Two types of traveling wave solutions to the osmosis K(2, 2) equation ut+(u2)x−(u2)xxx=0
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

ut+a(un)x+(un)xxx=0,n>1,
where a is a constant and both the convection term (un)x and the dispersion effect term (un)xxx 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 a>0, spikes, peaks, and cusps are the hallmark of the defocusing branch where a<0 which also supports the motion of kinks. Further, the negative branch, where a<0, 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 mK(n,n) equations and defined by

un-1ut+a(un)x+b(un)xxx=0,n>1,
where a,b 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

um-1ut+a(un)x+b(uk)xxx=0,nk≠1.
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

(ul)t+a(um)ux+b(un)xxx=0,
where a,b are constants, while l,m, and n 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 K(2,2,1) equation

ut+(u2)x-(u2)xxx+uxxxxx=0,
and the K(3,3,1) equation

ut+(u3)x-(u3)xxx+uxxxxx=0.
Recently, Xu and Tian [16] introduced the osmosis K(2,2) equation

ut+(u2)x-(u2)xxx=0,
where the positive convection term (u2)x means the convection moves along the motion direction, and the negative dispersive term (u2)xxx 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 K(2,2) 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 a and b. We show that there are four distinct periodic solutions corresponding to four different ranges of values of a and restricted ranges of values of b. 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:

(φφξ)2=ε2f(φ),
where

f(φ)=(φ-φ1)(φ-φ2)(φ3-φ)(φ4-φ),
and φ1, φ2, φ3, φ4 are chosen to be real constants with φ1≤φ2≤φ≤φ3≤φ4.

Following [20] we introduce ζ defined by

dξdζ=φε,
so that (2.1) becomes

(φζ)2=f(φ).

Equation (2.4) has two possible forms of solution. The first form is found using result 254.00 in [21]. Its parametric form is

φ=φ2-φ1nsn2(w∣m)1-nsn2(w∣m),ξ=1εp(wφ1+(φ2-φ1)∏(n;w∣m)),
with w as the parameter, where

m=(φ3-φ2)(φ4-φ1)(φ4-φ2)(φ3-φ1),p=12(φ4-φ2)(φ3-φ1),w=pζ,n=φ3-φ2φ3-φ1.
In (2.5) sn(w∣m) is a Jacobian elliptic function, where the notation is as used in [22, Chapter 16]. Π(n;w∣m) 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 w as the parameter. With respect to w, φ in (2.5) is periodic with period 2K(m), where K(m) is the complete elliptic integral of the first kind. It follows from (2.5) that the wavelength λ of the solution to (2.1) is

λ=2εp|φ1K(m)+(φ2-φ1)∏(n∣m)|,
where Π(n∣m) is the complete elliptic integral of the third kind.

3. Solitary and Periodic Wave Solutions to (<xref ref-type="disp-formula" rid="EEq1.7">1.7</xref>)

Equation (1.7) can also be written in the form

ut+2uux-6uxuxx-2uuxxx=0.
Let u=φ(ξ)+c with ξ=x-ct be a traveling wave solution to (3.1), then it follows that

-cφξ+2φφξ-6φξφξξ-2φφξξξ=0,
where φξ is the derivative of function φ with respect to ξ.

Integrating (3.2) twice with respect to ξ yields

(φφξ)2=14(φ4-4c3φ3+aφ2+b),
where a and b are two arbitrary integration constants.

Equation (3.3) is in the form of (2.1) with ε=1/2 and f(φ)=(φ4-(4c/3)φ3+aφ2+b). For convenience we define g(φ) and h(φ) by

f(φ)=φ2g(φ)+b,whereg(φ)=φ2-4c3φ+a,f′(φ)=2φh(φ),whereh(φ)=2φ2-2cφ+a,
and define φL, φR, bL, and bR by

φL=12(c-c2-2a),φR=12(c+c2-2a),bL=-φL2g(φL)=a24-12c2a+c46-16(c3-2ac)c2-2a,bR=-φL2g(φL)=a24-12c2a+c46+16(c3-2ac)c2-2a.
Obviously, φL, φR are the roots of h(φ)=0.

Without loss of generality, we suppose the wave speed c>0. In the following, suppose that a<c2/2 and a≠0 for each value c>0, such that f(φ) has three distinct stationary points: φL, φR, 0 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 a and b as follows.

a<0

In this case φL<0<φR and f(φL)>f(φR). For each value a<0 and 0<b<bL (a corresponding curve of f(φ) is shown in Figure 1(a)), there are periodic loop-like solutions to (3.3) given by (2.10) so that 0<m<1, and with wavelength given by (2.12). See Figure 2(a) for an example.

The curve of f(φ) for the wave speed c=2. (a) a=-40, b=200; (b) a=-40, b=226.0424; (c) a=1.5, b=-0.05; (d) a=1.5, b=0; (e) a=16/9, b=-0.1; (f) a=16/9, b=0; (g) a=17/9, b=-0.24; (h) a=17/9, b=-0.1842.

Periodic solutions to (3.3) with 0<m<1 and the wave speed c=2. (a) a=-40, b=200 so m=0.8978; (b) a=1.5, b=-0.05 so m=0.6893; (c) a=16/9, b=-0.1 so m=0.8254; (d) a=17/9, b=-0.24 so m=0.8412.

The case a<0 and b=bL (a corresponding curve of f(φ) is shown in Figure 1(b)) corresponds to the limit φ1=φ2=φL so that m=1, and then the solution is a loop-like solitary wave given by (2.13) with φ2≤φ<φR and

φ3=12c2-2a+c6-13c2+3c4-2a,φ4=12c2-2a+c6+13c2+3c4-2a.
See Figure 3(a) for an example.

Solutions to (3.3) with m=1 and the wave speed c=2. (a) a=-40, b=226.0424; (b) a=1.5, b=0; (c) a=16/9, b=0; (d) a=17/9, b=-0.1842.

0<a<4c2/9

In this case 0<φL<φR and f(φR)<f(0). For each value 0<a<4c2/9 and bL<b<0 (a corresponding curve of f(φ) is shown in Figure 1(c)), there are periodic valley-like solutions to (3.3) given by (2.10) so that 0<m<1, and with wavelength given by (2.12). See Figure 2(b) for an example.

The case 0<a<4c2/9 and b=0 (a corresponding curve of f(φ) is shown in Figure 1(d)) corresponds to the limit φ1=φ2=0 so that m=1, and then the solution can be given by (2.13) with φ3 and φ4 given by the roots of g(φ)=0, namely,

φ3=2c3-4c29-a,φ4=2c3+4c29-a.
In this case we obtain a weak solution, namely, the periodic downward-cusp wave

φ=φ(ξ-2jξm),(2j-1)ξm<ξ<(2j+1)ξm,j=0,±1,±2,…,
where

φ(ξ)=(φ3-φ4tanh2(ξ4))cosh2(ξ4),ξm=4tanh-1φ3φ4.
See Figure 3(b) for an example.

a=4c2/9

In this case 0<φL<φR and f(φR)=f(0). For a=4c2/9 and each value bL<b<0 (a corresponding curve of f(φ) is shown in Figure 1(e)), there are periodic valley-like solutions to (3.3) given by (2.5) so that 0<m<1, and with wavelength given by (2.8). See Figure 2(c) for an example.

The case a=4c2/9 and b=0 (a corresponding curve of f(φ) is shown in Figure 1(f)) corresponds to the limit φ3=φ4=φR=2c/3 and φ1=φ2=0 so that m=1. In this case neither (2.9) nor (2.13) is appropriate. Instead we consider (3.3) with f(φ)=(1/4)φ2(φ-(2c/3))2 and note that the bound solution has 0<φ<2c/3. On integrating (3.3) and setting φ=0 at ξ=0 we obtain a weak solution

φ=-2c3exp(-12|ξ|)+2c3,
that is, a single valley-like peaked solution with amplitude 2c/3. See Figure 3(c) for an example.

4c2/9<a<c2/2

In this case 0<φL<φR and f(φR)>f(0). For each value 4c2/9<a<c2/2 and bL<b<bR (a corresponding curve of f(φ) is shown in Figure 1(g)), there are periodic valley-like solutions to (3.3) given by (2.5) so that 0<m<1, and with wavelength given by (2.8). See Figure 2(d) for an example.

The case 4c2/9<a<c2/2 and b=bR (a corresponding curve of f(φ) is shown in Figure 1(h)) corresponds to the limit φ3=φ4=φR so that m=1, and then the solution is a velley-like solitary wave given by (2.10) with φL<φ≤φ3 and

φ1=c6-12c2-2a-13c2-3cc2-2a,φ2=c6-12c2-2a+13c2-3cc2-2a.
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 K(2,2) equation, that is, the soliton and periodic wave solutions. These solutions depend, in effect, on two parameters a and m. For m=1, there are loop-like (a<0), peakon (a=4c2/9), and smooth (4c2/9<a<c2/2) soliton solutions. For m=1,0<a<4c2/9 or 0<m<1,a<c2/2, and a≠0, 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).

RosenauP.HymanJ. M.Compactons: solitons with finite wavelengthWazwazA. M.Compactons and solitary patterns structures for variants of the KdV and the KP equationsHeJ.-H.Homotopy perturbation method for bifurcation of nonlinear problemsHeJ.-H.WuX.-H.Exp-function method for nonlinear wave equationsHeJ.-H.WuX.-H.Construction of solitary solution and compacton-like solution by variational iteration methodHeJ.-H.Some asymptotic methods for strongly nonlinear equationsXuL.Variational approach to solitons of nonlinear dispersive K(m,n)
equationsWazwazA. M.General compactons solutions and solitary patterns solutions for modified nonlinear dispersive equations mK(n,n) in higher dimensional spacesWazwazA. M.Compact and noncompact structures for a variant of KdV equation in higher dimensionsChenY.LiB.ZhangH. Q.New exact solutions for modified nonlinear dispersive equations mK(m,n)
in higher dimensions spacesHeB.MengQ.RuiW.LongY.Bifurcations of travelling wave solutions for the mK(n,n) equationYanZ. Y.Modified nonlinearly dispersive mK(m,n,k)
equations. I. New compacton solutions and solitary pattern solutionsYanZ. Y.Modified nonlinearly dispersive mK(m,n,k)
equations. II. Jacobi elliptic function solutionsBiswasA.1-soliton solution of the K(m,n)
equation with generalized evolutionZhuY. G.TongK.LuT. C.New exact solitary-wave solutions for the K(2,2,1) and K(3,3,1) equationsXuC.TianL.tianlx@ujs.edu.cnThe bifurcation and peakon for K(2,2)
equation with osmosis dispersionZhouJ.TianL.Soliton solution of the osmosis K(2,2) equationVakhnenkoV. O.ParkesE. J.Explicit solutions of the Camassa-Holm equationVakhnenkoV. O.ParkesE. J.Periodic and solitary-wave solutions of the Degasperis-Procesi equationParkesE. J.The stability of solutions of Vakhnenko's equationByrdP. F.FriedmanM. D.AbramowitzM.StegunI. A.