Research Article | Open Access

Jiangbo Zhou, Lixin Tian, Xinghua Fan, "Soliton and Periodic Wave Solutions to the Osmosis Equation", *Mathematical Problems in Engineering*, vol. 2009, Article ID 509390, 10 pages, 2009. https://doi.org/10.1155/2009/509390

# Soliton and Periodic Wave Solutions to the Osmosis Equation

**Academic Editor:**Shijun Liao

#### Abstract

Two types of traveling wave solutions to the osmosis *K*(2, 2) equation
are investigated. They are characterized by two parameters. The expresssions for
the soliton and periodic wave solutions are obtained.

#### 1. Introduction

In 1993, Rosenau and Hyman [1] introduced a genuinely nonlinear dispersive equation, a special type of KdV equation, of the form

where is a constant and both the convection term and the dispersion effect term are nonlinear. These equations arise in the process of understanding the role of nonlinear dispersion in the formation of structures like liquid drops. Rosenau and Hyman derived solutions called compactons to (1.1) and showed that while compactons are the essence of the focusing branch where , spikes, peaks, and cusps are the hallmark of the defocusing branch where which also supports the motion of kinks. Further, the negative branch, where , was found to give rise to solitary patterns having cusps or infinite slopes. The focusing branch and the defocusing branch represent two different models, each leading to a different physical structure. Many powerful methods were applied to construct the exact solutions to (1.1), such as Adomain method [2], homotopy perturbation method [3], Exp-function method [4], variational iteration method [5], and variational method [6, 7]. In [8], Wazwaz studied a generalized forms of (1.1), that is equations and defined by

where are constants. He showed how to construct compact and noncompact solutions to (1.2) and discussed it in higher-dimensional spaces in [9]. Chen et al. [10] showed how to construct the general solutions and some special exact solutions to (1.2) in higher-dimensional spatial domains. He et al. [11] considered the bifurcation behavior of traveling wave solutions to (1.2). Under different parametric conditions, smooth and nonsmooth periodic wave solutions, solitary wave solutions, and kink and antikink wave solutions were obtained. Yan [12] further extended (1.2) to be a more general form

And using some direct ansatze, some abundant new compacton solutions, solitary wave solutions and periodic wave solutions to (1.3) were obtained. By using some transformations, Yan [13] obtained some Jacobi elliptic function solutions to (1.3). Biswas [14] obtained 1-soliton solution of equation with the generalized evolution term

where are constants, while and are positive integers. Zhu et al. [15] applied the decomposition method and symbolic computation system to develop some new exact solitary wave solutions to the equation

and the equation

Recently, Xu and Tian [16] introduced the osmosis equation

where the positive convection term means the convection moves along the motion direction, and the negative dispersive term denotes the contracting dispersion. They obtained the peaked solitary wave solution and the periodic cusp wave solution to (1.7). In [17], the authors obtained the smooth soliton solutions to (1.7). In this paper, following Vakhnenko and Parkes's strategy [18, 19] we continue to investigate the traveling wave solutions to (1.7) and obtain soliton and periodic wave solutions. Our work in this paper covers and extends the results in [16, 17] and may help people to know deeply the described physical process and possible applications of the osmosis equation.

The remainder of this paper is organized as follows. In Section 2, for completeness and readability, we repeat [19, Appendix A], which discusses the solutions to a first-order ordinary differential equaion. In Section 3, we show that, for traveling wave solutions, (1.7) 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 [19, Appendix A]). For completeness and readability, we state it in the following.

Consider solutions to the following ordinary differential equation:

where

and , , , are chosen to be real constants with .

Following [20] 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 [21]. Its parametric form is

with as the parameter, where

In (2.5) is a Jacobian elliptic function, where the notation is as used in [22, Chapter 16]. is the elliptic integral of the third kind and the notation is as used in [22, 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 the solution to (2.4) is found using result 255.00 in [21]. 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. Solitary and Periodic Wave Solutions to (1.7)

Equation (1.7) can also be written in the form

Let with be a traveling wave solution to (3.1), then it follows that

where is the derivative of function with respect to .

Integrating (3.2) twice with respect to yields

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 .

Without loss of generality, we suppose the wave speed . In the following, suppose that and for each value , such that has three distinct stationary points: , , and comprise two minimums separated by a maximum. Under this assumption, (1.7) 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 loop-like solutions to (3.3) given by (2.10) so that , and with wavelength given by (2.12). See Figure 2(a) for an example.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(g)**

**(h)**

**(a)**

**(b)**

**(c)**

**(d)**

The case and (a corresponding curve of is shown in Figure 1(b)) corresponds to the limit so that , and then the solution is a loop-like solitary wave given by (2.13) with and

See Figure 3(a) for an example.

**(a)**

**(b)**

**(c)**

**(d)**

In this case and . For each value and (a corresponding curve of is shown in Figure 1(c)), there are periodic valley-like solutions to (3.3) given by (2.10) so that , and with wavelength given by (2.12). 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 (2.13) with and given by the roots of , namely,

In this case we obtain a weak solution, namely, the periodic downward-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 valley-like solutions to (3.3) given by (2.5) so that , and with wavelength given by (2.8). 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 valley-like peaked solution with amplitude . See Figure 3(c) for an example.

(4)In this case and . For each value and (a corresponding curve of is shown in Figure 1(g)), there are periodic valley-like solutions to (3.3) given by (2.5) so that , and with wavelength given by (2.8). 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 velley-like solitary wave given by (2.10) with and

See Figure 3(d) for an example.

#### 4. Conclusion

In this paper, we have found expressions for two types of traveling wave solutions to the osmosis equation, that is, the soliton and periodic wave solutions. These solutions depend, in effect, on two parameters and . For , there are loop-like (), peakon (), and smooth () soliton solutions. For or and , there are periodic wave solutions.

#### Acknowledgments

The authors are deeply grateful to the anonymous referees for careful reading of this paper and constructive comments. J. Zhou acknowledges funding from the Startup Fund for Advanced Talents of Jiangsu University (no. 09JDG013), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003) and Jiangsu Planned Projects for Postdoctoral Research Funds. L. Tian was partially supported by the National Natural Science Foundation of China (no. 10771088). X. Fan was supported by the Postdoctoral Foundation of China (no. 20080441071), the Postdoctoral Foundation of Jiangsu Province (no. 0802073c) and the High-level Talented Person Special Subsidizes of Jiangsu University (no. 08JDG013).

#### References

- P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,”
*Physical Review Letters*, vol. 70, no. 5, pp. 564–567, 1993. View at: Publisher Site | Google Scholar - A. M. Wazwaz, “Compactons and solitary patterns structures for variants of the KdV and the KP equations,”
*Applied Mathematics and Computation*, vol. 139, no. 1, pp. 37–54, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 6, no. 2, pp. 207–208, 2005. View at: Google Scholar - J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,”
*Chaos, Solitons & Fractals*, vol. 30, no. 3, pp. 700–708, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He and X.-H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,”
*Chaos, Solitons & Fractals*, vol. 29, no. 1, pp. 108–113, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He, “Some asymptotic methods for strongly nonlinear equations,”
*International Journal of Modern Physics B*, vol. 20, no. 10, pp. 1141–1199, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Xu, “Variational approach to solitons of nonlinear dispersive $K(m,n)$
equations,”
*Chaos, Solitons & Fractals*, vol. 37, no. 1, pp. 137–143, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. M. Wazwaz, “General compactons solutions and solitary patterns solutions for modified nonlinear dispersive equations $mK(n,n)$ in higher dimensional spaces,”
*Mathematics and Computers in Simulation*, vol. 59, no. 6, pp. 519–531, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. M. Wazwaz, “Compact and noncompact structures for a variant of KdV equation in higher dimensions,”
*Applied Mathematics and Computation*, vol. 132, no. 1, pp. 29–45, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Chen, B. Li, and H. Q. Zhang, “New exact solutions for modified nonlinear dispersive equations $mK(m,n)$
in higher dimensions spaces,”
*Mathematics and Computers in Simulation*, vol. 64, no. 5, pp. 549–559, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. He, Q. Meng, W. Rui, and Y. Long, “Bifurcations of travelling wave solutions for the $mK(n,n)$ equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 13, no. 10, pp. 2114–2123, 2008. View at: Publisher Site | Google Scholar | MathSciNet - Z. Y. Yan, “Modified nonlinearly dispersive $mK(m,n,k)$
equations. I. New compacton solutions and solitary pattern solutions,”
*Computer Physics Communications*, vol. 152, no. 1, pp. 25–33, 2003. View at: Google Scholar | MathSciNet - Z. Y. Yan, “Modified nonlinearly dispersive $mK(m,n,k)$
equations. II. Jacobi elliptic function solutions,”
*Computer Physics Communications*, vol. 153, no. 1, pp. 1–16, 2003. View at: Google Scholar | MathSciNet - A. Biswas, “1-soliton solution of the $K(m,n)$
equation with generalized evolution,”
*Physics Letters A*, vol. 372, no. 25, pp. 4601–4602, 2008. View at: Publisher Site | Google Scholar | MathSciNet - Y. G. Zhu, K. Tong, and T. C. Lu, “New exact solitary-wave solutions for the $K(2,2,1)$ and $K(3,3,1)$ equations,”
*Chaos, Solitons & Fractals*, vol. 33, no. 4, pp. 1411–1416, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Xu and L. Tian, “The bifurcation and peakon for $K(2,2)$
equation with osmosis dispersion,”
*Chaos, Solitons & Fractals*, vol. 40, no. 2, pp. 893–901, 2009. View at: Publisher Site | Google Scholar - J. Zhou and L. Tian, “Soliton solution of the osmosis $K(2,2)$ equation,”
*Physics Letters A*, vol. 372, no. 41, pp. 6232–6234, 2008. View at: Publisher Site | Google Scholar | MathSciNet - V. O. Vakhnenko and E. J. Parkes, “Explicit solutions of the Camassa-Holm equation,”
*Chaos, Solitons & Fractals*, vol. 26, no. 5, pp. 1309–1316, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. O. Vakhnenko and E. J. Parkes, “Periodic and solitary-wave solutions of the Degasperis-Procesi equation,”
*Chaos, Solitons & Fractals*, vol. 20, no. 5, pp. 1059–1073, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. J. Parkes, “The stability of solutions of Vakhnenko's equation,”
*Journal of Physics A*, vol. 26, no. 22, pp. 6469–6475, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. F. Byrd and M. D. Friedman,
*Handbook of Elliptic Integrals for Engineers and Scientists*, Die Grundlehren der Mathematischen Wissenschaften, vol. 67, Springer, Berlin, Germany, 2nd edition, 1971. View at: MathSciNet - M. Abramowitz and I. A. Stegun,
*Handbook of Mathematical Functions*, Dover, New York, NY, USA, 1972.

#### Copyright

Copyright © 2009 Jiangbo Zhou 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.