Research Article | Open Access
Soliton and Periodic Wave Solutions to the Osmosis Equation
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.
In 1993, Rosenau and Hyman  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 , homotopy perturbation method , Exp-function method , variational iteration method , and variational method [6, 7]. In , 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 . Chen et al.  showed how to construct the general solutions and some special exact solutions to (1.2) in higher-dimensional spatial domains. He et al.  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  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  obtained some Jacobi elliptic function solutions to (1.3). Biswas  obtained 1-soliton solution of equation with the generalized evolution term
where are constants, while and are positive integers. Zhu et al.  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  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 , 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:
and , , , are chosen to be real constants with .
Following  we introduce defined by
so that (2.1) becomes
with as the parameter, where
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
where are as in (2.6), and
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.
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.
See Figure 3(a) for an example.
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.
In this case we obtain a weak solution, namely, the periodic downward-cusp wave
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.
See Figure 3(d) for an example.
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.
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).
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