Some New Families of Exact Solitary Wave Solutions for Pseudo-Parabolic Type Nonlinear Models

,


Introduction
Nonlinear problems have always been of interest to researchers [1][2][3] due to various applications to practical problems.While several approximate and numerical methods are available, analytical solutions always provide a benchmark for such methods.Several ansatz-based methods like mapping method [4,5], the Jacobi elliptic function method [6], the new extended direct algebraic method [7], the tanh-coth method [8], the simple equation method [9], the symmetry method [10,11], and many others are used for handling exact solutions to nonlinear PDEs.Pseudo-parabolic equations, which appear in many branches of mathematics and physics, have a one-time derivative in the highest-order term.Tey arise, for example, in the study of the fow of fuid in fractured rocks, the consolidation of clay, the shear of second-order fuids, thermodynamics, and the propagation of long, lowamplitude waves.
To solve a nonlinear pseudo-parabolic equation, a numerical approach has been established [12].Te stability of numerical approximations to backward-time parabolic and pseudo-parabolic problems and a relationship between parabolic and pseudo-parabolic diference schemes were also explored, along with a relationship between parabolic and pseudo-parabolic diference schemes.Te approach suggested by Sobolev [13] may be used to identify the source of new issues in mathematical physics, and [14] explained why these problems are referred to as pseudo-parabolic problems.Pseudo-parabolic equations apply to the study of a variety of signifcant physical phenomena, including the seepage of homogeneous fuids through a fssured rock [15], the accumulation of populations, and the conduction of heat between bodies kept at two diferent temperatures.Tey are characterized by the occurrence of a time derivative appearing in the highest-order term [16].Such equations, which comprise the nonlinear pseudo-parabolic diferential, are used in many branches of physics and mathematics to explain many physical implications [17].In this direction, Camassa [18] has obtained some soliton solutions for the pseudo-parabolic equations.Wazwaz [19,20] has considered analytical solutions to the same class of equations.Johnson [21] discussed some models arising from water waves.Te authors [22,23] discussed the trigonometric and hyperbolictype soliton solutions of the same class.To provide readers with additional information, reference [24] is cited.
Te generalized pseudo-parabolic equations are one specifc instance of this class.We consider the following generalized form of Benjamin Bona Mahony equation: where g(q) is a C 2 -smooth nonlinear function, α is a positive constant, c is a real constant, and q(x, t) represents the velocity of fuid in the horizontal direction.Peregrine [25] and Benjamin et al. [26] suggested the regular long-wave equation for the widely used KdV equation as the specifc case of g(q) � qq x with α � 0, c � 1 in equation ( 1), i.e., Te equation for BBMPB is obtained by substituting g(q) � θqq x + βq xxx in the following equation (1) to get Te following equation is obtained for α � β � 0 in the above equation, which is the following BBM equation: in which c, θ ∈ R and θ ≠ 0 is a parameter.We also consider the one-dimensional OSK equation which models incompressible viscoelastic Kelvin-Voigt fuid [27] q t − λq xxt − αq xx + qq x � 0. ( Tis article aims to fnd new families of exact soliton solutions for the nonlinear pseudo-parabolic type models arising in mathematical physics using the Exp (-ϕ(ξ))-expansion method [28][29][30].Tis method is an extremely powerful tool for dealing with soliton solutions of nonlinear PDEs.It provides the hyperbolic, trigonometric, rational, exponential, and polynomial functions-based soliton solutions to the nonlinear PDEs.Tis is the actual limitation of the used methodology here.For the more general soliton solutions, we need to enhance our methodology also.Akcagil et al. [31] reported the solutions to the class of trigonometric, hyperbolic, and rational soliton solutions.We want to build on earlier research to advance our quest for a wealth of fresh traveling wave solutions.Our fndings include the dark, bright, rational, exponential, polynomial, and solitary wave solutions.Te solutions provided in this research are exclusively novel and valid and have not been previously presented for this class of equations.
Te structure of this paper is as follows: In Section 2, we present a layout for the Exp (-ϕ(ξ))-expansion method.In Sections 3-5, the BBMPB, the 1D OSK and BBM equations respectively have been studied to obtain various solutions using this method.Te graphical representations of these solutions are presented in Section 6.In Section 7 the conclusion and some possible directions of future study are mentioned.

Floor Plan for the Exp ( − ϕ(ξ))-Expansion Method
Tis section deals with the brief foor plan for the Exp (− ϕ(ξ))-expansion method [28][29][30] to fnd the explicit soliton solutions.We give here the main steps of the method.
We consider an explicit form of the nonlinear PDE as follows: P q, q x , q t , q xt . . . � 0, where q is the dependent variable.
Step 1.To reduce the number of independent variables of the equation ( 6), we introduce the following traveling wave transformation: where w is a nonzero real parameter indicating wave speed.Ten, by adopting the traveling wave transformation, the nonlinear PDE (6) becomes Note that for cases, the invariance of the transformation (7) serves as the existence criterion for the traveling wave solution.Also, the term traveling wave is due to the time behavior of the dependent variable.
Step 2. Te general solution to (8) is appropriated as a polynomial in exp (− ϕ(ξ)) where a i (0 < i ≤ M) are the coefcients to be determined later, and ϕ(ξ) is the solution of the following equation: where a i , λ 1 , and λ 2 are real constant parameters and M can be determined by the homogeneous balance principle.Tere are fve cases for the ϕ(ξ); Case II: For λ 2 1 − 4λ 2 < 0 and λ 1 ≠ 0, Case III: For λ 1 � 0 and λ 2 ≠ 0, Case IV: For Step 3. Substituting the equation ( 9) into (8), the left hand side of the ODE (8) becomes a polynomial in e − ϕ (ξ) .By comparing the coefcients of both sides, we get a system of algebraic equations that is solvable by some symbolic software like Mathematica 11.3 or Maple.
Step 4. Putting the values of ϕ(ξ) from equations ( 11)- (15) one by one in (9), we get solutions for ODE (8).Replacing ξ by x − wt, we get solutions for our PDE (6).Now we are going to apply this method for some important nonlinear PDEs to have explicit soliton solutions.

The Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) Equation
Here, we frst look at the equation BBMPB given by By using traveling wave transformation q(x, t) � V(ξ), ξ � x − wt, we obtain the following nonlinear ODE: By comparing the highest order of linear term V ″ with the highest degree of nonlinear term V 2 in (17), we decide the order of V as O(V) � 2. Based on this order, we deduce that the solution of ( 17) is of the following type: Such that ϕ(ξ) satisfes (10).By substituting (18) into (17), we transform the right-hand side of the equation into a polynomial in e − ϕ(ξ) , and then by comparing coefcients the following system arises:

Journal of Mathematics
We solve this system with Mathematica 11.3 to get the following set of parameters.
Tus, the solutions for (10) becomes

The One-Dimensional Oskolkov (OSK) Equation
Te second model, OSK, is given by By using the wave transformation q(x, t), ξ � x − wt, we obtain the following nonlinear ODE: By comparing the highest order of linear term V ″ with the highest degree of nonlinear term V 2 in (57), we decide the order of V as O(V) � 2. Based on this order, we deduce that the solution of (57) is of the type.

The Benjamin-Bona-Mahony (BBM) Equation
Te third equation, BBM, is given by By using the traveling wave transformation q(x, t) � V(ξ), ξ � x − wt, we obtain the following nonlinear ODE: By comparing the highest order of linear term V ″ with the highest degree of nonlinear term V 2 in (73), we decide the order of V as O(V) � 2. Based on this order, we deduce that the solution of ( 73) is of the following type: Such that ϕ(ξ) satisfes (10).By substituting (74) into (73), we transform the right-hand side of the equation into a polynomial in e − ϕ(ξ) , and then by comparing coefcients the following system arises: 10

Journal of Mathematics
We solve this system with Mathematica 11.3 for desired constants that lead to the following two sets of parameters.

Graphical Structures of Some Solitons
Tis section ofers a brief graphical summary of the solutions to the pseudo-parabolic equations discussed here.Te wave profle of the BBMPB equation, the BBM equation, and the OSK equation are the main topics of our discussion.Our newly developed families of soliton solutions for nonlinear pseudo-parabolic models represent hyperbolic, trigonometric, rational, exponential, and polynomial functions.Te waveform characteristics conform to the properties of some known solitons including the solitary waves, dark, bright, rational, exponential, and polynomial solutions provided in this research.For a particular set of parameters that are listed alongside, Wolfram Mathematica 11.3 simulations were used to create all of these visualizations.Te plots include 3D, 2D, contour plots, and density plots.Te plots of the solutions are shown in Figures 1-5.

Discussion of the Obtained Results
Using the Exp (− ϕ(ξ))-expansion method, new solitonic families for nonlinear pseudo-parabolic type models have been efectively found.Our focus was on the BBMPB equation, the BBM equation, and the OSK equation.In this study, we have successfully derived the new families of soliton solutions for the nonlinear pseudo-parabolic models.We have obtained the solutions hyperbolic, trigonometric, rational, exponential, and polynomial functions.Te dynamics of the solutions show that the obtained solitons are solitary waves, dark, bright, rational, exponential, and polynomial functions-based.Te results obtained from the authors [22,23] contain only dark and bright solitons.So,

. Conclusions
To integrate the pseudo-parabolic type equations in this study, new applications of the Exp (-ϕ(ξ))-expansion method were used.It proved to be efective to apply the Exp (-ϕ(ξ))-expansion method to fnd new analytical solutions to the pseudo-parabolic equations.Tis technique establishes the solutions of the pseudo-parabolic equations in terms of hyperbolic, trigonometric, rational, exponential, and polynomial functions.Previously, only hyperbolic function-based solutions were given by the authors [22,23], while the solutions found in [31] were trigonometric, hyperbolic, and rational functions.Tese solutions exhibited the characteristics of dark and bright soliton solutions.However, our results showed that the solitary waves, dark, bright, rational, exponential, and polynomial solutions provided in this research are exclusively novel and valid and have not been previously presented for this class of equations.Tese precise solutions capture the dynamics of various soliton wave shapes, and they may be used to evaluate, compare, and numerical studies in the area.Additionally, the approach utilized in this work can be applied to other mathematical and physics-related problems.It appears that more study is necessary for the advancement of fresh, efective analytical techniques for solving partial diferential equations.By exploiting the improved capacities of such techniques, more problems in science and engineering that occur in the real world can be solved efectively.Tis can serve as one of the fnest incentives for scientists to concentrate more on this remarkable area of study.