Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 507815 | 10 pages | https://doi.org/10.1155/2009/507815

Periodic and Solitary Wave Solutions to the Fornberg-Whitham Equation

Academic Editor: Katica R. (Stevanovic) Hedrih
Received27 Nov 2008
Revised04 Feb 2009
Accepted06 Mar 2009
Published02 Jun 2009

Abstract

New travelling wave solutions to the Fornberg-Whitham equation are investigated. They are characterized by two parameters. The expresssions for the periodic and solitary wave solutions are obtained.

1. Introduction

Recently, Ivanov [1] investigated the integrability of a class of nonlinear dispersive wave equations: where and are real constants.

The important cases of (1.1) are as follows. The hyperelastic-rod wave equation has been recently studied as a model, describing nonlinear dispersive waves in cylindrical compressible hyperelastic rods [27]. The physical parameters of various compressible materials put in the range from –29.4760 to 3.4174 [2, 4].

The Camassa-Holm equation describes the unidirectional propagation of shallow water waves over a flat bottom [8, 9]. It is completely integrable [1] and admits, in addition to smooth waves, a multitude of travelling wave solutions with singularities: peakons, cuspons, stumpons, and composite waves [912]. The solitary waves of (1.2) are smooth if and peaked if [9, 10]. Its solitary waves are stable solitons [13, 14], retaining their shape and form after interactions [15]. It models wave breaking [1618].

The Degasperis-Procesi equation models nonlinear shallow water dynamics. It is completely integrable [1] and has a variety of travelling wave solutions including solitary wave solutions, peakon solutions and shock waves solutions [1926].

The Fornberg-Whitham equation appeared in the study qualitative behaviors of wave-breaking [27]. It admits a wave of greatest height, as a peaked limiting form of the travelling wave solution [28], , where is an arbitrary constant. It is not completely integrable [1].

The regularized long-wave or BBM equation and the modified BBM equation have also been investigated by many authors [2937].

Many efforts have been devoted to study (1.2)–(1.4), (1.6), and (1.7), however, little attention was paid to study (1.5). In [38], we constructed two types of bounded travelling wave solutions to (1.5), which are defined on semifinal bounded domains and called kink-like and antikink-like wave solutions. In this paper, we continue to study the travelling wave solutions to (1.5). Following Vakhnenko and Parkes's strategy in [39], we obtain some periodic and solitary wave solutions to (1.5) which are defined on . The travelling wave solutions obtained in this paper are obviously different from those obtained in our previous work [38]. To the best of our knowledge, these solutions are new for (1.5). Our work may help people to know deeply the described physical process and possible applications of the Fornberg-Whitham equation.

The remainder of the paper is organized as follows. In Section 2, for completeness and readability, we repeat Appendix A in [39], which discusses the solutions to a first-order ordinary differential equaion. In Section 3, we show that, for travelling wave solutions, (1.5) may be reduced to a first-order ordinary differential equation involving two arbitrary integration constants and . We show that there are four distinct periodic solutions corresponding to four different ranges of values of and restricted ranges of values of . A short conclusion is given in Section 4.

2. Solutions to a First-Order Ordinary Differential Equaion

This section is due to Vakhnenko and Parkes (see Appendix A in [39]). For completeness and readability, we repeat it in the following.

Consider solutions to the following ordinary differential equation where and , , , are chosen to be real constants with .

Following [40] we introduce defined by so that (2.1) becomes

Equation(2.4) has two possible forms of solution. The first form is found using result 254.00 in [41]. Its parametric form is with as the parameter, where In (2.5) is a Jacobian elliptic function, where the notation is as used in [42, Chapter 16], and the notation is as used in [42, Section 17.2.15].

The solution to (2.1) is given in parametric form by (2.5) with as the parameter. With respect to , in (2.5) is periodic with period , where is the complete elliptic integral of the first kind. It follows from (2.5) that the wavelength of the solution to (2.1) is where is the complete elliptic integral of the third kind.

When , , (2.5) becomes

The second form of solution of (2.5) is found using result 255.00 in [41]. Its parametric form is where are as in (2.6), and

The solution to (2.1) is given in parametric form by (2.10) with as the parameter. The wavelength of the solution to (2.1) is

When , , (2.10) becomes

3. Periodic and Solitary Wave Solutions to Equaion (1.5)

Equation (1.5) can also be written in the form Let with be a travelling wave solution to (3.1), then it follows that Integrating (3.2) twice with respect to , we have where and are two arbitrary integration constants.

Equation (3.3) is in the form of (2.1) with and . For convenience we define and by and define , , , and by Obviously, , are the roots of .

In the following, suppose that and such that has three distinct stationary points: , , and comprise two minimums separated by a maximum. Under this assumption, (3.3) has periodic and solitary wave solutions that have different analytical forms depending on the values of and as follows.

(1)

In this case and . For each value and (a corresponding curve of is shown in Figure 1(a)), there are periodic inverted loop-like solutions to (3.3) given by (2.5) so that , and with wavelength given by (2.8); see Figure 2(a), for an example.

The case and (a corresponding curve of is shown in Figure 1(b)) corresponds to the limit so that , and then the solution is an inverted loop-like solitary wave given by (2.9) with and see Figure 3(a), for an example.

(2)

In this case and . For each value and (a corresponding curve of is shown in Figure 1(c)), there are periodic hump-like solutions to (3.3) given by (2.5) so that , and with wavelength given by (2.8); see Figure 2(b), for an example.

The case and (a corresponding curve of is shown in Figure 1(d)) corresponds to the limit so that , and then the solution can be given by (2.9) with and given by the roots of , namely In this case we obtain a weak solution, namely, the periodic upward-cusp wave where see Figure 3(b), for an example.

(3)

In this case and . For and each value (a corresponding curve of is shown in Figure 1(e)), there are periodic hump-like solutions to (3.3) given by (2.10) so that , and with wavelength given by (2.12); see Figure 2(c), for an example.

The case and (a corresponding curve of is shown in Figure 1(f)) corresponds to the limit and so that . In this case neither (2.9) nor (2.13) is appropriate. Instead we consider (3.3) with and note that the bound solution has . On integrating (3.3) and setting at we obtain a weak solution that is, a single peakon solution with amplitude , see Figure 3(c).

(4)

In this case and . For each value and (a corresponding curve of is shown in Figure 1(g)), there are periodic hump-like solutions to (3.3) given by (2.10) so that , and with wavelength given by (2.12); see Figure 2(d), for an example.

The case and (a corresponding curve of is shown in Figure 1(h)) corresponds to the limit so that , and then the solution is a hump-like solitary wave given by (2.13) with and see Figure 3(d), for an example.

On the above, we have obtained expressions of parametric form for periodic and solitary wave solutions to (3.3). So in terms of , we can get expressions for the periodic and solitary wave solutions to (1.5).

4. Conclusion

In this paper, we have found expressions for new travelling wave solutions to the Fornberg-Whitham equation. These solutions depend, in effect, on two parameters and . For , there are inverted loop-like (), single peaked (), and hump-like () solitary wave solutions. For or , and , there are periodic hump-like wave solutions.

Acknowledgments

The authors are deeply grateful to an anonymous referee for the important comments and suggestions. Zhou acknowledges funding from Startup Fund for Advanced Talents of Jiangsu University (No. 09JDG013). Tian's work was partially supported by NSF of China (No. 90610031).

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Copyright © 2009 Jiangbo Zhou and Lixin Tian. 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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