Solution by Yue et al. (2020)	Solution in this article
(i)	For $\delta =\frac{\rho }{2}=3,$ solution ${\psi }_{34}$ becomes $\frac{2}{3}-2{\tan }^2(10t+x+y+z)$	For $k=l=m=1, r=\sqrt{3}, q=2$ and $p=\frac{2}{\sqrt{3}},$ solution $u_{33}$ becomes $\frac{2}{3}-2{\tan }^2(10t+x+y+z)$
(ii)	For $\delta =\frac{\rho }{2}=3,$ solution ${\psi }_{30}$ becomes $-2\mathrm{se}{\mathrm{c}}^2 (10t+x+y+z)$	For $k=l=m=1, r=1, q=0$ and $p=1,$ solution $u_{9_1}$ becomes $-2\mathrm{se}{\mathrm{c}}^2 (10t+x+y+z)$

In addition to the solutions discussed in the above table, Yue et al. (2020) have found four more solutions that do not match our obtained results. On the other hand, we obtain more fresh exact soliton solutions in our study which are not been stated in the previous investigation. Nevertheless, we have explained the wave profile of the asserted solitons through 3D and 2D plots in details that are absent in the previous literature.

6. Conclusion

In this article, we have successfully applied two significant methods namely, IBSEFM and the NAEM to the Kadomtsev–Petviashvili equation and accomplished an innovative exact solution. There are hyperbolic function, exponential and trigonometric function type solutions have been found. The obtained solution may be very effective for analyzing the nonlinear phenomena that occur in water waves, wind waves, wind-generated waves as well as in fluid dynamics. 3D plots with particular values of the involved parameters have been considered to show the variations in the same solution and 2D plots explained the nature of the wave profile of the same solution regarding the displacement and time. We have provided an efficient wave profile that signifies both comparison and relationship among the results attained by the presented techniques in a comprehensive manner that might be helpful for future researchers to explain the parameters and velocity effect in this arena. Besides, we compared the obtained results with the results of previous literature to purity of this study. The results show that the proposed method is straighter, more effective, and can be applied to many other nonlinear evolution equations in mathematical physics.

Acknowledgments

Thank you very much to Mr. Md. Rawshan Yeazdani, Assistant Professor, Department of English, Pabna University of Science and Technology, Pabna, Bangladesh for checking the whole manuscript and correcting the errors.

Disclosure statement

The authors declare that they have no conflict of interest.

Data availability statement

No real data used here.