New closed form solutions of some nonlinear pseudo-parabolic models via a new extended direct algebraic method

Numerous instances of specific models or phenomena are described by nonlinear differential equations (DEs). These equations prove invaluable in capturing and analyzing the dynamic behaviors inherent to the models. Nonlinear PDEs, in particular, hold a remarkable ability to model a wide range of intriguing scenarios. Among these, pseudo-parabolic equations stand out. They incorporate time derivatives within their highest order term and are employed to describe specific physical phenomena. A noteworthy member of this category is the generalized Benjamin Bona Mahony Burgers equation, which (1) vt−vxxt−αvxx+γvx+g(v)x=0, \[{{v}_{t}}-{{v}_{xxt}}-\alpha {{v}_{xx}}+\gamma {{v}_{x}}+g{{(v)}_{x}}=0,\] where g(v) is a C²-smooth nonlinear function, α, γ are real constants, v(x,t) indicates velocity of fluid along (horizontal) x-direction. In the special case when we take g(v)_x = vv_x with α = 0, γ = 1 in equation (1), we get the well-known KdV equation (2) vt+vxxt+vx+vvx=0. \[{{v}_{t}}+{{v}_{xxt}}+{{v}_{x}}+v{{v}_{x}}=0.\]

Equation (2) was proposed by Peregrine [1] and Benjamin et al. [2] and interprets a different regular long-wave in comparison with (1). In the second special case if we take g(v)_x = θ vv_x + β v_xxx in (1), we get another well known BBMPB equation (3) vt−vxxt−αvxx+γvx+θvvx+βvxxx=0. \[{{v}_{t}}-{{v}_{xxt}}-\alpha {{v}_{xx}}+\gamma {{v}_{x}}+\theta v{{v}_{x}}+\beta {{v}_{xxx}}=0.\]

Next for the third special case we can take α = β = 0 in (3), we get the well known BBM equation (4) vt−vxxt+γvx+θvvx=0, \[{{v}_{t}}-{{v}_{xxt}}+\gamma {{v}_{x}}+\theta v{{v}_{x}}=0,\] where γ, θ ≠ 0 are real parameters. Equation (3) includes different types of the BBM equation, one may refer to [3, 4, 5, 6, 7, 8] for more details. For β = 0 in equation (3), we obtain the OBBMB equation (5) vt−vxxt−αvxx+γvx+θvvx=0. \[{{v}_{t}}-{{v}_{xxt}}-\alpha {{v}_{xx}}+\gamma {{v}_{x}}+\theta v{{v}_{x}}=0.\]

The above equation is a one-dimensional, nonlinear, pseudo-parabolic equation that characterizes the behavior of nonlinear surface waves propagating along the x-axis. The term αv_xx represents the viscosity term [9, 10, 11]. To uncover multiple soliton solutions for this equation, the inverse scattering method has been effectively employed in [12, 13]. The equation (6) vt−λvxxt−αvxx+vvx=0, \[{{v}_{t}}-\lambda {{v}_{xxt}}-\alpha {{v}_{xx}}+v{{v}_{x}}=0,\] is a 1D analog of the Oskolkov system (7) (1−κ∇2)vt=α∇2v−(v⋅∇)v−∇2p+g, ∇⋅v=0. \[(1-\kappa {{\nabla}^{2}}){{v}_{t}}=\alpha {{\nabla}^{2}}v-(v\cdot \nabla)v-{{\nabla}^{2}}p+g, \nabla \cdot v=0.\]

The described system serves as a model for capturing the motion of a viscoelastic, incompressible Kelvin-Voigt fluid. The possibility of the parameter κ taking on negative values is sightable. The generalized HERW is given by [14] (8) vt−vxxt+αvx+2βvvx+3θv2vx−γvxvxx−vvxxx=0. \[{{v}_{t}}-{{v}_{xxt}}+\alpha {{v}_{x}}+2\beta v{{v}_{x}}+3\theta {{v}^{2}}{{v}_{x}}-\gamma {{v}_{x}}{{v}_{xx}}-v{{v}_{xxx}}=0.\]

The equation given in Eq (8) involves real parameters α, β, θ, and γ. This equation holds relevance in describing numerous noteworthy physical models within the realm of mathematical physics.

When setting β=32 \[\beta =\frac{3}{2}\] , θ = 0, and γ = 2, the equation transforms into the Camassa Holm equation (9) vt−vxxt+αvx+3vvx−2vxvxx−vvxxx=0. \[{{v}_{t}}-{{v}_{xxt}}+\alpha {{v}_{x}}+3v{{v}_{x}}-2{{v}_{x}}{{v}_{xx}}-v{{v}_{xxx}}=0.\]

In this equation, v represents the velocity of the fluid in the x direction, and α is a constant associated with the speed of shallow water waves. Similarly, when β = 2, θ = 0, and γ = 3, equation (3) becomes the Degasperis Procesi equation (10) vt−vxxt+αvx+4vvx−3vxvxx−vvxxx=0. \[{{v}_{t}}-{{v}_{xxt}}+\alpha {{v}_{x}}+4v{{v}_{x}}-3{{v}_{x}}{{v}_{xx}}-v{{v}_{xxx}}=0.\]

It is worth mentioning that Wazwaz [15] extensively discussed the traveling wave solutions of both the Camassa Holm equation and the Degasperis Procesi equation.

By setting α = 1, β=12 \[\beta =\frac{1}{2}\] , θ = 0, and γ = 3, equation (3) transforms into the Fornberg-Whitham equation (11) vt−vxxt+vx+vvx−3vxvxx−vvxxx=0. \[{{v}_{t}}-{{v}_{xxt}}+{{v}_{x}}+v{{v}_{x}}-3{{v}_{x}}{{v}_{xx}}-v{{v}_{xxx}}=0.\]

The Fornberg-Whitham equation is particularly useful for analyzing wave-breaking phenomena. The solution to this equation, as demonstrated by Fornberg and Whitham, can be expressed as v(x,t)=Ae−12|x−43t| \[v(x,t)=A{{e}^{-\frac{1}{2}|x-\frac{4}{3}t|}}\] [16]

The objective of this article is to discover novel families of exact soliton solutions for nonlinear pseudo-parabolic type models in mathematical physics, utilizing the NEDA method. This approach serves as a remarkably potent tool for addressing soliton solutions in nonlinear PDEs. The NEDA method yields solutions based on hyperbolic, trigonometric, rational, exponential, and polynomial functions, catering to various forms of nonlinear PDEs. However, it is important to note that the current approach has its limitations, especially when seeking more general soliton solutions. To overcome this limitation, we aim to enhance our methodology. Here [17, 18, 19, 20, 21, 22, 23, 24, 25, 26] are several established techniques that are effectively applied to handle nonlinear PDEs.

Previous work by Akcagil et al. [27] has reported solutions involving trigonometric, hyperbolic, and rational solitons. We intend to build upon these earlier findings to push the boundaries in our pursuit of a diverse range of new traveling wave solutions. Our research endeavors encompass various types of solitons, including dark, bright, singular, combined dark-bright soliton, dark-singular-combined solitons, solitary wave solutions and many more. The solutions we present in this study are groundbreaking and unique, as they haven't been previously showcased for this specific class of equations.

The organization of this paper is outlined as follows: In Section 2, we introduce the framework of the NEDA method. Moving forward, Sections 3, 4, 5, and 6 are dedicated to the examination of specific equations-namely, the BBMPB, OBBMB equation, the 1D Oskolkov equation and generalized HERW equations, respectively. These sections delve into the application of the NEDA method to extract a variety of solutions. Graphical representations of these solutions are presented in Section 7. Finally, in Section 8, we provide a summary and conclusion, besides suggesting potential avenues for future investigations.

In this section, we outline the general methodology of the NEDA method [28]. We use the following main steps to find the traveling wave solutions of the nonlinear PDEs.

Step 1. Consider a nth order nonlinear PDE (12) Θ(ζ,v,v1,⋯,vn)=0, \[\Theta (\zeta,v,{{v}_{1}},\cdots,{{v}_{n}})=0,\] where ζ = ζ¹, ζ²,⋯, ζⁿ indicates independent variables, v is dependent variable and v₁ and v_n signifies first and nth order derivatives of v with respect to ϑ, respectively.

Step 2. We use the transformation v(x,t) = η(ξ), ξ = x − μt to convert (12) into a nonlinear ODE (13) Λ(η,η′,η″,⋯)=0. \[\Lambda (\eta,{\eta}',{{\eta}''},\cdots)=0.\]

Step 3. The ansatz solution of ODE (13) as a polynomial in hⁱ(ξ) is given by (14) η(ξ)=∑i=0Mbihi(ξ), \[\eta (\xi)=\sum\limits_{i=0}^{M}{}{{b}_{i}}{{h}^{i}}(\xi),\] where b_i(0 < i ≤ n) are the coefficients which can be decided later. Note that equation (14) accepts only a particular class of h(ξ)'s that satisfies the ODE condition given as (15) h′(ξ)=(lnφ)[k1+k2h(ξ)+k3h2(ξ)], φ≠0,1, \[{h}'(\xi)=(\ln \varphi)[{{k}_{1}}+{{k}_{2}}h(\xi)+{{k}_{3}}{{h}^{2}}(\xi)], \varphi \ne 0,1,\] where k₁, k₂, k₃ are nonzero constants. Equation (15) does not influence the solution of ODE (13), it just suggests a proper choice of h(ξ). Next, we substitute (14) and (15) into (13) and get a system of algebraic equations. Solving this system for desired constants leads to the solutions. If k22−4k1k3=Δ \[k_{2}^{2}-4{{k}_{1}}{{k}_{3}}=\Delta \] , then the solutions to (15) can be written as

(1) : If k22−4k1k3<0 \[k_{2}^{2}-4{{k}_{1}}{{k}_{3}}<0\] and k₃ ≠ 0, then (16) h1(x,t)=−k22k3+−Δ2k3tanφ(−Δ2ξ),h2(x,t)=−k22k3−−Δ2k3cotφ(−Δ2ξ),h3(x,t)=−k22k3+−Δ2k3(tanφ(−Δξ)±pqsecφ(−Δξ)),h4(x,t)=−k22k3−−Δ2k3(cotφ(−Δξ)±pqcscφ(−Δξ)),h5(x,t)=−k22k3+−Δ4k3(tanφ(−Δ4ξ)−cotφ(−Δ4ξ)), \[\begin{array}{*{35}{l}} {{h}_{1}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}+\frac{\sqrt{-\Delta}}{2{{k}_{3}}}\mathop{\tan}_{\varphi}(\frac{\sqrt{-\Delta}}{2}\xi), \\ {{h}_{2}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}-\frac{\sqrt{-\Delta}}{2{{k}_{3}}}\mathop{\cot}_{\varphi}(\frac{\sqrt{-\Delta}}{2}\xi), \\ {{h}_{3}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}+\frac{\sqrt{-\Delta}}{2{{k}_{3}}}(\mathop{\tan}_{\varphi}(\sqrt{-\Delta}\xi)\pm \sqrt{pq}\mathop{\sec}_{\varphi}(\sqrt{-\Delta}\xi)), \\ {{h}_{4}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}-\frac{\sqrt{-\Delta}}{2{{k}_{3}}}(\mathop{\cot}_{\varphi}(\sqrt{-\Delta}\xi)\pm \sqrt{pq}\mathop{\csc}_{\varphi}(\sqrt{-\Delta}\xi)), \\ {{h}_{5}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}+\frac{\sqrt{-\Delta}}{4{{k}_{3}}}(\mathop{\tan}_{\varphi}(\frac{\sqrt{-\Delta}}{4}\xi)-\mathop{\cot}_{\varphi}(\frac{\sqrt{-\Delta}}{4}\xi)), \\\end{array}\] where the generalized trigonometric and hyperbolic functions are defined in equation (28), and p, q are real constants known as deformation parameters.

(2) : If k22−4k1k3>0 \[k_{2}^{2}-4{{k}_{1}}{{k}_{3}}>0\] and k₃ ≠ 0, then (17) h6(x,t)=−k22k3−Δ2k3tanhφ(Δ2ξ),h7(x,t)=−k22k3−Δ2k3cothφ(Δ2ξ),h8(x,t)=−k22k3+Δ2k3(tanhφ(Δξ)±ipq sechφ(Δξ)),h9(x,t)=−k22k3−Δ2k3(cothφ(Δξ)±pq cschφ(Δξ)),h10(x,t)=−k22k3−Δ4k3(tanhφ(Δ4ξ)+cothφ(Δ4ξ))⋅ \[\begin{array}{*{35}{l}} {{h}_{6}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}-\frac{\sqrt{\Delta}}{2{{k}_{3}}}\mathop{\tanh}_{\varphi}(\frac{\sqrt{\Delta}}{2}\xi), \\ {{h}_{7}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}-\frac{\sqrt{\Delta}}{2{{k}_{3}}}\mathop{\coth}_{\varphi}(\frac{\sqrt{\Delta}}{2}\xi), \\ {{h}_{8}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}+\frac{\sqrt{\Delta}}{2{{k}_{3}}}(\mathop{\tanh}_{\varphi}(\sqrt{\Delta}\xi)\pm i\sqrt{pq} {{\rm sech}_{\varphi}}(\sqrt{\Delta}\xi)), \\ {{h}_{9}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}-\frac{\sqrt{\Delta}}{2{{k}_{3}}}(\mathop{\coth}_{\varphi}(\sqrt{\Delta}\xi)\pm \sqrt{pq} {{\rm csch}_{\varphi}}(\sqrt{\Delta}\xi)), \\ {{h}_{10}}(x,t) & =-\frac{{{k}_{2}}}{2{{k}_{3}}}-\frac{\sqrt{\Delta}}{4{{k}_{3}}}(\mathop{\tanh}_{\varphi}(\frac{\sqrt{\Delta}}{4}\xi)+\mathop{\coth}_{\varphi}(\frac{\sqrt{\Delta}}{4}\xi))\cdot \\\end{array}\]

(3) : If k₁k₃ > 0 and k₂ = 0, then (18) h11(x,t)=k1k3tanφ(k1k3ξ),h12(x,t)=−k1k3cotφ(k1k3ξ),h13(x,t)=k1k3(tanφ(2k1k3ξ)±pqsecφ(2k1k3ξ)),h14(x,t)=k1k3(−cotφ(2k1k3ξ)±pqcscφ(2k1k3ξ)),h15(x,t)=12k1k3(tanφ(k1k32ξ)−cotφ(k1k32ξ))⋅ \[\begin{array}{*{35}{l}} {{h}_{11}}(x,t) & =\sqrt{\frac{{{k}_{1}}}{{{k}_{3}}}}\mathop{\tan}_{\varphi}(\sqrt{{{k}_{1}}{{k}_{3}}}\xi), \\ {{h}_{12}}(x,t) & =-\sqrt{\frac{{{k}_{1}}}{{{k}_{3}}}}\mathop{\cot}_{\varphi}(\sqrt{{{k}_{1}}{{k}_{3}}}\xi), \\ {{h}_{13}}(x,t) & =\sqrt{\frac{{{k}_{1}}}{{{k}_{3}}}}(\mathop{\tan}_{\varphi}(2\sqrt{{{k}_{1}}{{k}_{3}}}\xi)\pm \sqrt{pq}\mathop{\sec}_{\varphi}(2\sqrt{{{k}_{1}}{{k}_{3}}}\xi)), \\ {{h}_{14}}(x,t) & =\sqrt{\frac{{{k}_{1}}}{{{k}_{3}}}}(-\mathop{\cot}_{\varphi}(2\sqrt{{{k}_{1}}{{k}_{3}}}\xi)\pm \sqrt{pq}\mathop{\csc}_{\varphi}(2\sqrt{{{k}_{1}}{{k}_{3}}}\xi)), \\ {{h}_{15}}(x,t) & =\frac{1}{2}\sqrt{\frac{{{k}_{1}}}{{{k}_{3}}}}(\mathop{\tan}_{\varphi}(\frac{\sqrt{{{k}_{1}}{{k}_{3}}}}{2}\xi)-\mathop{\cot}_{\varphi}(\frac{\sqrt{{{k}_{1}}{{k}_{3}}}}{2}\xi))\cdot \\\end{array}\]

(4) : If k₁k₃ < 0 and k₂ = 0, then (19) h16(x,t)=−−k1k3tanhφ(−k1k3ξ),h17(x,t)=−−k1k3cothφ(−k1k3ξ),h18(x,t)=−−k1k3(tanhφ(2−k1k3ξ)±ipq sechφ(2−k1k3ξ)),h19(x,t)=−−k1k3(cothφ(2−k1k3ξ)±pqcschφ(2−k1k3ξ)),h20(x,t)=−12−k1k3(tanhφ(−k1k32ξ)+cothφ(−k1k32ξ))⋅ \[\begin{array}{*{35}{l}} {{h}_{16}}(x,t) & =-\sqrt{-\frac{{{k}_{1}}}{{{k}_{3}}}}\mathop{\tanh}_{\varphi}(\sqrt{-{{k}_{1}}{{k}_{3}}}\xi), \\ {{h}_{17}}(x,t) & =-\sqrt{-\frac{{{k}_{1}}}{{{k}_{3}}}}\mathop{\coth}_{\varphi}(\sqrt{-{{k}_{1}}{{k}_{3}}}\xi), \\ {{h}_{18}}(x,t) & =-\sqrt{-\frac{{{k}_{1}}}{{{k}_{3}}}}(\mathop{\tanh}_{\varphi}(2\sqrt{-{{k}_{1}}{{k}_{3}}}\xi)\pm i\sqrt{pq} {{\rm sech}_{\varphi}}(2\sqrt{-{{k}_{1}}{{k}_{3}}}\xi)), \\ {{h}_{19}}(x,t) & =-\sqrt{-\frac{{{k}_{1}}}{{{k}_{3}}}}(\mathop{\coth}_{\varphi}(2\sqrt{-{{k}_{1}}{{k}_{3}}}\xi)\pm \sqrt{pq} {{\rm csch}_{\varphi}}(2\sqrt{-{{k}_{1}}{{k}_{3}}}\xi)), \\ {{h}_{20}}(x,t) & =-\frac{1}{2}\sqrt{-\frac{{{k}_{1}}}{{{k}_{3}}}}(\mathop{\tanh}_{\varphi}(\frac{\sqrt{-{{k}_{1}}{{k}_{3}}}}{2}\xi)+\mathop{\coth}_{\varphi}(\frac{\sqrt{-{{k}_{1}}{{k}_{3}}}}{2}\xi))\cdot \\\end{array}\]

(5) : If a₁ = a₃ and a₂ = 0, then (20) h21(x,t)=tanφ(k1ξ),h22(x,t)=−cotφ(k1ξ),h23(x,t)=tanφ(2k1ξ)±pqsecφ(2k1ξ),h24(x,t)=−cotφ(2k1ξ)±pqcscφ(2k1ξ),h25(x,t)=12(tanφ(k12ξ)−cotφ(k12ξ))⋅ \[\begin{array}{*{35}{l}} {{h}_{21}}(x,t) & =\mathop{\tan}_{\varphi}({{k}_{1}}\xi), \\ {{h}_{22}}(x,t) & =-\mathop{\cot}_{\varphi}({{k}_{1}}\xi), \\ {{h}_{23}}(x,t) & =\mathop{\tan}_{\varphi}(2{{k}_{1}}\xi)\pm \sqrt{pq}\mathop{\sec}_{\varphi}(2{{k}_{1}}\xi), \\ {{h}_{24}}(x,t) & =-\mathop{\cot}_{\varphi}(2{{k}_{1}}\xi)\pm \sqrt{pq}\mathop{\csc}_{\varphi}(2{{k}_{1}}\xi), \\ {{h}_{25}}(x,t) & =\frac{1}{2}(\mathop{\tan}_{\varphi}(\frac{{{k}_{1}}}{2}\xi)-\mathop{\cot}_{\varphi}(\frac{{{k}_{1}}}{2}\xi))\cdot \\\end{array}\]

(6) : If k₃ = −k₁ and k₂ = 0, then (21) h26(x,t)=−tanhφ(k1ξ),h27(x,t)=−cothφ(k1ξ),h28(x,t)=−tanhφ(2k1ξ)±ipq sechφ(2k1ξ),h29(x,t)=−cothφ(2k1ξ)±pqcschφ(2k1ξ),h30(x,t)=−12(tanhφ(k12ξ)−cothφ(k12ξ))⋅ \[\begin{array}{*{35}{l}} {{h}_{26}}(x,t) & =-\mathop{\tanh}_{\varphi}({{k}_{1}}\xi), \\ {{h}_{27}}(x,t) & =-\mathop{\coth}_{\varphi}({{k}_{1}}\xi), \\ {{h}_{28}}(x,t) & =-\mathop{\tanh}_{\varphi}(2{{k}_{1}}\xi)\pm i\sqrt{pq} {{\rm sech}_{\varphi}}(2{{k}_{1}}\xi), \\ {{h}_{29}}(x,t) & =-\mathop{\coth}_{\varphi}(2{{k}_{1}}\xi)\pm \sqrt{pq} {{\rm csch}_{\varphi}}(2{{k}_{1}}\xi), \\ {{h}_{30}}(x,t) & =-\frac{1}{2}(\mathop{\tanh}_{\varphi}(\frac{{{k}_{1}}}{2}\xi)-\mathop{\coth}_{\varphi}(\frac{{{k}_{1}}}{2}\xi))\cdot \\\end{array}\]

(7) : If k22=4k1k3 \[k_{2}^{2}=4{{k}_{1}}{{k}_{3}}\] , then (22) h31(x,t)=−2k1(k2ln(φ)ξ+2)k22ln(φ)ξ⋅ \[{{h}_{31}}(x,t)=-\frac{2{{k}_{1}}({{k}_{2}}\ln (\varphi)\xi +2)}{k_{2}^{2}\ln (\varphi)\xi}\cdot \]

(8) : If k₂ = k, k₁ = mk(m ≠ 0), and k₃ = 0, then (23) h32(x,t)=φkξ−m⋅ \[{{h}_{32}}(x,t)={{\varphi}^{k\xi}}-m\cdot \]

(9) : If k₂ = k₃ = 0, then (24) h33(x,t)=k1ξln(φ)⋅ \[{{h}_{33}}(x,t)={{k}_{1}}\xi \ln (\varphi)\cdot \]

(10) : If k₁ = k₂ = 0, then (25) h34(x,t)=−1k3ξln(φ)⋅ \[{{h}_{34}}(x,t)=\frac{-1}{{{k}_{3}}\xi \ln (\varphi)}\cdot \]

(11) : If k₁ = 0 and k₂ ≠ 0, then (26) h35(x,t)=−k2pk3(coshφ(k2ξ)−sinhφ(k2ξ)+p),h36(x,t)=−k2(coshφ(k2ξ)+sinhφ(k2ξ))k3(coshφ(k2ξ)+sinhφ(k2ξ)+q)⋅ \[\begin{array}{*{35}{l}} {{h}_{35}}(x,t) & =-\frac{{{k}_{2}}p}{{{k}_{3}}(\mathop{\cosh}_{\varphi}({{k}_{2}}\xi)-\mathop{\sinh}_{\varphi}({{k}_{2}}\xi)+p)}, \\ {{h}_{36}}(x,t) & =-\frac{{{k}_{2}}(\mathop{\cosh}_{\varphi}({{k}_{2}}\xi)+\mathop{\sinh}_{\varphi}({{k}_{2}}\xi))}{{{k}_{3}}(\mathop{\cosh}_{\varphi}({{k}_{2}}\xi)+\mathop{\sinh}_{\varphi}({{k}_{2}}\xi)+q)}\cdot \\\end{array}\]

(12) : If k₁ = 0, k₂ = k and k₃ = mk(m ≠ 0), then (27) h37(x,t)=pφkξp−mqφkξ⋅ \[{{h}_{37}}(x,t)=\frac{p{{\varphi}^{k\xi}}}{p-mq{{\varphi}^{k\xi}}}\cdot \]

The generalized hyperbolic and triangular functions appearing in the solutions (16–27) are defined as [6] (28) sinφ(ξ)=pφiξ−qφ−iξ2i, cosφ(ξ)=pφiξ+qφ−iξ2,tanφ(ξ)=−ipφiξ−qφ−iξpφiξ+qφ−iξ, cotφ(ξ)=ipφiξ+qφ−iξpφiξ−qφ−iξ,cscφ(ξ)=2ipφiξ−qφ−iξ, secφ(ξ)=2pφiξ+qφ−iξ,sinhφ(ξ)=pφξ−qφ−ξ2, coshφ(ξ)=pφξ+qφ−ξ2,tanhφ(ξ)=pφξ−qφ−ξpφξ+qφ−ξ, cothφ(ξ)=pφξ+qφ−ξpφξ−qφ−ξ,cschφ(ξ)=2pφξ−qφ−ξ, sechφ(ξ)=2pφξ+qφ−ξ, \[\begin{array}{*{35}{l}} \mathop{\sin}_{\varphi}(\xi) & =\frac{p{{\varphi}^{i\xi}}-q{{\varphi}^{-i\xi}}}{2i}, \mathop{\cos}_{\varphi}(\xi)=\frac{p{{\varphi}^{i\xi}}+q{{\varphi}^{-i\xi}}}{2}, \\ \mathop{\tan}_{\varphi}(\xi) & =-i\frac{p{{\varphi}^{i\xi}}-q{{\varphi}^{-i\xi}}}{p{{\varphi}^{i\xi}}+q{{\varphi}^{-i\xi}}}, \mathop{\cot}_{\varphi}(\xi)=i\frac{p{{\varphi}^{i\xi}}+q{{\varphi}^{-i\xi}}}{p{{\varphi}^{i\xi}}-q{{\varphi}^{-i\xi}}}, \\ \mathop{\csc}_{\varphi}(\xi) & =\frac{2i}{p{{\varphi}^{i\xi}}-q{{\varphi}^{-i\xi}}}, \mathop{\sec}_{\varphi}(\xi)=\frac{2}{p{{\varphi}^{i\xi}}+q{{\varphi}^{-i\xi}}}, \\ \mathop{\sinh}_{\varphi}(\xi) & =\frac{p{{\varphi}^{\xi}}-q{{\varphi}^{-\xi}}}{2}, \mathop{\cosh}_{\varphi}(\xi)=\frac{p{{\varphi}^{\xi}}+q{{\varphi}^{-\xi}}}{2}, \\ \mathop{\tanh}_{\varphi}(\xi) & =\frac{p{{\varphi}^{\xi}}-q{{\varphi}^{-\xi}}}{p{{\varphi}^{\xi}}+q{{\varphi}^{-\xi}}}, \mathop{\coth}_{\varphi}(\xi)=\frac{p{{\varphi}^{\xi}}+q{{\varphi}^{-\xi}}}{p{{\varphi}^{\xi}}-q{{\varphi}^{-\xi}}}, \\ {{\rm csch}_{\varphi}}(\xi) & =\frac{2}{p{{\varphi}^{\xi}}-q{{\varphi}^{-\xi}}}, {{\rm sech}_{\varphi}}(\xi)=\frac{2}{p{{\varphi}^{\xi}}+q{{\varphi}^{-\xi}}}, \\\end{array}\] where the nonzero parameters p, q are called deformation parameters and ξ indicates an independent variable.

The BBMPB equation (3) is given by vt−vxxt−αvx+γvx+θvvx+βvxxx=0, \[{{v}_{t}}-{{v}_{xxt}}-\alpha {{v}_{x}}+\gamma {{v}_{x}}+\theta v{{v}_{x}}+\beta {{v}_{xxx}}=0,\] where θ, β are nonzero real constants and α is a positive constant. We use the transformation v(x,t) = U (ξ), ξ = x − μt in (3) and integrate this equation to get (29) (−μ+γ)U−αU′+θ2U2+(μ+β)U″=0. \[(-\mu +\gamma)U-\alpha {U}'+\frac{\theta}{2}{{U}^{2}}+(\mu +\beta){{U}''}=0.\]

Now, we will discuss the traveling wave structure of the BBMPB equation by using (29) with the help of the NEDA method. Comparing the linear term U′′ with the nonlinear term U², we have the solution of the form: (30) η(ξ)=b0+b1h(ξ)+b2h2(ξ). \[\eta (\xi)={{b}_{0}}+{{b}_{1}}h(\xi)+{{b}_{2}}{{h}^{2}}(\xi).\]

Let us suppose that h(ξ) is the solution of the equation, then (31) h′(ξ)=(lnϕ)[ω1+ω2h(ξ)+ω3h2(ξ)]. \[{h}'(\xi)=(\ln \phi)[{{\omega}_{1}}+{{\omega}_{2}}h(\xi)+{{\omega}_{3}}{{h}^{2}}(\xi)].\]

Substituting (30) and (31) into the equation (29), and then comparing the coefficients of different powers of h(ξ), we arrive at a system of algebraic expressions. Solving this system for ω₁, ω₂, b₁, b₂ we get the set mentioned below (32) ω1=θb0(θb0+2b+2c)4b2ω3(ln(ϕ))2, ω2=−θb0+b+cbln(ϕ), b1=−2(ln(ϕ))bω3θ, b2=0, a=−b. \[{{\omega}_{1}}=\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{4{{b}^{2}}{{\omega}_{3}}{{(\ln (\phi))}^{2}}}, {{\omega}_{2}}=-\frac{\theta {{b}_{0}}+b+c}{b\ln (\phi)}, {{b}_{1}}=-\frac{2(\ln (\phi))b{{\omega}_{3}}}{\theta}, {{b}_{2}}=0, a=-b.\]

Case 1: When Δ < 0 and ω₃ ≠ 0, then (33) v1(x,t)=θb0+b+c2bω3ln(ϕ)+−Δ2ω3tanϕ(−Δ2ξ),v2(x,t)=θb0+b+c2bω3ln(ϕ)−−Δ2ω3cotϕ(−Δ2ξ),v3(x,t)=θb0+b+c2bω3ln(ϕ)−−Δ2ω3(tanϕ(−Δξ)±rssecϕ(−Δξ)),v4(x,t)=θb0+b+c2bω3ln(ϕ)−−Δ2ω3(cotϕ(−Δξ)±rscscϕ(−Δξ)),v5(x,t)=θb0+b+c2bω3ln(ϕ)+−Δ4ω3(tanϕ(−Δ4ξ)−cotϕ(−Δ4ξ))⋅ \[\begin{array}{*{35}{l}} {{v}_{1}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}+\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}\mathop{\tan}_{\phi}(\frac{\sqrt{-\Delta}}{2}\xi), \\ {{v}_{2}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}\mathop{\cot}_{\phi}(\frac{\sqrt{-\Delta}}{2}\xi), \\ {{v}_{3}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}(\mathop{\tan}_{\phi}(\sqrt{-\Delta}\xi)\pm \sqrt{rs}\mathop{\sec}_{\phi}(\sqrt{-\Delta}\xi)), \\ {{v}_{4}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}(\mathop{\cot}_{\phi}(\sqrt{-\Delta}\xi)\pm \sqrt{rs}\mathop{\csc}_{\phi}(\sqrt{-\Delta}\xi)), \\ {{v}_{5}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}+\frac{\sqrt{-\Delta}}{4{{\omega}_{3}}}(\mathop{\tan}_{\phi}(\frac{\sqrt{-\Delta}}{4}\xi)-\mathop{\cot}_{\phi}(\frac{\sqrt{-\Delta}}{4}\xi))\cdot \\\end{array}\] where Δ=ω22−ω1ω3 \[\Delta =\omega _{2}^{2}-{{\omega}_{1}}{{\omega}_{3}}\] , and r, s are constants greater than zero and called deformation parameters.

Case 2: When Δ > 0 and ω₃ ≠ 0, then v6(x,t)=θb0+b+c2bω3ln(ϕ)−Δ2ω3tanhϕ(Δ2ξ),v7(x,t)=θb0+b+c2bω3ln(ϕ)−Δ2ω3cothϕ(Δ2ξ), \[\begin{array}{*{35}{l}} {{v}_{6}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{\Delta}}{2{{\omega}_{3}}}\mathop{\tanh}_{\phi}(\frac{\sqrt{\Delta}}{2}\xi), \\ {{v}_{7}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{\Delta}}{2{{\omega}_{3}}}\mathop{\coth}_{\phi}(\frac{\sqrt{\Delta}}{2}\xi), \\\end{array}\] (34) v8(x,t)=θb0+b+c2bω3ln(ϕ)−Δ2ω3(tanhϕ(Δξ)±irssechϕ(Δξ)),v9(x,t)=θb0+b+c2bω3ln(ϕ)−Δ2ω3(cothϕ(Δξ)±pqcschϕ(Δξ)),v10(x,t)=θb0+b+c2bω3ln(ϕ)−Δ4ω3(tanhϕ(Δ4ξ)+cothϕ(Δ4ξ))⋅ \[\begin{array}{*{35}{l}} {{v}_{8}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{\Delta}}{2{{\omega}_{3}}}(\mathop{\tanh}_{\phi}(\sqrt{\Delta}\xi)\pm i\sqrt{rs}\sec {{h}_{\phi}}(\sqrt{\Delta}\xi)), \\ {{v}_{9}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{\Delta}}{2{{\omega}_{3}}}(\mathop{\coth}_{\phi}(\sqrt{\Delta}\xi)\pm \sqrt{pq}\csc {{h}_{\phi}}(\sqrt{\Delta}\xi)), \\ {{v}_{10}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{\Delta}}{4{{\omega}_{3}}}(\mathop{\tanh}_{\phi}(\frac{\sqrt{\Delta}}{4}\xi)+\mathop{\coth}_{\phi}(\frac{\sqrt{\Delta}}{4}\xi))\cdot \\\end{array}\]

Case 3: When ω₁ω₃ > 0 and ω₂ = 0, then (35) v11(x,t)=−θb0(θb0+2b+2c)2b(ln(ϕ))ω3tanϕ(θb0(θb0+2b+2c)2b(ln(ϕ))ξ),v12(x,t)=−θb0(θb0+2b+2c)2b(ln(ϕ))ω3cotϕ(θb0(θb0+2b+2c)2b(ln(ϕ))ξ),v13(x,t)=θb0(θb0+2b+2c)2b(ln(ϕ))ω3[tanϕ(θb0(θb0+2b+2c)b(ln(ϕ))ξ)±rssecϕ(θb0(θb0+2b+2c)b(ln(ϕ))ξ)],v14(x,t)=θb0(θb0+2b+2c)2b(ln(ϕ))ω3[−cotϕ(θb0(θb0+2b+2c)b(ln(ϕ))ξ)±rscscϕ(θb0(θb0+2b+2c)b(ln(ϕ))ξ)],v15(x,t)=θb0(θb0+2b+2c)4b(ln(ϕ))ω3[tanϕ(θb0(θb0+2b+2c)4b(ln(ϕ))ξ)−cotϕ(θb0(θb0+2b+2c)4b(ln(ϕ))ξ)]⋅ \[\begin{array}{*{35}{l}} {{v}_{11}}(x,t)=-\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}} & \mathop{\tan}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi))}\xi), \\ {{v}_{12}}(x,t)=-\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}} & \mathop{\cot}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi))}\xi), \\ {{v}_{13}}(x,t)=\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}}[ & \mathop{\tan}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{b(\ln (\phi))}\xi) \\ {} & \pm \sqrt{rs}\mathop{\sec}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{b(\ln (\phi))}\xi)], \\ {{v}_{14}}(x,t)=\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}}[ & -\mathop{\cot}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{b(\ln (\phi))}\xi) \\ {} & \pm \sqrt{rs}\mathop{\csc}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{b(\ln (\phi))}\xi)], \\ {{v}_{15}}(x,t)=\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{4b(\ln (\phi)){{\omega}_{3}}}[ & \mathop{\tan}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{4b(\ln (\phi))}\xi) \\ {} & -\mathop{\cot}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{4b(\ln (\phi))}\xi)]\cdot \\\end{array}\]

Case 4: When ω₁ω₃ < 0 and ω₂ = 0, then (36) v16(x,t)=−−θb0(θb0+2b+2c)2b(ln(ϕ))ω3tanhϕ(−θb0(θb0+2b+2c)2b(ln(ϕ))ξ),v17(x,t)=−−θb0(θb0+2b+2c)2b(ln(ϕ))ω3cothϕ(−θb0(θb0+2b+2c)2b(ln(ϕ))ξ),v18(x,t)=−−θb0(θb0+2b+2c)2b(ln(ϕ))ω3[tanhϕ(−θb0(θb0+2b+2c)b(ln(ϕ))ξ)±irssechϕ(−θb0(θb0+2b+2c)b(ln(ϕ))ξ)], \[\begin{array}{*{35}{l}} {{v}_{16}}(x,t)=-\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}} & \mathop{\tanh}_{\phi}(\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi))}\xi), \\ {{v}_{17}}(x,t)=-\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}} & \mathop{\coth}_{\phi}(\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi))}\xi), \\ {{v}_{18}}(x,t)=-\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}}[ & \mathop{\tanh}_{\phi}(\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{b(\ln (\phi))}\xi) \\ {} & \pm i\sqrt{rs}\sec {{h}_{\phi}}(\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{b(\ln (\phi))}\xi)], \\\end{array}\] v19(x,t)=−−θb0(θb0+2b+2c)2b(ln(ϕ))ω3[cothϕ(−θb0(θb0+2b+2c)b(ln(ϕ))ξ)±rscschϕ(−θb0(θb0+2b+2c)b(ln(ϕ))ξ)],v20(x,t)=−−θb0(θb0+2b+2c)2b(ln(ϕ))ω3[tanhϕ(−θb0(θb0+2b+2c)4b(ln(ϕ))ξ)+cothϕ(−θb0(θb0+2b+2c)4b(ln(ϕ))ξ)]⋅ \[\begin{array}{*{35}{l}} {{v}_{19}}(x,t)=-\frac{-\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}}[ & \mathop{\coth}_{\phi}(\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{b(\ln (\phi))}\xi) \\ {} & \pm \sqrt{rs}\csc {{h}_{\phi}}(\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{b(\ln (\phi))}\xi)], \\ {{v}_{20}}(x,t)=-\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{2b(\ln (\phi)){{\omega}_{3}}}[ & \mathop{\tanh}_{\phi}(\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{4b(\ln (\phi))}\xi) \\ {} & +\mathop{\coth}_{\phi}(\frac{\sqrt{-\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}}{4b(\ln (\phi))}\xi)]\cdot \\\end{array}\]

Case 5: If ω₁ = ω₃ and ω₂ = 0, then (37) v21(x,t)=tanhϕ(θb0(θb0+2b+2c)4b2(ln(ϕ))2ω3ξ),v22(x,t)=−cotϕ(θb0(θb0+2b+2c)4b2(ln(ϕ))2ω3ξ),v23(x,t)=tanϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ)±irssecϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ),v24(x,t)=−cotϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ)±rscscϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ),v25(x,t)=12[tanϕ(θb0(θb0+2b+2c)8b2(ln(ϕ))2ω3ξ)−cotϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ)]⋅ \[\begin{array}{*{35}{l}} {{v}_{21}}(x,t) & =\mathop{\tanh}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{4{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi), \\ {{v}_{22}}(x,t) & =-\mathop{\cot}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{4{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi), \\ {{v}_{23}}(x,t) & =\mathop{\tan}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi)\pm i\sqrt{rs}\mathop{\sec}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi), \\ {{v}_{24}}(x,t) & =-\mathop{\cot}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi)\pm \sqrt{rs}\mathop{\csc}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi), \\ {{v}_{25}}(x,t) & =\frac{1}{2}[\mathop{\tan}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{8{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi)-\mathop{\cot}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi)]\cdot \\\end{array}\]

Case 6: When ω₃ = −ω₁ and ω₂ = 0, then (38) v26(x,t)=−tanhϕ(θb0(θb0+2b+2c)4b2(ln(ϕ))2ω3ξ),v27(x,t)=−cothϕ(θb0(θb0+2b+2c)4b2(ln(ϕ))2ω3ξ),v28(x,t)=−tanhϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ)±ipqsechϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ),v29(x,t)=−cothϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ)±pqcschϕ(θb0(θb0+2b+2c)2b2(ln(ϕ))2ω3ξ),v30(x,t)=−12[tanhϕ(θb0(θb0+2b+2c)8b2(ln(ϕ))2ω3ξ)−cothϕ(θb0(θb0+2b+2c)8b2(ln(ϕ))2ω3ξ)]⋅ \[\begin{array}{*{35}{l}} {{v}_{26}}(x,t) & =-\mathop{\tanh}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{4{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi), \\ {{v}_{27}}(x,t) & =-\mathop{\coth}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{4{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi), \\ {{v}_{28}}(x,t) & =-\mathop{\tanh}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi)\pm i\sqrt{pq}\sec {{h}_{\phi}}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi), \\ {{v}_{29}}(x,t) & =-\mathop{\coth}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi)\pm \sqrt{pq}\csc {{h}_{\phi}}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi), \\ {{v}_{30}}(x,t) & =-\frac{1}{2}[\mathop{\tanh}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{8{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi)-\mathop{\coth}_{\phi}(\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{8{{b}^{2}}{{(\ln (\phi))}^{2}}{{\omega}_{3}}}\xi)]\cdot \\\end{array}\]

Case 7: When ω22=4ω1ω3 \[\omega _{2}^{2}=4{{\omega}_{1}}{{\omega}_{3}}\] , then (39) v31(x,t)=−θb0(θb0+2c+2b)[(θb0+c+b)ξ](ln(ϕ))2b(θb0+b+c)2ξ(ln(ϕ))2ω3⋅ \[{{v}_{31}}(x,t)=-\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2c+2b)[(\theta {{b}_{0}}+c+b)\xi ](\ln (\phi))}{2b{{(\theta {{b}_{0}}+b+c)}^{2}}\xi {{(\ln (\phi))}^{2}}{{\omega}_{3}}}\cdot \]

Case 8: When ω₂ = k, ω₁ = mk(m ≠ 0), and ω₃ = 0, then (40) v32(x,t)=ϕkξ−p⋅ \[{{v}_{32}}(x,t)={{\phi}^{k\xi}}-p\cdot \]

Case 9: When ω₂ = ω₃ = 0, then (41) v33(x,t)=θb0(θb0+2b+2c)2b2(ln(ϕ))ω3ξ⋅ \[{{v}_{33}}(x,t)=\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2b+2c)}{2{{b}^{2}}(\ln (\phi)){{\omega}_{3}}}\xi \cdot \]

Case 10: When ω₁ = ω₂ = 0, then (42) v34(x,t)=−1ω3ρln(ϕ)⋅ \[{{v}_{34}}(x,t)=\frac{-1}{{{\omega}_{3}}\rho \ln (\phi)}\cdot \]

Case 11: When ω₁ = 0 and ω₂ ≠ 0, then (43) v35 (x,t )=r (θb0+b+c )bω3 (ln(ϕ))coshϕ(− (θb0+b+c )b(ln (ϕ))ξ )+sinhϕ(− (θb0+b+c )b(ln (ϕ))ξ )+r,v36 (x,t )= (θb0+b+c )bω3 (ln(ϕ))(coshϕ (−(θb0+b+c )b(ln (ϕ))ξ )+sinhϕ(− (θb0+b+c )b(ln (ϕ))ξ))coshϕ(− (θb0+b+c )b(ln (ϕ))ξ )+sinhϕ(− (θb0+b+c )b(ln (ϕ))ξ )+s, \[\begin{array}{*{35}{l}} {{v}_{35}}(x,t) & =\frac{\frac{r(\theta {{b}_{0}}+b+c)}{b{{\omega}_{3}}(\ln (\phi))}}{\mathop{\cosh}_{\phi}(-\frac{(\theta {{b}_{0}}+b+c)}{b(\ln (\phi))}\xi)+\mathop{\sinh}_{\phi}(-\frac{(\theta {{b}_{0}}+b+c)}{b(\ln (\phi))}\xi)+r}, \\ {{v}_{36}}(x,t) & =\frac{\frac{(\theta {{b}_{0}}+b+c)}{b{{\omega}_{3}}(\ln (\phi))}(\mathop{\cosh}_{\phi}(-\frac{(\theta {{b}_{0}}+b+c)}{b(\ln (\phi))}\xi)+\mathop{\sinh}_{\phi}(-\frac{(\theta {{b}_{0}}+b+c)}{b(\ln (\phi))}\xi))}{\mathop{\cosh}_{\phi}(-\frac{(\theta {{b}_{0}}+b+c)}{b(\ln (\phi))}\xi)+\mathop{\sinh}_{\phi}(-\frac{(\theta {{b}_{0}}+b+c)}{b(\ln (\phi))}\xi)+s}, \\\end{array}\]

Case 12: When ω₁ = 0, ω₂ = k and ω₃ = mk(m ≠ 0), then (44) v37(x,t)=rϕkξs−prϕkξ, \[{{v}_{37}}(x,t)=\frac{r{{\phi}^{k\xi}}}{s-pr{{\phi}^{k\xi}}},\] where ξ = x − μt.

The OBBMB equation is given by vt−vxxt−αvxx+γvx+θvvx=0. \[{{v}_{t}}-{{v}_{xxt}}-\alpha {{v}_{xx}}+\gamma {{v}_{x}}+\theta v{{v}_{x}}=0.\]

In the above equation, the parameters α and θ are real. We use the transformation v(x,t) = U (ξ), ξ = x − μt in (4) then simplifying we obtain (45) (−μ+γ)U−αU′+θ2U2+μU″=0. \[(-\mu +\gamma)U-\alpha {U}'+\frac{\theta}{2}{{U}^{2}}+\mu {{U}''}=0.\]

Now we will discuss the traveling wave structure of the OBBMB equation by using equation (45) with the help of the NEDA method. Comparing U² with the last term U″, we have the solution of the form (46) η(ξ)=b0+b1h(ξ)+b2h2(ξ). \[\eta (\xi)={{b}_{0}}+{{b}_{1}}h(\xi)+{{b}_{2}}{{h}^{2}}(\xi).\]

Let us suppose that h(ξ) is the solution of the equation, then (47) h′(ξ)=(lnϕ)[ω1+ω2h(ξ)+ω3h2(ξ)]. \[{h}'(\xi)=(\ln \phi)[{{\omega}_{1}}+{{\omega}_{2}}h(\xi)+{{\omega}_{3}}{{h}^{2}}(\xi)].\]

Substituting (46) and (47) into the (45), and then comparing the different powers of h(ξ), we obtain a system of algebraic expressions. Solving this system for ω₁, ω₂, b₁, b₂ we get the set mentioned below (48) ω1=θb0(θb0+2γ)4α2ω3(ln(ϕ))2, ω2=−θb0+γαln(ϕ), b1=−2(ln(ϕ))αω3θ, b2=0, \[{{\omega}_{1}}=\frac{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}{4{{\alpha}^{2}}{{\omega}_{3}}{{(\ln (\phi))}^{2}}}, {{\omega}_{2}}=-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi)}, {{b}_{1}}=-\frac{2(\ln (\phi))\alpha {{\omega}_{3}}}{\theta}, {{b}_{2}}=0,\]

Case 1: When Δ < 0 and ω₃ ≠ 0, then (49) v1(x,t)=−θb0+γαln(ϕ)ω3+−Δ2ω3tanϕ(−Δ2ξ),v2(x,t)=−θb0+γαln(ϕ)ω3−−Δ2ω3cotϕ(−Δ2ξ),v3(x,t)=−θb0+γαln(ϕ)ω3+−Δ2ω3(tanϕ(−Δξ)±rssecϕ(−Δξ)),v4(x,t)=−θb0+γαln(ϕ)ω3−−Δ2ω3(cotϕ(−Δξ)±rscscϕ(−Δξ)),v5(x,t)=−θb0+γαln(ϕ)ω3+−Δ4ω3(tanϕ(−Δ4ξ)−cotϕ(−Δ4ξ)), \[\begin{array}{*{35}{l}} {{v}_{1}}(x,t) & =-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi){{\omega}_{3}}}+\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}\mathop{\tan}_{\phi}(\frac{\sqrt{-\Delta}}{2}\xi), \\ {{v}_{2}}(x,t) & =-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi){{\omega}_{3}}}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}\mathop{\cot}_{\phi}(\frac{\sqrt{-\Delta}}{2}\xi), \\ {{v}_{3}}(x,t) & =-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi){{\omega}_{3}}}+\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}(\mathop{\tan}_{\phi}(\sqrt{-\Delta}\xi)\pm \sqrt{rs}\mathop{\sec}_{\phi}(\sqrt{-\Delta}\xi)), \\ {{v}_{4}}(x,t) & =-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi){{\omega}_{3}}}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}(\mathop{\cot}_{\phi}(\sqrt{-\Delta}\xi)\pm \sqrt{rs}\mathop{\csc}_{\phi}(\sqrt{-\Delta}\xi)), \\ {{v}_{5}}(x,t) & =-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi){{\omega}_{3}}}+\frac{\sqrt{-\Delta}}{4{{\omega}_{3}}}(\mathop{\tan}_{\phi}(\frac{\sqrt{-\Delta}}{4}\xi)-\mathop{\cot}_{\phi}(\frac{\sqrt{-\Delta}}{4}\xi)), \\\end{array}\] where Δ=ω22−ω1ω3 \[\Delta =\omega _{2}^{2}-{{\omega}_{1}}{{\omega}_{3}}\] , and r, s are constants greater than zero and called deformation parameters.

Case 2: When Δ > 0 and ω₃ ≠ 0, then (50) v6(x,t)=−θb0+γαln(ϕ)ω3−−Δ2ω3tanhϕ(−Δ2ξ),v7(x,t)=−θb0+γαln(ϕ)ω3−−Δ2ω3cothϕ(−Δ2ξ),v8(x,t)=−θb0+γαln(ϕ)ω3−−Δ2ω3(tanhϕ(−Δξ)±irssechϕ(−Δξ)),v9(x,t)=θb0+b+c2bω3ln(ϕ)−−Δ2ω3(cothϕ(−Δξ)±pqcschϕ(−Δξ)),v10(x,t)=θb0+b+c2bω3ln(ϕ)−−Δ4ω3(tanhϕ(−Δ4ξ)+cothϕ(−Δ4ξ))⋅ \[\begin{array}{*{35}{l}} {{v}_{6}}(x,t) & =-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi){{\omega}_{3}}}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}\mathop{\tanh}_{\phi}(\frac{\sqrt{-\Delta}}{2}\xi), \\ {{v}_{7}}(x,t) & =-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi){{\omega}_{3}}}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}\mathop{\coth}_{\phi}(\frac{\sqrt{-\Delta}}{2}\xi), \\ {{v}_{8}}(x,t) & =-\frac{\theta {{b}_{0}}+\gamma}{\alpha \ln (\phi){{\omega}_{3}}}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}(\mathop{\tanh}_{\phi}(\sqrt{-\Delta}\xi)\pm i\sqrt{rs}\sec {{h}_{\phi}}(\sqrt{-\Delta}\xi)), \\ {{v}_{9}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{-\Delta}}{2{{\omega}_{3}}}(\mathop{\coth}_{\phi}(\sqrt{-\Delta}\xi)\pm \sqrt{pq}\csc {{h}_{\phi}}(\sqrt{-\Delta}\xi)), \\ {{v}_{10}}(x,t) & =\frac{\theta {{b}_{0}}+b+c}{2b{{\omega}_{3}}\ln (\phi)}-\frac{\sqrt{-\Delta}}{4{{\omega}_{3}}}(\mathop{\tanh}_{\phi}(\frac{\sqrt{-\Delta}}{4}\xi)+\mathop{\coth}_{\phi}(\frac{\sqrt{-\Delta}}{4}\xi))\cdot \\\end{array}\]

Case 3: When ω₁ω₃ > 0 and ω₂ = 0, then (51) v11(x,t)=θb0(θb0+2γ)2α(ln(ϕ))ω3tanϕ(θb0(θb0+2γ)2α(ln(ϕ))ξ),v12(x,t)=−θb0(θb0+2γ)2α(ln(ϕ))ω3cotϕ(θb0(θb0+2γ)2α(ln(ϕ))ξ),v13(x,t)=θb0(θb0+2γ)2α(ln(ϕ))ω3[tanϕ(θb0(θb0+2γ)α(ln(ϕ))ξ)±rssecϕ(θb0(θb0+2γ)α(ln(ϕ))ξ)],v14(x,t)=θb0(θb0+2γ)2α(ln(ϕ))ω3[−cotϕ(θb0(θb02+2γ)α(ln(ϕ))ξ)±rscscϕ(θb0(θb0+2γ)α(ln(ϕ))ξ)],v15(x,t)=θb0(θb0+2γ)4α(ln(ϕ))ω3[tanϕ(θb0(θb0+2γ)4α(ln(ϕ))ξ)−cotϕ(θb0(θb0+2γ)4α(ln(ϕ))ξ)]⋅ \[\begin{array}{*{35}{l}} {{v}_{11}}(x,t) & =\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{2\alpha (\ln (\phi)){{\omega}_{3}}}\mathop{\tan}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{2\alpha (\ln (\phi))}\xi), \\ {{v}_{12}}(x,t) & =-\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{2\alpha (\ln (\phi)){{\omega}_{3}}}\mathop{\cot}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{2\alpha (\ln (\phi))}\xi), \\ {{v}_{13}}(x,t) & =\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{2\alpha (\ln (\phi)){{\omega}_{3}}}[\mathop{\tan}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{\alpha (\ln (\phi))}\xi)\pm \sqrt{rs}\mathop{\sec}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{\alpha (\ln (\phi))}\xi)], \\ {{v}_{14}}(x,t) & =\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{2\alpha (\ln (\phi)){{\omega}_{3}}}[-\mathop{\cot}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}2+2\gamma)}}{\alpha (\ln (\phi))}\xi)\pm \sqrt{rs}\mathop{\csc}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{\alpha (\ln (\phi))}\xi)], \\ {{v}_{15}}(x,t) & =\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{4\alpha (\ln (\phi)){{\omega}_{3}}}[\mathop{\tan}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{4\alpha (\ln (\phi))}\xi)-\mathop{\cot}_{\phi}(\frac{\sqrt{\theta {{b}_{0}}(\theta {{b}_{0}}+2\gamma)}}{4\alpha (\ln (\phi))}\xi)]\cdot \\\end{array}\]