New Solutions for the Generalized BBM Equation in terms of Jacobi and Weierstrass Elliptic Functions

Salas, Alvaro H.; Martinez H, Lorenzo J.; Ocampo R, David L.

doi:https://doi.org/10.1155/2021/5513266

Abstract and Applied Analysis

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Abstract Introduction Analysis and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5513266 | https://doi.org/10.1155/2021/5513266

New Solutions for the Generalized BBM Equation in terms of Jacobi and Weierstrass Elliptic Functions

Alvaro H. Salas,¹Lorenzo J. Martinez H,²and David L. Ocampo R²

Academic Editor: J.-C. Cortés

Received06 Jan 2021

Revised30 Mar 2021

Accepted08 Apr 2021

Published23 Apr 2021

Abstract

The Jacobi elliptic function method is applied to solve the generalized Benjamin-Bona-Mahony equation (BBM). Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. A power series method is also applied in some particular cases. Some solutions are expressed in terms of the Weierstrass elliptic function.

1. Introduction

The regularized long-wave (RLW) equation is a famous nonlinear wave equation which gives the phenomena of dispersion and weak nonlinearity, including magneto hydrodynamic wave in plasma, phonon packets in nonlinear crystals, and nonlinear transverse waves in shallow water or in ion acoustic. This equation is also called the BBM (Benjamin-Bona-Mahony) equation and reads

It describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems, and it was proposed by Benjamin et al. in 1972 [1] as a more satisfactory model than the KdV equation [2]:

It is easy to see that Equation (1) can be derived from the equal width equation [3]:

by means of the change of variable , that is, by replacing with . This last equation is considered an equally valid and accurate model for the same wave phenomena simulated by (1) and (2). On the other hand, some researches analyzed the generalized equation with variable coefficients

because this model has important applications in several fields of science. Motivated by these facts, we will consider here the generalized equal width (RW) equation with constant coefficients

Our aim is to present solutions different from those in [2, 4–6].

1.1. Trigonometric and Soliton Solutions

In order to obtain new solutions to Equation (5), we make the traveling wave transformation

Inserting (6) into (5) gives

Integrating once with respect to to get where is the constant of integration. Now, we multiply Equation (8) by , and we integrate the resulting equation with respect to : where is the constant of integration. If Equation (9) takes the form

Let

Inserting Equation (11) into (10), we obtain

From Equation (12), if , then

A sech solution to Equation (13) is

It is clear that function is a solution to equation for any choice of the parameters and .

Finally, if , then taking into account that we obtain from (15) the trigonometric solution

2. The Novel Solutions

In this section, we will give analytical solutions to Equation (5) for two special cases: and First of all, observe that Equation (5) is equivalent to Equation (9).

2.1. The Case

If , Equation (9) reads

To solve this equation, we consider the reduced Duffing equation where the coefficient does not depend on . Observe that and integrating this equation w.r.t. gives

Suppose that is a solution to Equation (19) and let

We have

Taking into account Equations (18) and (23), we must have

Equating to zero the coefficients of , , and , we conclude that

We have proved that the general solution to the nonlinear ODE may be written in the form , where is a solution to the Duffing equation

According to [2], the general solution to Equation (29) may be expressed in terms of the Jacobian elliptic functions. More exactly, the general solution to Equation (19) is

The value of is determined by solving equation . In principle, we may consider that is any number as well as . Taking into account these facts, we have proved that the general solution to Equation (28) is

Finally, an exact solution to the EW-equation is

Solution (32) involves four arbitrary constants: , , , and (these do not depend on ). We choose so that .

2.2. The Case

This case corresponds to the so-called modified BBM or modified equal width equation. When , Equation (9) reads

This equation is harder to solve. We seek solution to Equation (33) in the form

Observe that function satisfies the nonlinear differential equation for any constants and . This equation is Duffing equation with coefficients and . The relationship between the frequency , the modulus , and the coefficients is given by the system

Solving this system for and gives

To solve the modified BBM equation, we will make use of the power series method (PSM). This is a promising method that also may be applicable to obtain an approximate solution for those ODE’s that do not admit solution in closed form. In our case, we consider the function where

The values of and are obtained from the initial conditions and , that is,

Thus,

Inserting the ansatz (39) into (38), we obtain an expression for in terms of the unknowns , , , and . Observe that function given by (39) is a solution to Equation (33), if and only if The series method consists in coefficients of Taylor series of around some point. Usually, this is the origin. Since the calculations are enormous, we cannot realize this work by hand. Instead, we may use Mathematica 12 or Maple 16. The Mathematica command has the form Series . This gives us the th degree Taylor polynomial of around the point . In our case, we will use the command Series for , since we need at least four equations to determine the five unknowns , , , and in terms of the coefficients of Equation (33). Of course, the series method is applicable to solve the BBM when . Our aim is to show two different methods. Let us proceed. We have that for all and where

Solving this last equation for and taking into account (45), (47), and (49), we obtain after long algebraic calculations following solution:

Direct calculations show that function defined by (39) with parameters given by (51)-(55) and (41) is a solution to Equation (41). Then, function is a solution to the modified BBM equation

Observe that It may be verified that the following function is a solution to Equation (56): for the choice

We obtain soliton solutions for and trigonometric solutions for

2.3. Solution for a Generalized BBM Equation

Let us consider the following generalized BBM equation:

In order to obtain new solutions to Equation (59), we make the travelling wave transformation

Inserting (60) into (59) gives

Equation (61) is hard to integrate. We seek for an exact solution of the form

Inserting ansatz (62) into (61) gives , where being

Equating to zero the coefficients of gives a nonlinear algebraic system. Solving it with the aid of Wolfram Mathematica, we obtain

Other solutions may be obtained choosing from the condition

Then, a solution to (61) is where , and are obtained from (65). The obtained solution is valid for

We have solved the generalized BBM equation for any such that The exact traveling wave solution is given by

The numbers and are arbitrary. Particular cases are (1)(2)(3)

2.4. Forbidden Values

Let us examine the cases when

Case 1 (). For this value, Equation (59) takes the form After doing a traveling wave transformation, we get the linear ode whose general solution is given by

Case 2 (). Making the traveling wave transformation and making use of the ansatz (62), we obtain that , which is a trivial solution. So, in the cases and , we cannot get nontrivial solutions to the generalized BBM equation.

3. Analysis and Discussion

We obtained the traveling wave solutions to BBM equation and modified BBM equation Other solutions were obtained in [7] using the extended Jacobi elliptic function expansion method. However, in our approach, we got nonzero integration constants. Most authors get zero integration constants to simplify matters, but this may cause loss of solutions. We also may consider the following combined BBM equation:

Making the traveling wave transformation , gives

We now integrate once with respect to , and denoting the constant of integration by , we obtain the ode

This last equation is a Duffing-Helmholtz equation. The general form of the undamped and unforced Duffing-Helmholtz equation reads

The general solution to Equation (79) may be expressed in the ansatz form

Indeed, given the initial conditions the solution to the initial value problem (80)-(81) is given by (80), where

The number is a solution to the quartic

In soliton theory, we are interested in sech or tanh solutions to Equation (79). The soliton solutions may occur only when

In this case, we will have the following soliton solution to Equation (79):

Let us examine condition (85) for (78). We must have

Solving this equation for , we will have the following soliton solution to equation (18):

In (88), the constants and are arbitrary. In the case when , Equation (18) converts into usual BBM equation

In a more general fashion, we solved Equation (59). This equation was solved in [2, 5, 8] using the substitution . In the present paper, we solved it by using the substitution . Let us examine the odes obtained by means of these different substitutions. Using gives

On the other hand, letting gives a different ode given by (61). So, our method gives other than that obtained in [2, 5, 8] solutions.

The authors in [5] claim that there are no periodic solutions for the parameter values of different form , , or . However, we obtained periodic solutions for .

On the other hand, we may solve the ode (89) by means of the ansatz (62). That was the approach the author employed in [8] by means of the substitution . Indeed, let us consider the equation [8]:

Let . Then, the ode (90) converts into the ode

In order to solve equation (91), we will assume the ansatz form (62). Plugging this ansatz into (91), we get where

Solving the system gives the following solution:

Thus, a traveling wave solution to Equation (90) reads

In a similar way, we may consider the following more general equation than that considered in [2]:

In [9], the author solved the following generalization of the BBM equation:

Let us consider the following more general variant of BBM equation:

We again may use the ansatz (62) to solve this equation. Making the traveling wave transformation gives the ode

Plugging the ansatz (62) into the ode (99) and equating to zero the coefficients of will give a very large algebraic system. After some hard algebraic calculations, we get the following solution:

The forbidden values are . The soliton solutions may be obtained under the condition

For , we have the following soliton solution to Equation (98):

In the case when , we have the following solution to Equation (98):

Finally, more solutions may be obtained by means of the transformation . We will not derive them here.

4. Conclusions

We successfully obtained exact solutions to regularized long-wave and generalized BBM equation by using different approaches. We showed the way to derive all traveling wave solutions to all known until now variants of BBM. We compared the known previously published solutions with the solutions obtained in this work. Other methods to find exact solutions to nonlinear differential equations may be found in [1–3, 7, 9–39].

Data Availability

Data statement is not applicable.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors aknowledge Dr. S.A. Tantawy at Al Baha University for additional comments.

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Copyright © 2021 Alvaro H. Salas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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