Study of compressive and rarefactive lump solitons structures in dusty plasma with double spectral (r,q) distributed electrons

Abstract

Nonlinear features are revealed by one of the most competent and powerful approaches, i.e. soliton theory. The present article starts with the dust collisionless magnetized plasma where electrons are following double spectral $(r,q)$ distribution. From the hydrodynamical governing equations set, Kadomtsev–Petviashvili (KP) equation is derived using the reductive perturbation technique. The derived KP equations are converted to standard KP equations by a suitable transformation to use the Hirota bilinear method (HBM) more conveniently. By using HBM, an auxiliary function is constructed, and through symbolic computation, three arbitrary constant sets are formed. These sets associate with three lump soliton solution sets. Finally, all these lump soliton solutions are returned back by the inverse transformation that was used to make the derived KP equation to the standard KP equation and get the lump soliton solution of the associated system. It is investigated that associated plasma parameters have a significant impact on the lump soliton structures while tracing figures using these plasma parameters within the justified range of the system. While lump soliton features' are analysed for various parameters' effect on it, one most important aspect is revealed. Double spectral index r and q play a significant role in the lump solitons structures and depict compressive and rarefactive lump soliton structures with the associated system. It is found that these double spectral indices r and q are the decision-making parameters for the lump soliton structures to be compressive or rarefactive.

Keywords:

• Lump solitons
• nonlinear structures
• KP equation
• rarefactive solitons
• dusty plasma

1. Introduction

Fascinating dusty plasmas are emerging with a great manifestation of nonlinear dynamics for their omnipresence in the solar system and space plasmas [1,2]. Although dusty plasma is a mixture of dust particles with conventional plasma, it has introduced many interesting features and numerous research workers [3–9] have revealed various wave features like dust acoustic solitary waves (DASWs), dust ion acoustic solitary waves (DIASWs), dust lattice (DL), etc., which are different characters compared to acoustic or ion acoustic modes in conventional plasma. Mathematicians are interested in finding new methods of solving problems, and at the same time, physicists are interested to obtain special features of physical phenomena. Soliton is one such nonlinear phenomenon that has been observed from water waves to optical fibres, also in energy transport, in protein molecules.

Several mathematical techniques have been developed to understand the different possible feature solitons [10,11]. In 1971, Hirota developed a method for obtaining the multi-soliton solutions of the Korteweg–de Vries (KdV) equation which is much more elegant but easy to handle. It is an algebraic method. The solutions come easily by simple algebraic calculation but at the same time, the results obtained are much more informative. The main problem of the nonlinear partial differential equation (PDE) is its integrability. Hirota fantastically handled this situation. He introduced a new derivative called Hirota D-operator [12–14] and converted the nonlinear PDE into a bilinear equation. With a suitable transformation, a nonlinear PDE with dependent variable $f(x,t)$ is converted into a linear equation where the dependent variable is f.f, and the resulting equation is called a Bilinear equation. If an equation can be put into a bilinear form, it is integrable in the Hirota sense. After that one needs a simple algebra to get the explicit expression of the solution. When two solitons collide, they regain their shape after a large time but the phase is changed. Using Hirota's bilinear method, the change of phase after a collision can also be obtained. For the last few decades, this method has been used for a large class of nonlinear evolution equations, including difference-differential and integrodifferential equations.

Kadomtsev–Petviashvili (KP) is a fully integrable nonlinear PDE which was introduced by Boris Kadomtsev and Vladimir Petviashvili [15] at the time of finding stability criteria of single soliton of KdV equation through transverse perturbations. One-dimensional wave structures and their interactions are not able to provide all of the nonlinear features related to nonlinear dynamics, the nature demands the study of nonlinear structures and their interactions in a higher-dimensional form that is close to reality. These are the limitations of the so far investigations.

We want to focus our study on the investigations of solitons interactions based on the higher-dimensional nonlinear evolution equation, i.e. KP equations [16–22] to get much more information from their interactions. In that case, we get soliton structures in all directions and can study their interactions in all directions which will depict many undisclosed features. Lump soliton structures, breathers solitons, and resonance phenomena are now investigated in the different nonlinear dynamical fields of study while the investigations are based on the higher dimensional KP equations. Using HBM, we will be able to study interactions between various solitons like lump–lump interactions, breather–breather interactions, lump–kink, lump–antikink, etc. interactions which contribution will enrich the plasma research field and may be extended to various research fields like oceanography, nonlinear optical fibres, laboratory plasmas, ferrite magnetic material, fluid dynamics, and microwave oscillation.

Lump soliton solutions have been obtained from many integrable equations [23,24]. KP equation is such type of equation, lump solutions have been derived using HBM by Ma et al. [25] from KP equation. Tang et al. [26] investigated lump soliton structures and their interactions for the most general case of two nonlinear evolution equations. Lump soliton structures are recently studied by many researchers [27–29] in various astrophysical plasma situations. In magnetized dusty plasmas, Kumar and Malik [17] have investigated the soliton propagation and they also have noticed the existence of compressive solitons only through KP equations. Mushtaq et al. [18] investigated magneto-acoustic waves through KP soliton with quantum diffraction effects by using a quantum hydrodynamical model in an electron-ion Fermi plasma.

Motivated by the above-mentioned investigations and various investigations regarding lump soliton solutions in many research fields with nonlinear dynamics, we investigate lump solitons structures in plasma dynamics which has not been investigated yet. Lump soliton is nothing but a higher amplitude potential structure compared to their nearby structures. It is a sudden amplification of potential. Tsunami type wave is such an example of lump solitons. The existence of special unpredictable lump solitons in the research fields of the plasma system.

After a careful reading of recent research works [30–39], we have organized our article in such a way.

2. Basic hydrodynamical model equations

We consider a collisionless, magnetized dusty plasma where electrons feature double spectral $(r,q)$ distribution. The basic equations are as follows: (1) $\begin{array}{rl}& \frac{\mathrm{\partial }{n}_i}{\mathrm{\partial }T}+{\mathrm{\nabla }}^{\mathrm{\prime }}.(n_i{v}_i)=0,\end{array}$(1) (2) $\begin{array}{rl}& \frac{\mathrm{\partial }{v}_i}{\mathrm{\partial }T}+(v_i.{\mathrm{\nabla }}^{\mathrm{\prime }})v_i=-\frac{e{\mathrm{\nabla }}^{\mathrm{\prime }}\psi }{m_i}+\frac{e{B}_0}{m_{i}c}{v}_i\times e_z,\end{array}$(2) (3) $\begin{array}{rl}& ({\mathrm{\nabla }}^{\mathrm{\prime }}{)}^2\psi =-4\pi [-e{n}_e+e{n}_i-e{z}_d{n}_d],\end{array}$(3) $n_e$, $n_i$, $n_d$ are the densities of electrons, ions, and dust, respectively. $v_i$, $m_i$ are the velocity and mass of ions and ψ is the electrostatic plasma potential. $z_d$ is the dust charge number so the charge of the dust is given by $q_d=-e{z}_d$, where e is the elementary charge of electrons. To get dimensionless governing equations, we normalized the basic variables involved in Equations (1(1) $\begin{array}{rl}& \frac{\mathrm{\partial }{n}_i}{\mathrm{\partial }T}+{\mathrm{\nabla }}^{\mathrm{\prime }}.(n_i{v}_i)=0,\end{array}$(1) )–(3(3) $\begin{array}{rl}& ({\mathrm{\nabla }}^{\mathrm{\prime }}{)}^2\psi =-4\pi [-e{n}_e+e{n}_i-e{z}_d{n}_d],\end{array}$(3) ) to $t=\mathrm{\Omega T},\mathrm{\nabla }=\frac{c_s}{\mathrm{\Omega }}{\mathrm{\nabla }}^{\mathrm{\prime }},v=\frac{v_i}{c_s},n=\frac{n_i}{n_{i0}},\varphi =\frac{\mathrm{e\psi }}{T_e}.$The collective behaviour of plasma species in different situations demands different power-law functions which can describe its nature more conveniently. When the plasma undergoes a uniform magnetic field, the features of the system are dramatically altered. Power law tails are generated naturally in space plasmas as a consequence of the reliance of the mean free path and collision fragmentary on the particle velocity. Nevertheless, the rapid declines of the collision fragmentary with the particle speed signify that the statistics of greater energy molecules in the tail of the distribution of the velocity can alter strongly from Maxwellian/Gaussian on periods much lesser than the time will be taken for the initial velocity profile to ease completely to a Maxwellian. According to the demand of dynamics in plasma situations like magnetosphere [40], ionosphere, magneto sheaths different power-law functions have been prescribed on the strong recommendation of space observation by AMPTE satellite [41], WIND 3-D experiments [42], and solar wind proton data [43] from CLUSTER. The well-mentioned acknowledgement suggests the best fitment of the most comprehensive double spectral $(r,q)$ distribution and was first introduced by [44]. Several researchers [45–48] have investigated various nonlinear structures with electrons featuring the most useful non-Maxwellian distribution and close to reality in the above-mentioned plasma situations. The electron distribution function is governed by [21] (4) $f_e^{r,q}(v)=\delta {[1+\frac{1}{(q-1)}{\left(\frac{v^2}{{\gamma }_e^2}\right)}^{r+1}]}^{-q},$(4) where δ is the normalization constant and is given by (5) $\delta =\frac{n_{e0}(q-1{)}^{-\frac{1}{2(r+1)}}\mathrm{\Gamma }(q)}{2{\gamma }_e\mathrm{\Gamma }(q-\frac{1}{2(r+1)})\mathrm{\Gamma }(1+\frac{1}{2(r+1)})}$(5) and ${\gamma }_e$ is given by (6) $\begin{array}{rl}{\gamma }_e& =\{\frac{3(q-1{)}^{-\frac{1}{(r+1)}}\mathrm{\Gamma }(q-\frac{1}{2(r+1)})\mathrm{\Gamma }(1+\frac{1}{2(r+1)})}{\mathrm{\Gamma }(q-\frac{3}{2(r+1)})\mathrm{\Gamma }(1+\frac{3}{2(r+1)})}{\}}^{\frac{1}{2}}\\ & \times \sqrt{\frac{T_e}{m_e}}.\end{array}$(6) Electron density distribution can be written in a compact form following the same process as given by [45] (7) $n_e=1+A_1\varphi +B_1{\varphi }^2+C_1{\phi }^3+D_1{\varphi }^4,$(7) where (8) $\begin{array}{rl}{A}_1& =\frac{(q-1{)}^{-\frac{1}{2(r+1)}}\mathrm{\Gamma }[q+\frac{1}{2(r+1)}]\mathrm{\Gamma }[1-\frac{1}{2(r+1)}]}{2\mathrm{\Gamma }[q-\frac{1}{2(r+1)}]\mathrm{\Gamma }[1+\frac{1}{2(r+1)}]},\end{array}$(8) (9) $\begin{array}{rl}{B}_1& =-\frac{(q-1{)}^{-\frac{2}{(r+1)}}\mathrm{\Gamma }[q+\frac{3}{2(r+1)}]\mathrm{\Gamma }[1-\frac{3}{2(r+1)}]}{8\mathrm{\Gamma }[q-\frac{1}{2(r+1)}]\mathrm{\Gamma }[1+\frac{1}{2(r+1)}]},\end{array}$(9) (10) $\begin{array}{rl}{C}_1& =\frac{(q-1{)}^{-\frac{3}{(r+1)}}\mathrm{\Gamma }[q+\frac{5}{2(r+1)}]\mathrm{\Gamma }[1-\frac{5}{2(r+1)}]}{16\mathrm{\Gamma }[q-\frac{1}{2(r+1)}]\mathrm{\Gamma }[1+\frac{1}{2(r+1)}]},\end{array}$(10) (11) $\begin{array}{rl}{D}_1& =-\frac{5(q-1{)}^{-\frac{4}{(r+1)}}\mathrm{\Gamma }[q+\frac{7}{2(r+1)}]\mathrm{\Gamma }[1-\frac{7}{2(r+1)}]}{128\mathrm{\Gamma }[q-\frac{1}{2(r+1)}]\mathrm{\Gamma }[1+\frac{1}{2(r+1)}]}\end{array}$(11) and ${\lambda }_e={(\frac{T_e}{4\pi n_{e0}{\mathrm{e}}^2})}^{1/2}$ is the electron Debye length, $c_s={(\frac{T_e}{m_i})}^{1/2}$ is the ion acoustic velocity, $\mathrm{\Omega }=\frac{e{B}_0}{m_{i}c}$ is the ion gyrofrequency. $n_{e0}$, $n_{i0}$ are the electron and ion densities respectively in the unperturbed state.

2.1. Normalized equations

Using the above-mentioned relations of the variables into Equations (1(1) $\begin{array}{rl}& \frac{\mathrm{\partial }{n}_i}{\mathrm{\partial }T}+{\mathrm{\nabla }}^{\mathrm{\prime }}.(n_i{v}_i)=0,\end{array}$(1) )–(11(11) $\begin{array}{rl}{D}_1& =-\frac{5(q-1{)}^{-\frac{4}{(r+1)}}\mathrm{\Gamma }[q+\frac{7}{2(r+1)}]\mathrm{\Gamma }[1-\frac{7}{2(r+1)}]}{128\mathrm{\Gamma }[q-\frac{1}{2(r+1)}]\mathrm{\Gamma }[1+\frac{1}{2(r+1)}]}\end{array}$(11) ) we get the normalized set of equations as (12) $\begin{array}{rl}& \frac{\mathrm{\partial }n}{\mathrm{\partial }t}+\mathrm{\nabla }.(\mathrm{nv})=0,\end{array}$(12) (13) $\begin{array}{rl}& \frac{\mathrm{\partial }v}{\mathrm{\partial }t}+(v.\mathrm{\nabla }).v=-\mathrm{\nabla \varphi }+(v\times e_z),\end{array}$(13) (14) $\begin{array}{rl}& {\mathrm{\nabla }}^2\varphi =\beta [(1+A_1\varphi +B_1{\varphi }^2+C_1{\varphi }^3+\cdots )-{\delta }_{1}n+{\delta }_2],\end{array}$(14) where $\beta =\frac{4\pi {\mathrm{e}}^2}{T_e}{r}_g^2$, ${\delta }_1$ and ${\delta }_2$ are the ratio of ions and dust grains to the electrons number density and $r_g=\frac{c_s}{\mathrm{\Omega }}$ is the ion gyroradius.

3. Derivation of KP equation from the governing equation

We assume that the propagation plane of the wave is the x–z plane. Accordingly, the normalized equations set is taken as (15) $\begin{array}{rl}& \frac{\mathrm{\partial }n}{\mathrm{\partial }t}+\frac{\mathrm{\partial }(n{v}_x)}{\mathrm{\partial }x}+\frac{\mathrm{\partial }(n{v}_z)}{\mathrm{\partial }z}=0,\end{array}$(15) (16) $\begin{array}{rl}& \frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }t}+(v_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}+v_z\frac{\mathrm{\partial }}{\mathrm{\partial }z})v_x=-\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}+v_y,\end{array}$(16) (17) $\begin{array}{rl}& \frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }t}+(v_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}+v_z\frac{\mathrm{\partial }}{\mathrm{\partial }z})v_y=-v_x,\end{array}$(17) (18) $\begin{array}{rl}& \frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }t}+(v_x\frac{\mathrm{\partial }}{\mathrm{\partial }x}+v_z\frac{\mathrm{\partial }}{\mathrm{\partial }z})v_z=-\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }z},\end{array}$(18) (19) $\begin{array}{rl}& (\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2})\varphi =\beta [(1+A_1\varphi +B_1{\varphi }^2+C_1{\varphi }^3+\cdots )\\ & -{\delta }_{1}n+{\delta }_2].\end{array}$(19) To linearize the normalized governing Equations (15(15) $\begin{array}{rl}& \frac{\mathrm{\partial }n}{\mathrm{\partial }t}+\frac{\mathrm{\partial }(n{v}_x)}{\mathrm{\partial }x}+\frac{\mathrm{\partial }(n{v}_z)}{\mathrm{\partial }z}=0,\end{array}$(15) )–(19(19) $\begin{array}{rl}& (\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2})\varphi =\beta [(1+A_1\varphi +B_1{\varphi }^2+C_1{\varphi }^3+\cdots )\\ & -{\delta }_{1}n+{\delta }_2].\end{array}$(19) ), let us write the dependent variable as the sum of equilibrium and perturbed parts so that we write $n=1+\overline{n}$, $v_x=\overline{v_x}$ , $v_y=\overline{v_y}$, $v_z=\overline{v_z}$ and $\varphi =\overline{\varphi }$. After obtaining a linearized set of equations, we take all perturbed variables of the form ${\mathrm{e}}^{i(k_{x}x+k_{z}z-\mathrm{\omega t})}$, where $k_x$, $k_z$ are the wave numbers in x and z directions respectively and ω is the wave frequency $(\omega \ll \mathrm{\Omega })$ , this leads to (20) $\left|\begin{array}{ccccc}-\mathrm{i\omega }& i{k}_x& 0& i{k}_z& 0\\ 0& -\mathrm{i\omega }& -1& 0& i{k}_x\\ 0& 1& -\mathrm{i\omega }& 0& 0\\ 0& 0& 0& -\mathrm{i\omega }& i{k}_z\\ \beta {\delta }_1& 0& 0& 0& -(k_x^2+k_z^2+A_1\beta )\end{array}\right|=0.$(20) This provides us with the dispersion relation (21) $\omega =k_z{[\frac{a_1}{{\delta }_1}+(1+\frac{1}{\beta {\delta }_1})k_x^2+(1+\frac{1}{\beta {\delta }_1})k_z^2]}^{-1/2}.$(21)

3.1. Stretched co-ordinates and perturbation

The dispersion relation guides us on which stretching should be adopted for the fluid hydrodynamical model to obtain nonlinear evolution equations. Based on the dispersion relation, according to the reductive perturbation technique (RPT), independent variables are stretched as (22) $\begin{array}{rl}X& ={ϵ}^{2}x,\end{array}$(22) (23) $\begin{array}{rl}\xi & =ϵ(z-\mathrm{Vt}),\end{array}$(23) (24) $\begin{array}{rl}\tau & ={ϵ}^{3}t,\end{array}$(24) where V is the phase velocity of DASWs and ϵ measures the strength of the nonlinearity.

The dependent variables follow the expansion: (25) $\begin{array}{rl}n& =1+{ϵ}^2{n}_1+{ϵ}^4{n}_2+\cdots ,\end{array}$(25) (26) $\begin{array}{rl}{v}_x& ={ϵ}^3{v}_{x_1}+{ϵ}^5{v}_{x_2}+\cdots ,\end{array}$(26) (27) $\begin{array}{rl}{v}_y& ={ϵ}^3{v}_{y_1}+{ϵ}^5{v}_{y_2}+\cdots ,\end{array}$(27) (28) $\begin{array}{rl}{v}_z& ={ϵ}^2{v}_{z_1}+{ϵ}^4{v}_{z_2}+\cdots ,\end{array}$(28) (29) $\begin{array}{rl}\varphi & ={ϵ}^2{\varphi }_1+{ϵ}^4{\varphi }_2+\cdots .\end{array}$(29) Normalized basic set of Equations (15(15) $\begin{array}{rl}& \frac{\mathrm{\partial }n}{\mathrm{\partial }t}+\frac{\mathrm{\partial }(n{v}_x)}{\mathrm{\partial }x}+\frac{\mathrm{\partial }(n{v}_z)}{\mathrm{\partial }z}=0,\end{array}$(15) )–(19(19) $\begin{array}{rl}& (\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2})\varphi =\beta [(1+A_1\varphi +B_1{\varphi }^2+C_1{\varphi }^3+\cdots )\\ & -{\delta }_{1}n+{\delta }_2].\end{array}$(19) ) can be expressed in terms of X, ξ and τ as (30) $\begin{array}{rl}& -\mathrm{ϵV}\frac{\mathrm{\partial }n}{\mathrm{\partial }\xi }+{ϵ}^3\frac{\mathrm{\partial }n}{\mathrm{\partial }\tau }+{ϵ}^2\frac{\mathrm{\partial }(n{v}_x)}{\mathrm{\partial }X}+ϵ\frac{\mathrm{\partial }(n{v}_z)}{\mathrm{\partial }\xi }=0,\end{array}$(30) (31) $\begin{array}{rl}& -\mathrm{ϵV}\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }\xi }+{ϵ}^3\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }\tau }+({ϵ}^2{v}_x\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }X}+ϵv_z\frac{\mathrm{\partial }{v}_x}{\mathrm{\partial }\xi })\\ & =-{ϵ}^2\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }X}+v_y,\end{array}$(31) (32) $\begin{array}{rl}& -\mathrm{ϵV}\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }\xi }+{ϵ}^3\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }\tau }+({ϵ}^2{v}_x\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }X}+ϵv_z\frac{\mathrm{\partial }{v}_y}{\mathrm{\partial }\xi })=-v_x,\end{array}$(32) (33) $\begin{array}{rl}& -\mathrm{ϵV}\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }\xi }+{ϵ}^3\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }\tau }+({ϵ}^2{v}_x\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }X}+ϵv_z\frac{\mathrm{\partial }{v}_z}{\mathrm{\partial }\xi })=-ϵ\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\xi },\end{array}$(33) (34) $\begin{array}{rl}& ({ϵ}^4\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{X}^2}+{ϵ}^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\xi }^2})\varphi =\beta [(1+A_1\varphi +B_1{\varphi }^2+C_1{\varphi }^3\\ & +\cdots )-{\delta }_{1}n+{\delta }_2].\end{array}$(34) Substituting the depended variables expressions (25(25) $\begin{array}{rl}n& =1+{ϵ}^2{n}_1+{ϵ}^4{n}_2+\cdots ,\end{array}$(25) )–(29(29) $\begin{array}{rl}\varphi & ={ϵ}^2{\varphi }_1+{ϵ}^4{\varphi }_2+\cdots .\end{array}$(29) ) in Equations (30(30) $\begin{array}{rl}& -\mathrm{ϵV}\frac{\mathrm{\partial }n}{\mathrm{\partial }\xi }+{ϵ}^3\frac{\mathrm{\partial }n}{\mathrm{\partial }\tau }+{ϵ}^2\frac{\mathrm{\partial }(n{v}_x)}{\mathrm{\partial }X}+ϵ\frac{\mathrm{\partial }(n{v}_z)}{\mathrm{\partial }\xi }=0,\end{array}$(30) )–(34(34) $\begin{array}{rl}& ({ϵ}^4\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{X}^2}+{ϵ}^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\xi }^2})\varphi =\beta [(1+A_1\varphi +B_1{\varphi }^2+C_1{\varphi }^3\\ & +\cdots )-{\delta }_{1}n+{\delta }_2].\end{array}$(34) ). Starting from the lowest order ϵ's coefficient to zero, we get equations containing variables. For each next order of ϵ, there corresponds an equation. All the variables are expressed in the form of potential. Eliminating all other variables and converting all second-order dependent variables into first-order potential terms we get the KP equation after simplification as (35) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\tau }+A{\varphi }_1\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\xi }+B\frac{{\mathrm{\partial }}^3{\varphi }_1}{\mathrm{\partial }{\xi }^3}]-C\frac{{\mathrm{\partial }}^2{\varphi }_1}{\mathrm{\partial }{X}^2}=0,$(35) where $A=\frac{-2B_1{V}^2+3A_1}{2A_{1}V}$, $B=\frac{V}{2A_1\beta }$ and $C=\frac{V}{2}$ and $A_1$ and $B_1$ are given by (7(7) $n_e=1+A_1\varphi +B_1{\varphi }^2+C_1{\phi }^3+D_1{\varphi }^4,$(7) )–(9(9) $\begin{array}{rl}{B}_1& =-\frac{(q-1{)}^{-\frac{2}{(r+1)}}\mathrm{\Gamma }[q+\frac{3}{2(r+1)}]\mathrm{\Gamma }[1-\frac{3}{2(r+1)}]}{8\mathrm{\Gamma }[q-\frac{1}{2(r+1)}]\mathrm{\Gamma }[1+\frac{1}{2(r+1)}]},\end{array}$(9) ) $V^2=\frac{{\delta }_1}{A_1}$ is the phase velocity. Equations (35(35) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\tau }+A{\varphi }_1\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\xi }+B\frac{{\mathrm{\partial }}^3{\varphi }_1}{\mathrm{\partial }{\xi }^3}]-C\frac{{\mathrm{\partial }}^2{\varphi }_1}{\mathrm{\partial }{X}^2}=0,$(35) ) is known as the KP equation.

3.2. Standard form of KP equation

Using the transformations, (36) $\xi =\overline{\xi }{B}^{\frac{1}{3}},{\varphi }_1=6\overline{{\varphi }_1}{A}^{-1}{B}^{\frac{1}{3}},X=\frac{1}{\sqrt{\sigma }}\overline{X}{B}^{\frac{1}{6}}{C}^{\frac{1}{2}},\tau =\overline{\tau }.$(36) Equation (35(35) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\tau }+A{\varphi }_1\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\xi }+B\frac{{\mathrm{\partial }}^3{\varphi }_1}{\mathrm{\partial }{\xi }^3}]-C\frac{{\mathrm{\partial }}^2{\varphi }_1}{\mathrm{\partial }{X}^2}=0,$(35) ) is converted to $\frac{\mathrm{\partial }}{\mathrm{\partial }\overline{\xi }}[\frac{\mathrm{\partial }\overline{{\varphi }_1}}{\mathrm{\partial }\overline{\tau }}+6\overline{{\varphi }_1}\frac{\mathrm{\partial }\overline{{\varphi }_1}}{\mathrm{\partial }\overline{\xi }}+\frac{{\mathrm{\partial }}^3\overline{{\varphi }_1}}{\mathrm{\partial }{\overline{\xi }}^3}]-\sigma \frac{{\mathrm{\partial }}^2\overline{{\varphi }_1}}{\mathrm{\partial }{\overline{X}}^2}=0$Removing the bar and subscript, we have the standard KP equation as (37) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\tau }+6\varphi \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\xi }+\frac{{\mathrm{\partial }}^3\varphi }{\mathrm{\partial }{\xi }^3}]-\sigma \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{X}^2}=0.$(37)

4. Lump solutions of KP equation using HBM

In this section, we derive the lump soliton solutions by using HBM.

When $\sigma =1$, Equation (37(37) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\tau }+6\varphi \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\xi }+\frac{{\mathrm{\partial }}^3\varphi }{\mathrm{\partial }{\xi }^3}]-\sigma \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{X}^2}=0.$(37) ) is called KPI type equation and when $\sigma =-1$, Equation (37(37) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\tau }+6\varphi \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\xi }+\frac{{\mathrm{\partial }}^3\varphi }{\mathrm{\partial }{\xi }^3}]-\sigma \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{X}^2}=0.$(37) ) is called the KPII-type equation.

According to the HBM, the auxiliary function is the frame as (38) $\begin{array}{rl}\varphi (\xi ,X,\tau )& =2{(\ln f)}_{\mathrm{\xi \xi }}\\ & =2\left[\frac{f_{\mathrm{\xi \xi }}f-f_{\xi }^2}{f^2}\right].\end{array}$(38) On implementation of the auxiliary function in the standard KP equation, the SKP equation takes its bilinear form as (39) $\begin{array}{rl}& (f_{\mathrm{\xi \tau }}f-f_{\xi }{f}_{\tau })-\sigma (f_{\mathrm{XX}}f-f_X^2)+f_{\mathrm{\xi \xi \xi \xi }}f-4f_{\mathrm{\xi \xi \xi }}{f}_{\xi }\\ & +3f_{\mathrm{\xi \xi }}^2=0.\end{array}$(39) The Hirota D operator of Equation (39(39) $\begin{array}{rl}& (f_{\mathrm{\xi \tau }}f-f_{\xi }{f}_{\tau })-\sigma (f_{\mathrm{XX}}f-f_X^2)+f_{\mathrm{\xi \xi \xi \xi }}f-4f_{\mathrm{\xi \xi \xi }}{f}_{\xi }\\ & +3f_{\mathrm{\xi \xi }}^2=0.\end{array}$(39) ) is (40) $(D_{\xi }{D}_{\tau }+D_{\xi }^4)(f.f)-\sigma D_X^2(f.f)=0,\sigma =\pm 1.$(40) The (2+1)-dimensional bilinear KPI equation (for $\sigma =1$ in (39(39) $\begin{array}{rl}& (f_{\mathrm{\xi \tau }}f-f_{\xi }{f}_{\tau })-\sigma (f_{\mathrm{XX}}f-f_X^2)+f_{\mathrm{\xi \xi \xi \xi }}f-4f_{\mathrm{\xi \xi \xi }}{f}_{\xi }\\ & +3f_{\mathrm{\xi \xi }}^2=0.\end{array}$(39) ) is given as follows: (41) $(f_{\mathrm{\xi \tau }}f-f_{\xi }{f}_{\tau })-(f_{\mathrm{XX}}f-f_X^2)+f_{\mathrm{\xi \xi \xi \xi }}f-4f_{\mathrm{\xi \xi \xi }}{f}_{\xi }+3f_{\mathrm{\xi \xi }}^2=0.$(41) To search for quadratic function solutions to the (41(41) $(f_{\mathrm{\xi \tau }}f-f_{\xi }{f}_{\tau })-(f_{\mathrm{XX}}f-f_X^2)+f_{\mathrm{\xi \xi \xi \xi }}f-4f_{\mathrm{\xi \xi \xi }}{f}_{\xi }+3f_{\mathrm{\xi \xi }}^2=0.$(41) ), we begin with (42) $\begin{array}{rl}& f=g^2+h^2+A_9,g=A_1\xi +A_{2}X+A_3\tau +A_4,\\ h& =A_5\xi +A_{6}X+A_7\tau +A_8,\end{array}$(42) where $A_i,1\le i\le 9$, are real parameters to be determined.

Putting the value of f obtained from (42(42) $\begin{array}{rl}& f=g^2+h^2+A_9,g=A_1\xi +A_{2}X+A_3\tau +A_4,\\ h& =A_5\xi +A_{6}X+A_7\tau +A_8,\end{array}$(42) ) and its partial derivatives, in (41(41) $(f_{\mathrm{\xi \tau }}f-f_{\xi }{f}_{\tau })-(f_{\mathrm{XX}}f-f_X^2)+f_{\mathrm{\xi \xi \xi \xi }}f-4f_{\mathrm{\xi \xi \xi }}{f}_{\xi }+3f_{\mathrm{\xi \xi }}^2=0.$(41) ), we get the set of constraining equations for the parameters: (43) $\left\{\begin{array}{rl}& A_1=A_1,A_2=A_2,A_3=\frac{A_1{A}_2^2-A_1{A}_6^2+2A_2{A}_5{A}_6}{A_1^2+A_5^2},\\ & A_4=A_4,A_5=A_5,A_6=A_6,A_7\\ & =\frac{2A_1{A}_2{A}_6-A_2^2{A}_5+A_5{A}_6^2}{A_1^2+A_5^2},A_8=A_8,A_9\\ & =\frac{3(A_1^2+A_5^2{)}^3}{(A_1{A}_6-A_2{A}_5{)}^2}\end{array}\right\}$(43) which needs to satisfy a determinant condition (44) $\mathrm{\Delta }:=A_1{A}_6-A_2{A}_5=\left|\begin{array}{cc}{A}_1& A_2\\ A_5& A_6\end{array}\right|\ne 0.$(44) This set leads to a class of positive quadratic function solutions to the bilinear KPI Equation (41(41) $(f_{\mathrm{\xi \tau }}f-f_{\xi }{f}_{\tau })-(f_{\mathrm{XX}}f-f_X^2)+f_{\mathrm{\xi \xi \xi \xi }}f-4f_{\mathrm{\xi \xi \xi }}{f}_{\xi }+3f_{\mathrm{\xi \xi }}^2=0.$(41) ): (45) $\begin{array}{rl}f& ={[A_1\xi +A_{2}X+\frac{A_1{A}_2^2-A_1{A}_6^2+2A_2{A}_5{A}_6}{A_1^2+A_5^2}\tau +A_4]}^2\\ & +{[A_5\xi +A_{6}X+\frac{2A_1{A}_2{A}_6-A_2^2{A}_5+A_5{A}_6^2}{A_1^2+A_5^2}\tau +A_8]}^2\\ & +\frac{3(A_1^2+A_5^2{)}^3}{(A_1{A}_6-A_2{A}_5{)}^2}\end{array}$(45) and the resulting class of quadratic function solutions, in turn, yields a class of lump solutions to the (2+1)-dimensional KPI Equation (41(41) $(f_{\mathrm{\xi \tau }}f-f_{\xi }{f}_{\tau })-(f_{\mathrm{XX}}f-f_X^2)+f_{\mathrm{\xi \xi \xi \xi }}f-4f_{\mathrm{\xi \xi \xi }}{f}_{\xi }+3f_{\mathrm{\xi \xi }}^2=0.$(41) ) through the transformation (38(38) $\begin{array}{rl}\varphi (\xi ,X,\tau )& =2{(\ln f)}_{\mathrm{\xi \xi }}\\ & =2\left[\frac{f_{\mathrm{\xi \xi }}f-f_{\xi }^2}{f^2}\right].\end{array}$(38) ): (46) $\varphi (\xi ,X,\tau )=\frac{4(A_1^2+A_5^2)f-8(A_{1}g+A_{5}h{)}^2}{f^2},$(46) where the function f is defined by Equation (45(45) $\begin{array}{rl}f& ={[A_1\xi +A_{2}X+\frac{A_1{A}_2^2-A_1{A}_6^2+2A_2{A}_5{A}_6}{A_1^2+A_5^2}\tau +A_4]}^2\\ & +{[A_5\xi +A_{6}X+\frac{2A_1{A}_2{A}_6-A_2^2{A}_5+A_5{A}_6^2}{A_1^2+A_5^2}\tau +A_8]}^2\\ & +\frac{3(A_1^2+A_5^2{)}^3}{(A_1{A}_6-A_2{A}_5{)}^2}\end{array}$(45) ), and the functions of g and h are given as follows: (47) $\begin{array}{rl}g& =A_1\xi +A_{2}X+\frac{A_1{A}_2^2-A_1{A}_6^2+2A_2{A}_5{A}_6}{A_1^2+A_5^2}\tau +A_4,\end{array}$(47) (48) $\begin{array}{rl}h& =A_5\xi +A_{6}X+\frac{2A_1{A}_2{A}_6-A_2^2{A}_5+A_5{A}_6^2}{A_1^2+A_5^2}\tau +A_8.\end{array}$(48) In this class of lump solutions, all six involved parameters of $A_1,A_2,A_4,A_5,A_6,A_8$ are arbitrarily provided that the solutions are well defined, i.e. if the determinant condition Equation (44(44) $\mathrm{\Delta }:=A_1{A}_6-A_2{A}_5=\left|\begin{array}{cc}{A}_1& A_2\\ A_5& A_6\end{array}\right|\ne 0.$(44) ) is satisfied. That determinant condition precisely means that two directions $(A_1,A_2)$ and $(A_5,A_6)$ in the $\mathrm{\xi X}$-plane are not parallel.

Now a class of Lump solutions to the $(2+1)$ -dimensional KPI equation in Equation (35(35) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\tau }+A{\varphi }_1\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\xi }+B\frac{{\mathrm{\partial }}^3{\varphi }_1}{\mathrm{\partial }{\xi }^3}]-C\frac{{\mathrm{\partial }}^2{\varphi }_1}{\mathrm{\partial }{X}^2}=0,$(35) ) by going back to its original coefficient and omitting the subscript of ${\varphi }_1$, is obtained as (49) $\varphi (\xi ,X,\tau )=4\left(\frac{6B^{1/3}}{A}\right)\left[\frac{(A_1^2+A_5^2)f-2(A_{1}g+A_{5}h{)}^2}{f^2}\right],$(49) where the functions g, h and f are given as follows: (50) $\begin{array}{rl}g& =\frac{1}{B^{1/3}{C}^{1/2}}[C^{1/2}{A}_1\xi +B^{1/6}{A}_{2}X+B^{1/3}{C}^{1/2}{A}_3\tau \\ & +B^{1/3}{C}^{1/2}{A}_4],\end{array}$(50) (51) $\begin{array}{rl}h& =\frac{1}{B^{1/3}{C}^{1/2}}[C^{1/2}{A}_5\xi +B^{1/6}{A}_{6}X+B^{1/3}{C}^{1/2}{A}_7\tau \\ & +B^{1/3}{C}^{1/2}{A}_8]\end{array}$(51) and (52) $\begin{array}{rl}f& =\frac{1}{B^{2/3}C}((C^{1/2}{A}_1\xi +B^{1/6}{A}_{2}X\\ & +B^{1/3}{C}^{1/2}{A}_3\tau +B^{1/3}{C}^{1/2}{A}_4{)}^2\\ & +(C^{1/2}{A}_5\xi +B^{1/6}{A}_{6}X+B^{1/3}{C}^{1/2}{A}_7\tau \\ & +B^{1/3}{C}^{1/2}{A}_8{)}^2+B^{2/3}C{A}_9),\end{array}$(52) where $\begin{array}{rl}{A}_3& =\frac{A_1{A}_2^2-A_1{A}_6^2+2A_2{A}_5{A}_6}{A_1^2+A_5^2},\\ A_7& =\frac{2A_1{A}_2{A}_6-A_2^2{A}_5+A_5{A}_6^2}{A_1^2+A_5^2},\\ A_9& =\frac{3(A_1^2+A_5^2{)}^3}{(A_1{A}_6-A_2{A}_5{)}^2}\end{array}$and $\begin{array}{rl}A& =\frac{-2b_1{V}^2+3a_1}{2a_{1}V},B=\frac{V}{2a_1\beta },C=\frac{-V}{2},\\ a_1& =\frac{(1+q)}{2},b_1=\frac{(1+q)(3-q)}{8}.\end{array}$But it is not possible to provide the explanation of Equation (38(38) $\begin{array}{rl}\varphi (\xi ,X,\tau )& =2{(\ln f)}_{\mathrm{\xi \xi }}\\ & =2\left[\frac{f_{\mathrm{\xi \xi }}f-f_{\xi }^2}{f^2}\right].\end{array}$(38) ) because this expression is coming due to the proper guess of transformation of the dependent variables when applying the Hirota bilinear method (HBM). According to the HBM, a suitable transformation has to be set so that it can change the nonlinear partial differential equations that are in quadratic form in dependent variables into two linear equations, i.e. ”Bilinear form” and this is why the method is named as HBM. This is the most challenging part to set the transformation to get the soliton solution of a nonlinear evolution equation using the HBM. To obtain the lump soliton solution of the derived KP Equation (35(35) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\tau }+A{\varphi }_1\frac{\mathrm{\partial }{\varphi }_1}{\mathrm{\partial }\xi }+B\frac{{\mathrm{\partial }}^3{\varphi }_1}{\mathrm{\partial }{\xi }^3}]-C\frac{{\mathrm{\partial }}^2{\varphi }_1}{\mathrm{\partial }{X}^2}=0,$(35) ) rather to solve the standard KP Equation (37(37) $\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }[\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\tau }+6\varphi \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }\xi }+\frac{{\mathrm{\partial }}^3\varphi }{\mathrm{\partial }{\xi }^3}]-\sigma \frac{{\mathrm{\partial }}^2\varphi }{\mathrm{\partial }{X}^2}=0.$(37) ) using HBM could be done by constructing the auxiliary function separately.

4.1. Special choice for the parameters

Set 1: $A_1=1,A_2=a,A_5=0,A_6=b,A_4=c,A_8=d$. Then (53) $\begin{array}{rl}g& =\frac{\xi }{B^{1/3}}+\frac{\mathrm{aX}}{B^{1/6}{C}^{1/2}}+(a^2-b^2)\tau +c,\end{array}$(53) (54) $\begin{array}{rl}h& =\frac{\mathrm{bX}}{B^{1/6}{C}^{1/2}}+2\mathrm{ab\tau }+d,\end{array}$(54) (55) $\begin{array}{rl}f& ={(\frac{\xi }{B^{1/3}}+\frac{\mathrm{aX}}{B^{1/6}{C}^{1/2}}+(a^2-b^2)\tau +c)}^2,\\ & +{(\frac{\mathrm{bX}}{B^{1/6}{C}^{1/2}}+2\mathrm{ab\tau }+d)}^2+\frac{3}{b^2}.\end{array}$(55) Hence the lump solutions are given as follows: (56) $\begin{array}{rl}\varphi (\xi ,X,\tau )& =4\left(\frac{6B^{1/3}}{A}\right)\\ & \times \frac{\begin{array}{c}-(\frac{\xi }{B^{1/3}}+\frac{\mathrm{aX}}{B^{1/6}{C}^{1/2}}+(a^2-b^2)\tau +c{)}^2\\ +(\frac{\mathrm{bX}}{B^{1/6}{C}^{1/2}}+2\mathrm{ab\tau }+d{)}^2+\frac{3}{b^2}\end{array}}{\begin{array}{c}((\frac{\xi }{B^{1/3}}+\frac{\mathrm{aX}}{B^{1/6}{C}^{1/2}}+(a^2-b^2)\tau +c{)}^2\\ +(\frac{\mathrm{bX}}{B^{1/6}{C}^{1/2}}+2\mathrm{ab\tau }+d{)}^2+\frac{3}{b^2}{)}^2\end{array}}\end{array}$(56) under c = d = 0, this solution reduces to the lump solution (57) $\begin{array}{rl}\varphi (\xi ,X,\tau )& =4\left(\frac{6B^{1/3}}{A}\right)\frac{\begin{array}{c}-(\frac{\xi }{B^{1/3}}+\frac{\mathrm{aX}}{B^{1/6}{C}^{1/2}}+(a^2-b^2)\tau {)}^2\\ +b^2(\frac{X}{B^{1/6}{C}^{1/2}}+2\mathrm{a\tau }{)}^2+\frac{3}{b^2}\end{array}}{\begin{array}{c}((\frac{\xi }{B^{1/3}}+\frac{\mathrm{aX}}{B^{1/6}{C}^{1/2}}+(a^2-b^2)\tau {)}^2\\ +b^2(\frac{X}{B^{1/6}{C}^{1/2}}+2\mathrm{a\tau }{)}^2+\frac{3}{b^2}{)}^2\end{array}}.\end{array}$(57) Set 2: $A_1=1,A_2=2,A_4=0,A_5=1,A_6=-1,A_8=0$, then (58) $\begin{array}{rl}g& =\frac{\xi }{B^{1/3}}+\frac{2X}{B^{1/6}{C}^{1/2}}-\frac{1}{2}\tau ,\end{array}$(58) (59) $\begin{array}{rl}h& =\frac{\xi }{B^{1/3}}-\frac{X}{B^{1/6}{C}^{1/2}}-\frac{7}{2}\tau ,\end{array}$(59) (60) $\begin{array}{rl}f& =\frac{25}{2}{\tau }^2-8\frac{\mathrm{\xi \tau }}{B^{1/3}}+5\frac{\mathrm{X\tau }}{B^{1/6}{C}^{1/2}}+2\frac{{\xi }^2}{B^{2/3}}+5\frac{X^2}{B^{1/3}C}\\ & +2\frac{\mathrm{\xi X}}{B^{1/2}{C}^{1/2}}+\frac{8}{3}.\end{array}$(60) Hence, the lump solution is given as follows: (61) $\begin{array}{rl}\varphi (\xi ,X,\tau )& =48\left(\frac{6B^{1/3}}{A}\right)\\ & \times \frac{\begin{array}{c}(-21{\tau }^2+48\frac{\mathrm{\xi \tau }}{B^{1/3}}+78\frac{\mathrm{X\tau }}{B^{1/6}{C}^{1/2}}\\ -12\frac{{\xi }^2}{B^{2/3}}+24\frac{X^2}{B^{1/3}C}-12\frac{\mathrm{\xi X}}{B^{1/2}{C}^{1/2}}+16)\end{array}}{\begin{array}{c}(75{\tau }^2-48\frac{\mathrm{\xi \tau }}{B^{1/3}}+30\frac{\mathrm{X\tau }}{B^{1/6}{C}^{1/2}}\\ +12\frac{{\xi }^2}{B^{2/3}}+30\frac{X^2}{B^{1/3}C}+12\frac{\mathrm{\xi X}}{B^{1/2}{C}^{1/2}}+16{)}^2\end{array}}.\end{array}$(61) Set 3: $A_1=1,A_2=-2,A_4=0,A_5=-2,A_6=1,A_8=0,$ then (62) $\begin{array}{rl}g& =\frac{\xi }{B^{1/3}}-\frac{2X}{B^{1/6}{C}^{1/2}}+\frac{11}{5}\tau ,\end{array}$(62) (63) $\begin{array}{rl}h& =\frac{-2\xi }{B^{1/3}}+\frac{X}{B^{1/6}{C}^{1/2}}+\frac{2}{5}\tau ,\end{array}$(63) (64) $\begin{array}{rl}f& =\frac{5{\xi }^2}{B^{2/3}}+\frac{5X^2}{B^{1/3}C}+5{\tau }^2-\frac{8\mathrm{\xi X}}{B^{1/2}{C}^{1/2}}-\frac{8\mathrm{X\tau }}{B^{1/6}{C}^{1/2}}\\ & +\frac{14}{5}\frac{\mathrm{\xi \tau }}{B^{1/3}}+\frac{125}{3}.\end{array}$(64) Hence the lump solution is given as follows: (65) $\begin{array}{rl}\varphi (\xi ,X,\tau )& =12\left(\frac{6B^{1/3}}{A}\right)\\ & \times \frac{\begin{array}{c}1581{\tau }^2-\frac{1875{\xi }^2}{B^{2/3}}-\frac{525X^2}{B^{1/3}C}+\frac{3000\mathrm{\xi X}}{B^{1/2}{C}^{1/2}}\\ -\frac{1320\mathrm{X\tau }}{B^{1/6}{C}^{1/2}}-\frac{1050\mathrm{\xi \tau }}{B^{1/3}}+15625\end{array}}{\begin{array}{c}(\frac{75{\xi }^2}{B^{2/3}}+\frac{75X^2}{B^{1/3}C}+75{\tau }^2\\ -\frac{120\mathrm{\xi X}}{B^{1/2}{C}^{1/2}}-\frac{120\mathrm{X\tau }}{B^{1/6}{C}^{1/2}}+\frac{42\mathrm{\xi \tau }}{B^{1/3}}+625{)}^2\end{array}}.\end{array}$(65) Three sets of lump soliton solutions have been obtained for three sets of particular values of coefficients which are coming as the solution of a system of algebraic equations. The HBM leads to the solitons solutions of KdV, and MKdV equations in a compact form, known as ${\mathrm{Sech}}^2$ , Sech profile for the appropriate choice of auxiliary functions. But during the derivation of lump soliton solutions, it is seen that the auxiliary function is chosen as the rational function of two quadratic polynomials of independent variables. The main key of the method is to find out the exact bilinear form, once it is obtained appropriately, then we get its iterative solution by using series expansion. If the exact solutions are obtained by terminating the iterative procedure to finite series at a particular definite succession, then the soliton exists, therefore we get soliton solutions that lead to lump solitons. Obviously, a quadratic polynomial of more than one independent variable gives surface as a solution to the nonlinear partial differential equations. Thus, we get the lump soliton solution that represents the surface. For a particular parameter set, we get lump soliton structures, i.e. surface structure that passes through a particular curve obtained by setting a specific coefficient set out of these parameter sets. Thus solutions in all directions fulfill the requirement of possession of lump soliton due to the KP equation.

5. Results and discussions

In this article, starting from dust hydrodynamical equations we have derived the KP equation by employing the extended RPT. The derived KP equations from basic governing equations have been put in the standard KP equation by using the transformation (36(36) $\xi =\overline{\xi }{B}^{\frac{1}{3}},{\varphi }_1=6\overline{{\varphi }_1}{A}^{-1}{B}^{\frac{1}{3}},X=\frac{1}{\sqrt{\sigma }}\overline{X}{B}^{\frac{1}{6}}{C}^{\frac{1}{2}},\tau =\overline{\tau }.$(36) ). With the help of the HBM, we have obtained lump soliton solutions. Three different sets of real parameters have been obtained using symbolic computation and consequently, we get three lump soliton solutions. Now we plot our obtained solutions to understand the features associated with the solutions for various parameters that occurred in the solutions expressions.

First, we consider the soliton solution profile associated with parameter set-1, and taking the suitable choice of real parameters as a = 1, b = 1, we have traced the lump solution for different double spectral index values q and r. At the time t = 0, set-1 lump soliton solutions are plotted to see the effect of non-extensive parameter q on the potential structure profiles keeping all parameters constant. In this regard, Figure 1(a–c) are plotted for r = 0, and q = 1.75, q = 1.85, q = 2.0, respectively, to show the effect of non-extensivity on the lump soliton structure profile, whereas Figure 2(a–c) are also drawn for the same respective value of non-extensive parameters to clarify the features through the phase diagram. The lump soliton structures that appear in those figures indicate the existence of compressive lump soliton structures for this chosen parameter set. It is also seen that with the increment of the non-extensive parameter range, the corresponding amplitude of the lump soliton structures are gradually decreasing. The phase diagram Figure 2(a–c) for r = 0, and q = 1.75, q = 1.85, $q=2.0,$ respectively, shows that the region where the solution is confined is closed as a consequence, the system is conservative for this parameter regime. To extract more information embedded with the associated lump soliton solutions, we have plotted figures to see the amplitude profile concerning two separate axes, i.e. in one dimension plot explicitly. Figure 3(a ,b) will serve our purpose in this regard. These two figures are traced for the same set of a parameter value of Figure 1 set along with for q = 1.55 in particular. Figure 3(a) is drawn to show the profile nature of solitons for different frames, i.e. for $\chi =0.4$ (green line), $\chi =1$ (red line), $\chi =1.4$ (blue line) regarding the coordinate axis ξ. On the other hand, Figure 3(b) is traced to show the profile nature of solitons for different frames, i.e. for $\xi =0.4$ (green line), $\xi =1$ (red line), $\xi =1.4$ (blue line) with reference to the coordinate axis χ.

Figure 1. Figures are plotted to show the structure of lump solitons, the parameters are a = 1, b = 1, $\beta =0.2$, ${\delta }_1=1.005$, r = 0, q = 1.75, q = 1.85, q = 2.0 at time t = 0.

Figure 2. Contour plot for lump solitons the parameters are a = 1, b = 1, $\beta =0.2$, ${\delta }_1=1.005$, r = 0, q = 1.75, q = 1.85, q = 2.0 at time t = 0.

Figure 3. (a) ϕ vs. ξ and (b) ϕ vs. χ the structure of lump solitons for a = 1, b = 1, $\beta =0.2$, ${\delta }_1=1.005$, r = 0, q = 1.85 at time t = 0.

Now we consider the soliton solution profile associated with parameter set-2, we have traced the lump solution for different double spectral index values q and r. At the time t = 0, set-2 lump soliton solutions are plotted to see the effect of non-extensive parameter q on the potential structure profiles keeping all parameters constant. In this regard, Figure 4(a–c) are plotted for r = 0.85, and q = 1.5, q = 1.75, q = 1.85, respectively, to show the effect of non-extensivity on the lump soliton structure profile, whereas Figure 5(a–c) are also drawn for the same respective value of non-extensive parameters to clarify the features through the phase diagram. The lump soliton structures that appear in those figures indicate the existence of compressive lump soliton structures for this chosen parameter set. It is also seen that with the increment of the non-extensive parameter range, the corresponding amplitude of the lump soliton structures are gradually increasing. On comparison with the previous set solution plot, we can see that the second parameter of the double spectral index, i.e. r has a significant impact on lump soliton structures. The phase diagram Figure 5(a–c) for r = 0.85, and q = 1.5, q = 1.75, $q=1.85,$ respectively, show that the region where the solution is confined is closed with saddle path as a consequence, the system is conservative for this parameter regime but not forever. The paths are closed circle near the origin and it changes their nature from circle to ellipse and finally reaches to saddle path. To extract more information embedded with the associated lump soliton solutions we have plotted figures to see the amplitude profile concerning two separate axes, i.e. in one dimension plot explicitly. Figure 6(a,b) will serve our purpose in this regard. These two figures are traced for the same set of a parameter values of Figure 1 set along with for q = 1.75, r = 0.85 in particular. Figure 6(a) is drawn to show the profile nature of solitons for different frames, i.e. for $\chi =0.4$ (green line), $\chi =1$ (red line), $\chi =1.4$ (blue line) with reference to the coordinate axis ξ. On the other hand, Figure 6(b) is traced to show the profile nature of solitons for different frames, i.e. for $\xi =0.4$ (green line), $\xi =1$ (red line), $\xi =1.4$ (blue line) with reference to the coordinate axis χ.

Figure 4. Figures are plotted to show the structure of lump solitons, the parameters are $\beta =0.2$, ${\delta }_1=1.005$, r = 0.85, q = 1.5, q = 1.75, q = 1.85 at time t = 0.

Figure 5. Contour plot for lump solitons the parameters are $\beta =0.2$, ${\delta }_1=1.005$, r = 0.85, q = 1.5, q = 1.75, q = 1.85 at time t = 0.

Figure 6. (a) ϕ vs. ξ and (b) ϕ vs. χ the structure of lump solitons for $\beta =0.2$, ${\delta }_1=1.005$, r = 0.85, q = 1.75 at time t = 0.

Here, we consider the soliton solution profile associated with parameter set-3, and we have traced the lump solution for different double spectral index values q and r. At the time t = 0, set-1 lump soliton solutions are plotted to see the effect of non-extensive parameter q on the potential structure profiles keeping all parameters constant. In this regard, Figure 7(a–c) are plotted for r = 0, and q = 1.75, q = 1.85, $q=2.0,$ respectively, to show the effect of non-extensivity on the lump soliton structure profile, whereas Figure 8(a–c) are also drawn for the same respective value of non-extensive parameters to clarify the features through the phase diagram. The lump soliton structures that appear in those figures indicate the existence of compressive lump soliton structures for this chosen parameter set. It is also seen that with the increment of the non-extensive parameter range, the corresponding amplitude of the lump soliton structures are gradually decreasing. The phase diagram Figure 8(a–c) for r = 0, and q = 1.75, q = 1.85, and $q=2.0,$ respectively, show that the region where the solution is confined is closed as a consequence, the system is conservative for this parameter regime. To extract more information embedded with the associated lump soliton solutions we have plotted figures to see the amplitude profile concerning two separate axes, i.e. in one dimension plot explicitly. Figure 9(a ,b) will serve our purpose in this regard. These two figures are traced for the same set of a parameter values of Figure 1 set along with for q = 1.55 in particular. Figure 9(a) is drawn to show the profile nature of solitons for different frames, i.e. for $\chi =0.4$ (green line), $\chi =1$ (red line), $\chi =1.4$ (blue line) with reference to the coordinate axis ξ. On the other hand, Figure 9(b) is traced to show the profile nature of solitons for different frames, i.e. for $\xi =0.4$ (green line), $\xi =1$ (red line), $\xi =1.4$ (blue line) with reference to the coordinate axis χ.

Figure 7. Figures are plotted to show the structure of lump solitons, the parameters are $\beta =0.2$, ${\delta }_1=1.005$, r = 0.55, q = 1.1, q = 1.25, q = 1.35 at time t = 0.

Figure 8. Contour plot for lump solitons the parameters are $\beta =0.2$, ${\delta }_1=1.005$, r = 0.55, q = 1.1, q = 1.25, q = 1.35 at time t = 0.

Figure 9. Figure 10(a) ϕ vs. ξ and Figure 10(b) ϕ vs. χ the structure of lump solitons for $\beta =0.2$, ${\delta }_1=1.005$, r = 0.55, q = 1.25 at time t = 0.

Figure 10. Figures are plotted to show the structure of lump solitons, the parameters are $\beta =0.2$, ${\delta }_1=1.005$, r = 0.15, q = 1.1, q = 1.15, q = 1.25 at time t = 0.

Finally, we consider the soliton solution profile associated with parameter set-3, and we have traced the lump solution for different double spectral index values q and r. At the time t = 0, set-3 lump soliton solutions are plotted to see the effect of non-extensive parameter q on the potential structure profiles keeping all parameters constant. In this regard, Figure 10(a–c) are plotted for r = 0.15, q = 1.1, q = 1.15, $q=1.25,$ respectively, to show the effect of non-extensivity on the lump soliton structure profile, whereas Figure 11(a–c) are also drawn for the same respective value of non-extensive parameters to clarify the features through the phase diagram. The lump soliton structures that appear in those figures indicate the existence of rarefactive lump soliton structures for this chosen parameter set. It is also seen that with the increment of the non-extensive parameter range, the corresponding amplitude of the lump soliton structures are gradually decreasing. The phase diagram Figure 11(a–c) for r = 0.15, q = 1.1, q = 1.15, $q=1.25,$ respectively, show that the region where the solution is confined is closed as a consequence, the system is conservative for this parameter regime.

Figure 11. Contour plot for lump solitons the parameters are $\beta =0.2$, ${\delta }_1=1.005$, r = 0.15, q = 1.1, q = 1.15, q = 1.25 at time t = 0.

We have plotted a figure to compare our investigation with the literature of lump soliton structures studied by Tang et al. [26]. In this plot, we have drawn the surface and contour plot of the solution equation (11) of case 1 for the same parameter taken as of [26] to justify the validity of our numerical programme and we have got the same traced as that of the investigation carried out by Tang et al. [26]. Figure 12 represents the solutions corresponding to the lump soliton surface and contour structures of equation (11) of case 1. It is depicted that lump soliton structures of equation (11), both surface and contour structures are the same as that of equation (11) of Tang et al. [26].

Figure 12. Surface and contour plot for lump solitons of equation (Yang et al. [26]) the parameters are $\beta =1/2$, $\alpha =1$, $a_3=4$, $a_4=0$, $a_5=2$, $a_8=0$, $a_9=1$ at time t = 0.

6. Conclusions

It can be concluded that the lump soliton structures depend on the physical parameters, i.e. the amplitude is varying with system parameters. Double spectral index q and r play a significant role in the lump solitons structures and they are the decision-making factor for generating compressive or rarefactive lump soliton structures. The investigation has been carried out regarding various wave structures and wave interactions mainly, the existence of special unpredictable lump solitons in the research fields of oceanography, nonlinear optical fibres, plasmas, ferrite magnetic material, fluid dynamics, and atmosphere, microwave oscillation, and financial system. The above-mentioned investigations help physicists to understand the relations with nature, which are dealing with the most interesting wave appearance that is coming from nowhere and passing out of the site without any trace. Disastrous, which is a consequence of destructions of nonlinear systems in nonlinear optical fibres, plasmas, fluid dynamics and atmosphere, and financial systems, the harmful effect of the generation of lump solitons. Its appearance and disappearance in these fields of study are too significant to predict so by amplifying signals physicists can control the intensity of the disasters and their effects. This mechanism has been used in ferrite magnetic materials and optical fibres research fields. Thus our relevant investigation may enrich the plasma research field and may give a direction to open a new interesting research area.

Disclosure statement

No potential conflict of interest was reported by the author(s).