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Symmetry Analysis and Exact Solutions of Extended Kadomtsev-Petviashvili Equation in Fluids

Pubblicato online: 25 Nov 2022
Volume & Edizione: AHEAD OF PRINT
Pagine: -
Ricevuto: 23 Jun 2022
Accettato: 18 Oct 2022
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License
Formato
Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
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Inglese
Introduction

Computing solutions of nonlinear partial differential equations (NLPDEs) is of importance to solve and understand the physical meaning of a given problem. To obtain the exact solutions of NLPDEs, researchers have come up with several methods, such as Kudryshov method [1], double reduction method [2], simplest equation method [35], homotopy analysis transform method [6], direct method [7], etc.

The Lie’s theory [8, 9], introduced by Sophus Lie (1842–1899), is used to construct a symmetry reduction, and it can also play a major role in finding the exact solutions of partial differential equations (PDEs). The application of the Lie’s method can also be extended to fields such as relativity, classical mechanics, numerical analysis and so on [1012].

Conservation laws [1318] play a significant role in finding the solutions of PDEs. Noether’s theorem helps us with an innovation formula for deriving conservation laws by appealing to the symmetries of the action.

The Kadomtsev-Petviashvili (KP) equation emerged pursuant to the efforts made when Kadomtsev and Petviashvili were learning the growth of lengthy ion-acoustic surfs with minor amplitude broadcasting in plasmas falling short of the result of lengthy diagonal worries [19]. The KP equation defines the development of non-linear, lengthy surfs of minor amplitude with sluggish requirement on the transversal coordinate. It is the simplification of the Kortewegde Vries (KdV) equation.

In this paper, we investigate the extended Kadomtsev-Petviashvili (eKP) equation [20], which reads as: (ut+6uux+uxxx)xuyyα24utt+αuty+βuxt=0, $$\matrix{ {{{({u_t} + 6u{u_x} + {u_{xxx}})}_x} - {u_{yy}} - {{{\alpha ^2}} \over 4}{u_{tt}} + \alpha {u_{ty}} + \beta {u_{xt}} = 0,} \cr } $$

where α and β are system parameters. Eq. (1) describes the wave propaganda in the more general nonlinear fluids.

The Lie symmetry technique is used to achieve symmetry reduction of the eKP equation and solve the resulting ordinary differential equation we obtain by applying straightforward integration in conjunction with simplest equation technique to derive results for the eKP equation. Furthermore, we derive the conservation laws of the eKP equation using multiplier methods, and then give concluding remarks.

Exact solutions of Eq. (1) by using symmetry reductions

The direction ground can be expressed as follows: P=S1t+S2x+S3y+Gu $$\matrix{ {P = {S^1}{\partial \over {\partial t}} + {S^2}{\partial \over {\partial x}} + {S^3}{\partial \over {\partial y}} + G{\partial \over {\partial u}}} \cr } $$

where S1, S2, S3 and G depend on t, x and y, and u is a Lie point symmetry of Eq. (1) if the following expression stands fulfilled: P[4]{(ut+6uux+uxxx)xuyyα24utt+αuty+βuxt}|(1)=0, $$\matrix{ {{P^{[4]}}{{\left. {\left\{ {{{({u_t} + 6u{u_x} + {u_{xxx}})}_x} - {u_{yy}} - {{{\alpha ^2}} \over 4}{u_{tt}} + \alpha {u_{ty}} + \beta {u_{xt}}} \right\}} \right|}_{(1)}} = 0,} \cr } $$

where P[4] is defined as the fourth prolongation of Eq. (2). Writing Eq. (3) in detail and splitting the resulting determining equations on derivatives of u leads to linear PDEs, and with the aid of Maple, these can be used to solve the infinitesimal, which gives four symmetries, as follows: P1=F1(t,y)t,P2=F2(t,y)y,P3=[ F3(t,y)t+βF3(t,y)t ]u+6F3(t,y)x,P4=[ 12uF4(t,y)βxF4(t,y)txF4(t,y)t ]u6xF4(t,y)x, $$\eqalign{ & {P_1} = {F_1}(t,y){\partial \over {\partial t}}, \cr & {P_2} = {F_2}(t,y){\partial \over {\partial y}}, \cr & {P_3} = \left[ {{{\partial {F_3}(t,y)} \over {\partial t}} + \beta {{\partial {F_3}(t,y)} \over {\partial t}}} \right]{\partial \over {\partial u}} + 6{{\partial {F_3}(t,y)} \over {\partial x}}, \cr & {P_4} = \left[ {12u{F_4}(t,y) - \beta x{{\partial {F_4}(t,y)} \over {\partial t}} - x{{\partial {F_4}(t,y)} \over {\partial t}}} \right]{\partial \over {\partial u}} - 6x{F_4}(t,y){\partial \over {\partial x}}, \cr} $$

where F1 to F4 are arbitrary functions of t and y. Considering the symmetry (P = P1 +aP2), where a is a constant, with F1 = 1 and F2 = 1, we perform symmetry reduction. By solving the individual equations of PDE, we contract the following invariants: g=x,h=aty,u=Θ(g,h), $$\matrix{ {g = x,\quad h = at - y,\quad u = {\rm{\Theta }}(g,h),} \cr } $$

which, when used, reduce the eKP equation to the following form: (a+aβ)2hgΘ+6(gΘ)2+6Θ2g2Θ+4g4Θ2h2Θa2α242h2Θaα2h2θ=0. $$\matrix{ {(a + a\beta ){{{\partial ^2}} \over {\partial h\partial g}}\Theta + 6{{\left( {{\partial \over {\partial g}}\Theta } \right)}^2} + 6\Theta {{{\partial ^2}} \over {\partial {g^2}}}\Theta + {{{\partial ^4}} \over {\partial {g^4}}}\Theta - {{{\partial ^2}} \over {\partial {h^2}}}\Theta - {{{a^2}{\alpha ^2}} \over 4}{{{\partial ^2}} \over {\partial {h^2}}}\Theta - a\alpha {{{\partial ^2}} \over {\partial {h^2}}}\theta = 0.} \cr } $$

We further perform symmetry reductions by utilising the Lie symmetries of Eq. (4). Eq. (4) has three Lie symmetries, namely: ϒ1=g,ϒ2=h,ϒ3=(6a2α2Θ+24aαΘ+a2+2aαβ+aβ2+24Θ)Θ+(6a2α2h24aαh24h)g+(3a2α2h12aαg+6ah+6aβh12g)h. $$\eqalign{ & {\Upsilon _1} = {\partial \over {\partial g}}, \cr & {\Upsilon _2} = {\partial \over {\partial h}}, \cr & {\Upsilon _3} = (6{a^2}{\alpha ^2}{\rm{\Theta }} + 24a\alpha {\rm{\Theta }} + {a^2} + 2a\alpha \beta + a{\beta ^2} + 24{\rm{\Theta }}){\partial \over {\partial {\rm{\Theta }}}} + ( - 6{a^2}{\alpha ^2}h - 24a\alpha h - 24h){\partial \over {\partial g}} \cr & + ( - 3{a^2}{\alpha ^2}h - 12a\alpha g + 6ah + 6a\beta h - 12g){\partial \over {\partial h}}. \cr} $$

By considering the symmetry (Υ = Υ1 +µΥ), where µ is a constant, two invariants are resultantly obtained, i.e. ξ=hμg,Θ=Φ(ξ) $$\matrix{ {\xi = h - \mu g,\quad {\rm{\Theta }} = {\rm{\Phi }}(\xi )} \cr } $$

which reduces the eKP equation to the following ODE: aμΦ +6μ2(Φ)2+6μ2ΦΦ+μ4ΦΦa2α24ΦaαΦaβμΦ=0 $$\matrix{ { - a\mu {\rm{\Phi ''}} + 6{\mu ^2}{{({\rm{\Phi '}})}^2} + 6{\mu ^2}{\rm{\Phi \Phi ''}} + {\mu ^4}{{\rm{\Phi }}^{\prime \prime \prime \prime }} - {\rm{\Phi ''}} - {{{a^2}{\alpha ^2}} \over 4}{\rm{\Phi ''}} - a\alpha {\rm{\Phi ''}} - a\beta \mu {\rm{\Phi ''}} = 0.} \cr } $$

Eq. (5) can be written as AΦ+6μ2(Φ2+ΦΦ)+μ4Φ=0 $$\matrix{ {A{\rm{\Phi ''}} + 6{\mu ^2}({{\rm{\Phi }}^{\prime 2}} + {\rm{\Phi \Phi ''}}) + {\mu ^4}{{\rm{\Phi }}^{\prime \prime \prime \prime }} = 0} \cr } $$

where A=aμ1aαaβμ14a2α2 $A = - a\mu - 1 - a\alpha - a\beta \mu - {1 \over 4}{a^2}{\alpha ^2}$ .

Solutions of Eq. (1) by straightforward integration

Integrating Eq. (6) yields the following expression: AΦ+6μΦΦ+μ4Φ+C1=0 $$\matrix{ {A{\rm{\Phi '}} + 6\mu {\rm{\Phi \Phi '}} + {\mu ^4}{\rm{\Phi '''}} + {C_1} = 0} \cr } $$

where C1 is a constant of integration.

Furthermore, the integral of Eq. (7) is expressed as: AΦ+3μ2Φ2+μ4Φ+C1ξ+C2 $$\matrix{ {A{\rm{\Phi }} + 3{\mu ^2}{{\rm{\Phi }}^2} + {\mu ^4}{\rm{\Phi ''}} + {C_1}{\rm{\xi }} + {C_2}} \cr } $$

where C2 is an integration constant.

Taking C1 = 0 and multiplying Eq. (8) by Φ1 yields the following result: AΦΦ+3μ2Φ2Φ+μ4ΦΦ+C2Φ=0. $$\matrix{ {A{\rm{\Phi \Phi '}} + 3{\mu ^2}{{\rm{\Phi }}^2}{\rm{\Phi '}} + {\mu ^4}{\rm{\Phi '\Phi ''}} + {C_2}{\rm{\Phi '}} = 0.} \cr } $$

The integral of Eq. (9) yields: 12AΦ2+μ2Φ3+12μ4Φ2+C2Φ+C3=0 $$\matrix{ {{1 \over 2}A{{\rm{\Phi }}^2} + {\mu ^2}{{\rm{\Phi }}^3} + {1 \over 2}{\mu ^4}{{\rm{\Phi }}^{\prime 2}} + {C_2}{\rm{\Phi }} + {C_3} = 0} \cr } $$

which in its simplest form can be expressed as: Φ2+1μΦ3+Aμ2Φ2+2C2μ4Φ+2C3μ4=0. $$\matrix{ {{{\rm{\Phi }}^{\prime 2}} + {1 \over \mu }{{\rm{\Phi }}^3} + {A \over {{\mu ^2}}}{{\rm{\Phi }}^2} + {{2{C_2}} \over {{\mu ^4}}}{\rm{\Phi }} + {{2{C_3}} \over {{\mu ^4}}} = 0.} \cr } $$

Assuming that the cubic equation 1μΦ3+Aμ2Φ2+2C2μ4Φ+2C3μ4=0 $$\matrix{ {{1 \over \mu }{{\rm{\Phi }}^3} + {A \over {{\mu ^2}}}{{\rm{\Phi }}^2} + {{2{C_2}} \over {{\mu ^4}}}{\rm{\Phi }} + {{2{C_3}} \over {{\mu ^4}}} = 0} \cr } $$

has real roots a1, a2, a3 such that a1 > a2 > a3, the solution for Eq. (10) would then be: Φ(ξ)=a2+(a1a2)cn2{(a1a2)μ2ξ,M2},M2=a1a2a1a3 $$\matrix{ {{\rm{\Phi }}(\xi ) = {a_2} + ({a_1} - {a_2}){\rm{c}}{{\rm{n}}^2}\left\{ {\sqrt {{{({a_1} - {a_2})} \over {{\mu ^2}}}\xi } ,{M^2}} \right\},{M^2} = {{{a_1} - {a_2}} \over {{a_1} - {a_3}}}} \cr } $$

where cn is the Jacobi cosine function. Subsequently, the solution for Eq. (1) is obtained as: u(t,x,y)=a2+(a1a2)cn2{(a1a2)μ2(atyμx),M2}, $$\matrix{ {u(t,x,y) = {a_2} + ({a_1} - {a_2}){\rm{c}}{{\rm{n}}^2}\left\{ {\sqrt {{{({a_1} - {a_2})} \over {{\mu ^2}}}(at - y - \mu x)} ,{M^2}} \right\},} \cr } $$

where M2=a1a2a1a3 $${{M^2} = {{{a_1} - {a_2}} \over {{a_1} - {a_3}}}}$$

Exact solutions using the simplest equation method

We now solve Eq. (5) by employing the simplest equation method [21]. By doing so, we are facilitated to ascertain the exact solutions for Eq. (1). We will use the Bernoulli and Riccati equations as our simplest equations.

Recalling the simplest equation method briefly, let Φ(q)=r=0NAr(K(q))r, $$\Phi (q) = \mathop \sum \limits_{r = 0}^N {A_r}{(K(q))^r},$$

be the solutions of Eq. (1). In this case, K(q) fulfils the Bernoulli or Riccati equation. The positive integer N can be calculated by using the matching procedure. A0, ···, AN are the parameters needing to be calculated.

By considering the Riccati equation K(q)=eK2(q)+dK(q)+p, $$\matrix{ {K'(q) = e{K^2}(q) + dK(q) + p,} \cr } $$

where e, d and p are just constants, this gives the solutions K(q)=d2eΘ2etanh[ Θ2(q+B) ] $$\matrix{ {K(q) = - {d \over {2e}} - {{\rm{\Theta }} \over {2e}}\tanh \left[ {{{\rm{\Theta }} \over 2}(q + B)} \right]} \cr } $$

and K(q)=d2eΘ2etanh(Θq2)+sech(Θq2)cosh(Θq2)2eΘsinh(Θq2) $$\matrix{ {K(q) = - {d \over {2e}} - {{\rm{\Theta }} \over {2e}}\tanh \left( {{{{\rm{\Theta }}q} \over 2}} \right) + {{sech\left( {{{{\rm{\Theta }}q} \over 2}} \right)} \over {\cosh \left( {{{{\rm{\Theta }}q} \over 2}} \right) - {{2e} \over {\rm{\Theta }}}\sinh \left( {{{{\rm{\Theta }}q} \over 2}} \right)}}} \cr } $$

where Θ2 = d2 –4ep and B is an integration constant.

The Bernoulli equation is expressed as: K(q)=eK(q)2+dK(q), $$\matrix{ {K'(q) = eK{{(q)}^2} + dK(q),} \cr } $$

where e and d are constants. This gives the solution of the equation as: K(q)=e{cosh[e(q+B)]+sinh[e(q+B)]1dcosh[e(q+B)]dsinh[e(q+B)]}. $$\matrix{ {K(q) = e\left\{ {{{\cosh [e(q + B)] + \sinh [e(q + B)]} \over {1 - d\cosh [e(q + B)] - d\sinh [e(q + B)]}}} \right\}.} \cr } $$

Solution by Bernoulli equation

The matching technique produces N = 2, and this yields the solutions of Eq. (5) as: Φ(q)=A0+A1K+A2K2. $$\matrix{ {{\rm{\Phi }}(q) = {A_0} + {A_1}K + {A_2}{K^2}.} \cr } $$

We now proceed to replace Φ(q) into Eq. (5), concomitant with the usage of Eq. (15), and then compare the factors of Kr to zero; then, with the aid of Maple, we obtain systems of equations in relation to A0, A1 and A2: μ4A1d4A1d214α2a2A1d2αaA1d2μaA1d2+6μ2A0A1d2βμaA1d2=0,60μ4A1e3d+36μ2A0A2e26μaA2e2+330μ4A2e2d232α2a2A2e26αaA2e2+18μ2A12e2+48μ2A22d26A2e26βμaA2e2+126μ2A1eA2d=0,16μ4A2d44μaA2d2+24μ2A0A2d2+15μ4A1ed3α2a2A2d24αaA2d23A1ed+12μ2A12d23βμaA1ed4A2d234α2a2A1ed3αaA1ed4βμaA2d23μaA1ed+18μ2A0A1ed=0,2μaA1e2+30μ2A12ed+54μ2A1d2A2+12μ2A0A1e2+50μ4A1e2d2+130μ4A2ed312α2a2A1e22αaA1e210A2ed10βμaA2ed2A1e252α2a2A2ed10αaA2ed2βμaA1e210μaA2ed+60μ2A0A2ed=0,120e4μ4A2+60e2μ2A22=0,336μ4A2e3d+24μ4A1e4+108μ2A22ed+72μ2A1e2A2=0. $$\eqalign{ & {\mu ^4}{A_1}{d^4} - {A_1}{d^2} - {1 \over 4}{\mkern 1mu} {\alpha ^2}{a^2}{A_1}{d^2} - \alpha {\mkern 1mu} a{A_1}{d^2} - \mu {\mkern 1mu} a{A_1}{d^2} + 6{\mkern 1mu} {\mu ^2}{A_0}{A_1}{d^2} - \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_1}{d^2} = 0, \cr & 60{\mkern 1mu} {\mu ^4}{A_1}{e^3}d + 36{\mkern 1mu} {\mu ^2}{A_0}{A_2}{e^2} - 6{\mkern 1mu} \mu {\mkern 1mu} a{A_2}{e^2} + 330{\mkern 1mu} {\mu ^4}{A_2}{e^2}{d^2} - {3 \over 2}{\mkern 1mu} {\alpha ^2}{a^2}{A_2}{e^2} - 6{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}{e^2} + 18{\mkern 1mu} {\mu ^2}{A_1}^2{e^2} + 48{\mkern 1mu} {\mu ^2}{A_2}^2{d^2} \cr & - 6{\mkern 1mu} {A_2}{e^2} - 6{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}{e^2} + 126{\mkern 1mu} {\mu ^2}{A_1}e{A_2}d = 0, \cr & 16{\mkern 1mu} {\mu ^4}{A_2}{d^4} - 4{\mkern 1mu} \mu {\mkern 1mu} a{A_2}{d^2} + 24{\mkern 1mu} {\mu ^2}{A_0}{A_2}{d^2} + 15{\mkern 1mu} {\mu ^4}{A_1}e{d^3} - {\alpha ^2}{a^2}{A_2}{d^2} - 4{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}{d^2} - 3{\mkern 1mu} {A_1}ed + 12{\mkern 1mu} {\mu ^2}{A_1}^2{d^2} \cr & - 3{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_1}ed - 4{\mkern 1mu} {A_2}{d^2} - {3 \over 4}{\mkern 1mu} {\alpha ^2}{a^2}{A_1}ed - 3{\mkern 1mu} \alpha {\mkern 1mu} a{A_1}ed - 4{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}{d^2} - 3{\mkern 1mu} \mu {\mkern 1mu} a{A_1}ed + 18{\mkern 1mu} {\mu ^2}{A_0}{A_1}ed = 0, \cr & - 2{\mkern 1mu} \mu {\mkern 1mu} a{A_1}{e^2} + 30{\mkern 1mu} {\mu ^2}{A_1}^2ed + 54{\mkern 1mu} {\mu ^2}{A_1}{d^2}{A_2} + 12{\mkern 1mu} {\mu ^2}{A_0}{A_1}{e^2} + 50{\mkern 1mu} {\mu ^4}{A_1}{e^2}{d^2} + 130{\mkern 1mu} {\mu ^4}{A_2}e{d^3} - {1 \over 2}{\mkern 1mu} {\alpha ^2}{a^2}{A_1}{e^2} \cr & - 2{\mkern 1mu} \alpha {\mkern 1mu} a{A_1}{e^2} - 10{\mkern 1mu} {A_2}ed - 10{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}ed - 2{\mkern 1mu} {A_1}{e^2} - {5 \over 2}{\mkern 1mu} {\alpha ^2}{a^2}{A_2}ed - 10{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}ed - 2{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_1}{e^2} - 10{\mkern 1mu} \mu {\mkern 1mu} a{A_2}ed \cr & + 60{\mkern 1mu} {\mu ^2}{A_0}{A_2}ed = 0, \cr & 120{\mkern 1mu} {e^4}{\mu ^4}{A_2} + 60{\mkern 1mu} {e^2}{\mu ^2}{A_2}^2 = 0, \cr & 336{\mkern 1mu} {\mu ^4}{A_2}{e^3}d + 24{\mkern 1mu} {\mu ^4}{A_1}{e^4} + 108{\mkern 1mu} {\mu ^2}{A_2}^2ed + 72{\mkern 1mu} {\mu ^2}{A_1}{e^2}{A_2} = 0. \cr} $$

Solving the system of equations, we get A0=4μ4d2+α2a2+4βμa+4αa+4μa+424μ2,A1=2deμ2,A2=2e2μ2. $$\eqalign{ & {A_0} = {{ - 4{\mu ^4}{d^2} + {\alpha ^2}{a^2} + 4\beta \mu a + 4\alpha a + 4\mu a + 4} \over {24{\mu ^2}}}, \cr & {A_1} = - 2de{\mu ^2}, \cr & {A_2} = - 2{e^2}{\mu ^2}. \cr} $$

Fig. 1

Solution of Eq. (21)

As a result, a solution for Eq. (1) is obtained, as the following: u(x,t,y)=A0+A1e{ cosh[e(q+B)]+sinhe[(q+B)]1dcosh[e(q+B)]dsinh[e(q+B)] }+A2{ ecosh[e(q+B)]+sinh[e(q+B)]1dcosh[e(q+B)]dsinh[e(q+B)] }2, $$\displaylines{ u(x,t,y) = {A_0} + {A_1}e\left\{ {{{\cosh [e(q + B)] + \sinh e[(q + B)]} \over {1 - d\cosh [e(q + B)] - d\sinh [e(q + B)]}}} \right\} \cr + {A_2}{\left\{ {e{{\cosh [e(q + B)] + \sinh [e(q + B)]} \over {1 - d\cosh [e(q + B)] - d\sinh [e(q + B)]}}} \right\}^2}, \cr} $$

where q = atµxy and B represent the integration constant.

Solution by Ricatti equation

We obtain N = 2 as our matching technique, and so the solutions of Eq. (5) assume the form: Φ(q)=A0+A1K+A2K2. $$\matrix{ {{\rm{\Phi }}(q) = {A_0} + {A_1}K + {A_2}{K^2}.} \cr } $$

We replace Φ(q) into Eq. (5), concomitant with the usage of Eq. (15), and then compare the factors of Kr to zero; then, with the aid of Maple, we obtain systems of equations in relation to A0, A1 and A2:

18μ2A12e2+48μ2A22d26A2e2+36μ2A0A2e2+60μ4A1e3d+240μ4A2e3p+330μ4A2e2d232α2a2A2e26αaA2e26μaA2e2+96μ2A22ep+126μ2A1eA2d6βμaA2e22μaA2p2+12μ2A0A2p2+μ4A1d3p+14μ4A2d2p2+16μ4A2ep312α2a2A2p22αaA2p2+6μ2A12p2A1dp2A2p2+6μ2A0A1dp+8μ4A1edp214α2a2A1dpαaA1dp2βμaA2p2μaA1dpβμaA1dp=0,10A2ed2A1e2+40μ4A1e3p+50μ4A1e2d2+130μ4A2ed312α2a2A1e22αaA1e2+54μ2A1d2A2+84μ2A22dp+12μ2A0A1e22μaA1e2+30μ2A12ed52α2a2A2ed10αaA2ed2βμaA1e210μaA2ed+108μ2A1eA2p+60μ2A0A2ed+440μ4A2e2dp10βμaA2ed=0,14α2a2A1d2αaA1d2+18μ2A12dp+36μ2A1p2A2+6μ2A0A1d2μaA1d2+120μ4A2ep2d6βμaA2dp2βμaA1ep+16μ4A1e2p2+30μ4A2d3p+22μ4A1ed2p+12μ2A0A1ep+36μ2A0A2dp6μaA2dpβμaA1d22μaA1ep2αaA1ep6αaA2dp1/2α2a2A1ep32α2a2A2dp2A1ep+μ4A1d46A2dpA1d2=0,60μ4A1e2dp+48μ2A0A2ep+90μ2A1dA2p+18μ2A0A1ed8μaA2ep4βμaA2d23μaA1ed8αaA2ep2α2a2A2ep3αaA1ed+232μ4A2ed2p34α2a2A1ed4A2d2+36μ2A22p2+12μ2A12d28A2ep3A1ed+16μ4A2d43βμaA1ed8βμaA2ep+24μ2A12ep+24μ2A0A2d24μaA2d2+15μ4A1ed3+136μ4A2e2p2α2a2A2d24αaA2d2,120e4μ4A2+60e2μ2A22=0,336de3μ4A2+24e4μ4A1+108,deμ2A22+72e2μ2A1A2=0. $$\eqalign{ & 18{\mkern 1mu} {\mu ^2}{A_1}^2{e^2} + 48{\mkern 1mu} {\mu ^2}{A_2}^2{d^2} - 6{\mkern 1mu} {A_2}{e^2} + 36{\mkern 1mu} {\mu ^2}{A_0}{A_2}{e^2} + 60{\mkern 1mu} {\mu ^4}{A_1}{e^3}d + 240{\mkern 1mu} {\mu ^4}{A_2}{e^3}p + 330{\mkern 1mu} {\mu ^4}{A_2}{e^2}{d^2} \cr & - {3 \over 2}{\mkern 1mu} {\alpha ^2}{a^2}{A_2}{e^2} - 6{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}{e^2} - 6{\mkern 1mu} \mu {\mkern 1mu} a{A_2}{e^2} + 96{\mkern 1mu} {\mu ^2}{A_2}^2ep + 126{\mkern 1mu} {\mu ^2}{A_1}e{A_2}d - 6{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}{e^2} \cr & - 2{\mkern 1mu} \mu {\mkern 1mu} a{A_2}{p^2} + 12{\mkern 1mu} {\mu ^2}{A_0}{A_2}{p^2} + {\mu ^4}{A_1}{d^3}p + 14{\mkern 1mu} {\mu ^4}{A_2}{d^2}{p^2} + 16{\mkern 1mu} {\mu ^4}{A_2}e{p^3} - {1 \over 2}{\mkern 1mu} {\alpha ^2}{a^2}{A_2}{p^2} - 2{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}{p^2} \cr & + 6{\mkern 1mu} {\mu ^2}{A_1}^2{p^2} - {A_1}dp - 2{\mkern 1mu} {A_2}{p^2} + 6{\mkern 1mu} {\mu ^2}{A_0}{A_1}dp + 8{\mkern 1mu} {\mu ^4}{A_1}ed{p^2} - {1 \over 4}{\mkern 1mu} {\alpha ^2}{a^2}{A_1}dp - \alpha {\mkern 1mu} a{A_1}dp \cr & - 2{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}{p^2} - \mu {\mkern 1mu} a{A_1}dp - \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_1}dp = 0, \cr & - 10{\mkern 1mu} {A_2}ed - 2{\mkern 1mu} {A_1}{e^2} + 40{\mkern 1mu} {\mu ^4}{A_1}{e^3}p + 50{\mkern 1mu} {\mu ^4}{A_1}{e^2}{d^2} + 130{\mkern 1mu} {\mu ^4}{A_2}e{d^3} - {1 \over 2}{\mkern 1mu} {\alpha ^2}{a^2}{A_1}{e^2} - 2{\mkern 1mu} \alpha {\mkern 1mu} a{A_1}{e^2} \cr & + 54{\mkern 1mu} {\mu ^2}{A_1}{d^2}{A_2} + 84{\mkern 1mu} {\mu ^2}{A_2}^2dp + 12{\mkern 1mu} {\mu ^2}{A_0}{A_1}{e^2} - 2{\mkern 1mu} \mu {\mkern 1mu} a{A_1}{e^2} + 30{\mkern 1mu} {\mu ^2}{A_1}^2ed - {5 \over 2}{\mkern 1mu} {\alpha ^2}{a^2}{A_2}ed \cr & - 10{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}ed - 2{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_1}{e^2} - 10{\mkern 1mu} \mu {\mkern 1mu} a{A_2}ed + 108{\mkern 1mu} {\mu ^2}{A_1}e{A_2}p + 60{\mkern 1mu} {\mu ^2}{A_0}{A_2}ed + 440{\mkern 1mu} {\mu ^4}{A_2}{e^2}dp \cr & - 10{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}ed = 0, \cr & - {1 \over 4}{\mkern 1mu} {\alpha ^2}{a^2}{A_1}{d^2} - \alpha {\mkern 1mu} a{A_1}{d^2} + 18{\mkern 1mu} {\mu ^2}{A_1}^2dp + 36{\mkern 1mu} {\mu ^2}{A_1}{p^2}{A_2} + 6{\mkern 1mu} {\mu ^2}{A_0}{A_1}{d^2} - \mu {\mkern 1mu} a{A_1}{d^2} + 120{\mkern 1mu} {\mu ^4}{A_2}e{p^2}d \cr & - 6{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}dp - 2{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_1}ep + 16{\mkern 1mu} {\mu ^4}{A_1}{e^2}{p^2} + 30{\mkern 1mu} {\mu ^4}{A_2}{d^3}p + 22{\mkern 1mu} {\mu ^4}{A_1}e{d^2}p + 12{\mkern 1mu} {\mu ^2}{A_0}{A_1}ep \cr & + 36{\mkern 1mu} {\mu ^2}{A_0}{A_2}dp - 6{\mkern 1mu} \mu {\mkern 1mu} a{A_2}dp - \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_1}{d^2} - 2{\mkern 1mu} \mu {\mkern 1mu} a{A_1}ep - 2{\mkern 1mu} \alpha {\mkern 1mu} a{A_1}ep - 6{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}dp - 1/2{\mkern 1mu} {\alpha ^2}{a^2}{A_1}ep \cr & - {3 \over 2}{\mkern 1mu} {\alpha ^2}{a^2}{A_2}dp - 2{\mkern 1mu} {A_1}ep + {\mu ^4}{A_1}{d^4} - 6{\mkern 1mu} {A_2}dp - {A_1}{d^2} = 0, \cr & 60{\mkern 1mu} {\mu ^4}{A_1}{e^2}dp + 48{\mkern 1mu} {\mu ^2}{A_0}{A_2}ep + 90{\mkern 1mu} {\mu ^2}{A_1}d{A_2}p + 18{\mkern 1mu} {\mu ^2}{A_0}{A_1}ed - 8{\mkern 1mu} \mu {\mkern 1mu} a{A_2}ep - 4{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}{d^2} - 3{\mkern 1mu} \mu {\mkern 1mu} a{A_1}ed \cr & - 8{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}ep - 2{\mkern 1mu} {\alpha ^2}{a^2}{A_2}ep - 3{\mkern 1mu} \alpha {\mkern 1mu} a{A_1}ed + 232{\mkern 1mu} {\mu ^4}{A_2}e{d^2}p - {3 \over 4}{\mkern 1mu} {\alpha ^2}{a^2}{A_1}ed - 4{\mkern 1mu} {A_2}{d^2} + 36{\mkern 1mu} {\mu ^2}{A_2}^2{p^2} \cr & + 12{\mkern 1mu} {\mu ^2}{A_1}^2{d^2} - 8{\mkern 1mu} {A_2}ep - 3{\mkern 1mu} {A_1}ed + 16{\mkern 1mu} {\mu ^4}{A_2}{d^4} - 3{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_1}ed - 8{\mkern 1mu} \beta {\mkern 1mu} \mu {\mkern 1mu} a{A_2}ep + 24{\mkern 1mu} {\mu ^2}{A_1}^2ep \cr & + 24{\mkern 1mu} {\mu ^2}{A_0}{A_2}{d^2} - 4{\mkern 1mu} \mu {\mkern 1mu} a{A_2}{d^2} + 15{\mkern 1mu} {\mu ^4}{A_1}e{d^3} + 136{\mkern 1mu} {\mu ^4}{A_2}{e^2}{p^2} - {\alpha ^2}{a^2}{A_2}{d^2} - 4{\mkern 1mu} \alpha {\mkern 1mu} a{A_2}{d^2}, \cr & 120{\mkern 1mu} {e^4}{\mu ^4}{A_2} + 60{\mkern 1mu} {e^2}{\mu ^2}{A_2}^2 = 0, \cr & 336{\mkern 1mu} d{e^3}{\mu ^4}{A_2} + 24{\mkern 1mu} {e^4}{\mu ^4}{A_1} + 108,de{\mu ^2}{A_2}^2 + 72{\mkern 1mu} {e^2}{\mu ^2}{A_1}{A_2} = 0. \cr} $$

By solving the resultant system of equations, we acquire: A0=4μ4d232eμ2p+α2a2+4βμa+4αa+4μa+424μ2,A1=2deμ2,A2=2e2μ2. $$\eqalign{ & {A_0} = {{ - 4{\mu ^4}{d^2} - 32e{\mu ^2}p + {\alpha ^2}{a^2} + 4\beta \mu a + 4\alpha a + 4\mu a + 4} \over {24{\mu ^2}}}, \cr & {A_1} = - 2de{\mu ^2}, \cr & {A_2} = - 2{e^2}{\mu ^2}. \cr} $$

As a consequence, results for Eq. (1) are obtained in the following form: u(x,t,y)=A0+A1{ d2eΘ2etanh[ Θ2(q+B) ] }+A2{ d2eΘ2etanh[ Θ2(q+B) ] }2, $$\matrix{ {u(x,t,y) = {A_0} + {A_1}\left\{ { - {d \over {2e}} - {\Theta \over {2e}}\tanh \left[ {{\Theta \over 2}(q + B)} \right]} \right\} + {A_2}{{\left\{ { - {d \over {2e}} - {\Theta \over {2e}}\tanh \left[ {{\Theta \over 2}(q + B)} \right]} \right\}}^2},} \cr } $$

and u(x,t,y)=A0+A1e{d2eΘ2etanh(Θq2)+sech(Θq2)cosh(Θq2)2eΘsinh(Θq2)}+A2{d2eΘ2etanh(Θq2)+sech(Θq2)cosh(Θq2)2eΘsinh(Θq2)]}2 $$\matrix{ {u(x,t,y)} & { = {A_0} + {A_1}e\left\{ { - {d \over {2e}} - {{\rm{\Theta }} \over {2e}}\tanh ({{{\rm{\Theta }}q} \over 2}) + {{sech({{{\rm{\Theta }}q} \over 2})} \over {\cosh ({{{\rm{\Theta }}q} \over 2}) - {{2e} \over {\rm{\Theta }}}\sinh ({{{\rm{\Theta }}q} \over 2})}}} \right\}} \cr {} & { + {A_2}{{\left\{ { - {d \over {2e}} - {{\rm{\Theta }} \over {2e}}\tanh ({{{\rm{\Theta }}q} \over 2}) + {{sech({{{\rm{\Theta }}q} \over 2})} \over {\cosh ({{{\rm{\Theta }}q} \over 2}) - {{2e} \over {\rm{\Theta }}}\sinh ({{{\rm{\Theta }}q} \over 2})}}]} \right\}}^2}} \cr } $$

where q = atµzy and B represent the integration constant.

Fig. 2

Solution of Eq. (23)

Fig. 3

Solution of Eq. (24)

Conservation laws of eKP Eq. (1)

The resident conservation law for eKP Eq. (1) can be given as: G=(ut+6uux+uxxx)xuyyα24utt+αuty+βuxt $$G = {({u_t} + 6u{u_x} + {u_{xxx}})_x} - {u_{yy}} - {{{\alpha ^2}} \over 4}{u_{tt}} + \alpha {u_{ty}} + \beta {u_{xt}}$$

being a space-time deviation. DtTt+DxTx+DyTy|(1)=0, $$\matrix{ {{D_t}{T^t} + {D_x}{T^x} + {D_y}{T^y}{|_{(1)}} = 0,} \cr } $$

which holds good for all official results of Eq. (1). We achieve a multiplier Λ, which is expressed in terms of the equation: Λ(t,x,y,u)=14C1α2tx+14C1α2βtx+C4x+12C1t2+C2t+C3. $$\matrix{ {{\rm{\Lambda }}(t,x,y,u) = {1 \over 4}{\mkern 1mu} {C_1}{\alpha ^2}tx + {1 \over 4}{\mkern 1mu} {C_1}{\alpha ^2}\beta tx + {C_4}x + {1 \over 2}{\mkern 1mu} {C_1}{t^2} + {C_2}t + {C_3}.} \cr } $$

Corresponding to Eq. (27), we then obtain the resulting conservation laws: T1t=18+8β [ (2αt2+2αβt2+αtx)uy(α2t2+α2βt212α4tx)ut+(2t2+4βt2+α2tx+2β2t2+α2βtx)ux +(α2t+α2βt+12α4x)u ],T1x18+8β [ (24t2+24βt2+12α2x)uux+(4βt2+2β2t2+2t2+α2βx+α2tx)ut +(4t2+4βt2+2α2tx)uxxx2α2tuxx(4β2+8β+4+6α2u+α2x+α2βx)u ],T1y=18+8β[ (2αt2+2αβt2+α3xt)ut(4t2+4βt22α2tx)uy(4αt+4αβt+α2x)u ]; $$\matrix{ {T_1^t = {1 \over {8 + 8\beta }}\left[ {(2\alpha {t^2} + 2\alpha \beta {t^2} + \alpha tx){u_y} - ({\alpha ^2}{t^2} + {\alpha ^2}\beta {t^2} - {1 \over 2}{\alpha ^4}tx){u_t} + (2{t^2} + 4\beta {t^2} + {\alpha ^2}tx + 2{\beta ^2}{t^2} + {\alpha ^2}\beta tx){u_x}} \right.} \hfill \cr {\left. { + ({\alpha ^2}t + {\alpha ^2}\beta t + {1 \over 2}{\alpha ^4}x)u} \right],} \hfill \cr {T_1^x{1 \over {8 + 8\beta }}\left[ {(24{t^2} + 24\beta {t^2} + 12{\alpha ^2}x)u{u_x} + (4\beta {t^2} + 2{\beta ^2}{t^2} + 2{t^2} + {\alpha ^2}\beta x + {\alpha ^2}tx){u_t}} \right.} \hfill \cr {\left. { + (4{t^2} + 4\beta {t^2} + 2{\alpha ^2}tx){u_{xxx}} - 2{\alpha ^2}t{u_{xx}} - (4{\beta ^2} + 8\beta + 4 + 6{\alpha ^2}u + {\alpha ^2}x + {\alpha ^2}\beta x)u} \right],} \hfill \cr {T_1^y = {1 \over {8 + 8\beta }}\left[ {(2\alpha {t^2} + 2\alpha \beta {t^2} + {\alpha ^3}xt){u_t} - (4{t^2} + 4\beta {t^2} - 2{\alpha ^2}tx){u_y} - (4\alpha t + 4\alpha \beta t + {\alpha ^2}x)u} \right];} \hfill \cr } $$ T2t=14[ α2tut+2αtuy+2t(1+β)ux+α2u ],T2x=12[ 12tuux+βtut+tut+2tuxxx(1+β)u ],T2y=12α(tut+u)tuy; $$\eqalign{ & T_2^t = {1 \over 4}\left[ {{\alpha ^2}t{u_t} + 2\alpha t{u_y} + 2t(1 + \beta ){u_x} + {\alpha ^2}u} \right], \cr & T_2^x = {1 \over 2}\left[ {12tu{u_x} + \beta t{u_t} + t{u_t} + 2t{u_{xxx}} - (1 + \beta )u} \right], \cr & T_2^y = {1 \over 2}\alpha (t{u_t} + u) - t{u_y}; \cr} $$ T3t=12[ 12α2ut+αuy+(1+β)ux ],T3x=12(1+β)ut+6uux+uxxx,T3y=12αutuy; $$\eqalign{ & T_3^t = {1 \over 2}\left[ {{1 \over 2}{\alpha ^2}{u_t} + \alpha {u_y} + (1 + \beta ){u_x}} \right], \cr & T_3^x = {1 \over 2}(1 + \beta ){u_t} + 6u{u_x} + {u_{xxx}}, \cr & T_3^y = {1 \over 2}\alpha {u_t} - {u_y}; \cr} $$ T4t=12αxut+αxuy+(1+β)xux12(1+β)u,T4x=12(1+β)xut+6xuux+xuxxx3u2uxx,T4y=xutxuy. $$\eqalign{ & T_4^t = {1 \over 2}\alpha x{u_t} + \alpha x{u_y} + (1 + \beta )x{u_x} - {1 \over 2}(1 + \beta )u, \cr & T_4^x = {1 \over 2}(1 + \beta )x{u_t} + 6xu{u_x} + x{u_{xxx}} - 3{u^2} - {u_{xx}}, \cr & T_4^y = x{u_t} - x{u_y}. \cr} $$

Conclusion

In this research paper, we studied the eKP equation by using the Lie symmetry analysis, which reduced the eKP equation to an ODE; furthermore, we used straightforward integration and the simplest equation method to obtain exact solutions for the eKP equation. Straightforward integration gave us the Jacobi cosine function solution, whereas using the simplest equation method, we obtained the trigonometric hyperbolic solutions. Finally, using the multiplier approach and with the aid of Maple, conservation laws were derived for the eKP equation.

Fig. 1

Solution of Eq. (21)
Solution of Eq. (21)

Fig. 2

Solution of Eq. (23)
Solution of Eq. (23)

Fig. 3

Solution of Eq. (24)
Solution of Eq. (24)

N. Kadkhoda, H. Jafari, Kudryshov method for solutions of isothermal magnetostatic atmospheres, Iranian Journal of Numerical Analysis and Optimization, 6 (2016) 43- 52 Kadkhoda N. Jafari H., Kudryshov method for solutions of isothermal magnetostatic atmospheres, Iranian Journal of Numerical Analysis and Optimization, 6 (2016) 43- 52Search in Google Scholar

A. Sjöberg, Double reduction of PDEs from the association of symmetries with conservation laws with applications, Applied Mathematics and Computation 184 (2007) 608-616. Sjöberg A., Double reduction of PDEs from the association of symmetries with conservation laws with applications, Applied Mathematics and Computation 184 (2007) 608-616.Search in Google Scholar

L.D. Moleleki, T. Motsepaa, C.M. Khalique, Solutions and conservation laws of a generalized second extended (3+1)- dimensional Jimbo-Miwa equation, Applied Mathe- matics and Nonlinear Sciences 3(2) (2018) 459-474. Moleleki L.D. Motsepaa T. Khalique C.M., Solutions and conservation laws of a generalized second extended (3+1)-dimensional Jimbo-Miwa equation, Applied Mathe- matics and Nonlinear Sciences 3(2) (2018) 459-474.Search in Google Scholar

N. A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1217-1231, 2005 Kudryashov N. A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1217-1231, 2005Search in Google Scholar

K.O. Alweil, K.A. Gepreel, T.A. Nofal, Simplest equation method for nonlinear evolution equations in mathematical physics, WSEAS Transactions on Mathematics 15 (2016) 2224-2880. Alweil K.O. Gepreel K.A. Nofal T.A., Simplest equation method for nonlinear evolution equations in mathematical physics, WSEAS Transactions on Mathematics 15 (2016) 2224-2880.Search in Google Scholar

M.R. Alharthi, R.A. Alotabi, K.A. Gepreel, M.S. Mohamed, Homotopy Analysis Transform Method for Integro- Differential Equations, General Mathematics Notes 32 (2016) 32-48. Alharthi M.R. Alotabi R.A. Gepreel K.A. Mohamed M.S., Homotopy Analysis Transform Method for Integro-Differential Equations, General Mathematics Notes 32 (2016) 32-48.Search in Google Scholar

N.S. Al-Sayali, K.A. Gepreel, T.A. Nofal, Direct method for solving nonlinear strain wave equation in microstructure solids, International Journal of Physical Sciences 11 (2016) 121-131. Al-Sayali N.S. K.A. Nofal T.A., Direct method for solving nonlinear strain wave equation in microstructure solids, International Journal of Physical Sciences 11 (2016) 121-131.Search in Google Scholar

D. Baleanu, N. Kadkhoda, H. Jafari, G. M. Moremedi, Optimal system and symmetry reduction of the (1+1) dimensional Sawada-Kotera equation, Internatonal Journal of Pure and Applied Mathematics 108(2) (2016) 215-226. Baleanu D. Kadkhoda N. Jafari H. Moremedi G. M., Optimal system and symmetry reduction of the (1+1) dimensional Sawada-Kotera equation, Internatonal Journal of Pure and Applied Mathematics 108(2) (2016) 215-226.Search in Google Scholar

M. Azadi, H. Jafari, H. G. Sun, Lie symmetry reductions of coupled KdV system of fractional order, Nonlinear Dynamics and Systems Theory, 18(1) (2018) 22-28. Azadi M. Jafari H. Sun H. G., Lie symmetry reductions of coupled KdV system of fractional order, Nonlinear Dynamics and Systems Theory, 18(1) (2018) 22-28.Search in Google Scholar

G.W. Bluman, S. Kumei,Symmetries and differential equations, Springer-Verlag, New York (1989). Bluman G.W. Kumei S., Symmetries and differential equations, Springer-Verlag, New York (1989).Search in Google Scholar

P.J. Olver, Applications of lie group to differential equations, second ed, Springer-Verlag (1993). Olver P.J., Applications of lie group to differential equations, second ed, Springer-Verlag (1993).Search in Google Scholar

N.H. Ibragimov, Elemetary lie group analysis and ordinary differential equations, John Wiley & Sons, New York (1999). Ibragimov N.H., Elemetary lie group analysis and ordinary differential equations, John Wiley & Sons, New York (1999).Search in Google Scholar

H. Steudel, Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen, Zeitschrift fr Naturforschung, 17A (1962) 129-132. Steudel H., Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen, Zeitschrift fr Naturforschung, 17A (1962) 129-132.Search in Google Scholar

A. Sjöberg, Double reduction of partial differential equations from the association of symmetries with conservation laws with application, Applied Mathematics and Computation, 184 (2007) 608-616. Sjöberg A., Double reduction of partial differential equations from the association of symmetries with conservation laws with application, Applied Mathematics and Computation, 184 (2007) 608-616.Search in Google Scholar

S.C. Anco, G.W. Bluman, Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications, European Journal of Applied Mathematics, 13 (2002) 545-566. Anco S.C. Bluman G.W., Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications, European Journal of Applied Mathematics, 13 (2002) 545-566.Search in Google Scholar

A.H. Kara, F.M. Mahomed, Relationship between symmetries and conservation laws, International Journal Theoretical Physics, 39 (2000) 23-40. Kara A.H. Mahomed F.M., Relationship between symmetries and conservation laws, International Journal Theoretical Physics, 39 (2000) 23-40.Search in Google Scholar

L.D. Moleleki, C.M. Khalique, Symmetries, traveling wave solutions and conservation laws of a (3+1)-dimensional Boussinesq equation, Advances in Mathematical Physics, vol. 2014, Article ID 672679, 8 pages, 2014. Moleleki L.D. Khalique C.M., Symmetries, traveling wave solutions and conservation laws of a (3+1)-dimensional Boussinesq equation, Advances in Mathematical Physics, vol. 2014, Article ID 672679, 8 pages, 2014.Search in Google Scholar

L.D. Moleleki, Solutions and conservation laws of a generalized 3D Kawahara equation, Europian Physical Journal Plus 133, 496 (2018). Moleleki L.D., Solutions and conservation laws of a generalized 3D Kawahara equation, Europian Physical Journal Plus 133, 496 (2018).Search in Google Scholar

M. Kumar, A.K. Tiwari, R. Kumar, Some more solutions of Kadomtsev-Petviashvili equation, Computers and Mathematics with Applications 74 (2017) 2599-2607. Kumar M. Tiwari A.K. Kumar R., Some more solutions of Kadomtsev-Petviashvili equation, Computers and Mathematics with Applications 74 (2017) 2599-2607.Search in Google Scholar

Y. Ma, A. Wazwazb, B. Li, Novel bifurcation solitons for an extended Kadomtsev-Petviashvili equation in fluids, Physics letter A 413 (2021) 127585. Ma Y. Wazwazb A. Li B., Novel bifurcation solitons for an extended Kadomtsev-Petviashvili equation in fluids, Physics letter A 413 (2021) 127585.Search in Google Scholar

N. K. Vitanov, On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simples equation, Communications in Nonlinear Science and Numerical Simulation 16 (2011) 4215-4231. Vitanov N. K., On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simples equation, Communications in Nonlinear Science and Numerical Simulation 16 (2011) 4215-4231.Search in Google Scholar

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