Soliton and Periodic Wave Solutions to the Osmosis𝐾(2,2) Equation

Abstract

Two types of traveling wave solutions to the osmosis K(2, 2) equation 𝑢_𝑡+(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

𝑢_𝑡+𝑎(𝑢^𝑛)_𝑥+(𝑢^𝑛)_𝑥𝑥𝑥=0,𝑛>1, (1.1) where 𝑎 is a constant and both the convection term (𝑢^𝑛)_𝑥 and the dispersion effect term (𝑢^𝑛)_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

𝑢^𝑛−1𝑢_𝑡+𝑎(𝑢^𝑛)_𝑥+𝑏(𝑢^𝑛)_𝑥𝑥𝑥=0,𝑛>1, (1.2) 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

𝑢^𝑚−1𝑢_𝑡+𝑎(𝑢^𝑛)_𝑥+𝑏(𝑢^𝑘)_𝑥𝑥𝑥=0,𝑛𝑘≠1. (1.3) 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

(𝑢^𝑙)_𝑡+𝑎(𝑢^𝑚)𝑢_𝑥+𝑏(𝑢^𝑛)_𝑥𝑥𝑥=0, (1.4) 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

𝑢_𝑡+(𝑢²)_𝑥−(𝑢²)_𝑥𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑥𝑥}=0, (1.5) and the 𝐾(3,3,1) equation

𝑢_𝑡+(𝑢³)_𝑥−(𝑢³)_𝑥𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑥𝑥}=0. (1.6) Recently, Xu and Tian [16] introduced the osmosis 𝐾(2,2) equation

𝑢_𝑡+(𝑢²)_𝑥−(𝑢²)_𝑥𝑥𝑥=0, (1.7) 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:

(𝜑𝜑_𝜉)²=𝜀²𝑓(𝜑), (2.1) where

𝑓(𝜑)=(𝜑−𝜑₁)(𝜑−𝜑₂)(𝜑₃−𝜑)(𝜑₄−𝜑), (2.2) and 𝜑₁, 𝜑₂, 𝜑₃, 𝜑₄ are chosen to be real constants with 𝜑₁≤𝜑₂≤𝜑≤𝜑₃≤𝜑₄.

Following [20] we introduce 𝜁 defined by

𝑑𝜉=𝜑𝑑𝜁𝜀, (2.3) so that (2.1) becomes

(𝜑_𝜁)²=𝑓(𝜑). (2.4)

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

𝜑𝜑=²−𝜑₁𝑛sn²(𝑤∣𝑚)1−𝑛sn²,1(𝑤∣𝑚)𝜉=𝜀𝑝(𝑤𝜑₁+(𝜑₂−𝜑₁)(𝑛;𝑤∣𝑚)), (2.5) with 𝑤 as the parameter, where

𝑚=(𝜑₃−𝜑₂)(𝜑₄−𝜑₁)(𝜑₄−𝜑₂)(𝜑₃−𝜑₁)1,𝑝=2(𝜑₄−𝜑₂)(𝜑₃−𝜑₁),𝑤=𝑝𝜁,𝜑𝑛=³−𝜑₂𝜑₃−𝜑₁. (2.6)(2.7) 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

2𝜆=𝜀𝑝|𝜑₁𝐾(𝑚)+(𝜑₂−𝜑₁)(𝑛∣𝑚)|, (2.8) where Π(𝑛∣𝑚) is the complete elliptic integral of the third kind.

When 𝜑₃=𝜑₄, 𝑚=1, (2.5) becomes

𝜑𝜑=²−𝜑₁𝑛tanh²𝑤1−𝑛tanh²𝑤,1𝜉=𝜀(𝑤𝜑₃𝑝−2tanh⁻¹(√𝑛tanh𝑤)). (2.9)

The second form of the solution to (2.4) is found using result 255.00 in [21]. Its parametric form is

𝜑𝜑=³−𝜑₄𝑛sn²(𝑤∣𝑚)1−𝑛sn²,1(𝑤∣𝑚)𝜉=𝜀𝑝(𝑤𝜑₄−(𝜑₄−𝜑₃)(𝑛;𝑤∣𝑚)), (2.10) where 𝑚,𝑝,𝑤 are as in (2.6), and

𝜑𝑛=³−𝜑₂𝜑₄−𝜑₂. (2.11)

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

2𝜆=𝜀𝑝|𝜑₄𝐾(𝑚)−(𝜑₄−𝜑₃)(𝑛∣𝑚)|. (2.12)

When 𝜑₁=𝜑₂, 𝑚=1, (2.10) becomes

𝜑𝜑=³−𝜑₄𝑛tanh²𝑤1−𝑛tanh²𝑤,1𝜉=𝜀(𝑤𝜑₂𝑝+2tanh⁻¹(√𝑛tanh𝑤)). (2.13)

3. Solitary and Periodic Wave Solutions to (1.7)

Equation (1.7) can also be written in the form

𝑢_𝑡+2𝑢𝑢_𝑥−6𝑢_𝑥𝑢_𝑥𝑥−2𝑢𝑢_𝑥𝑥𝑥=0. (3.1) Let 𝑢=𝜑(𝜉)+𝑐 with 𝜉=𝑥−𝑐𝑡 be a traveling wave solution to (3.1), then it follows that

−𝑐𝜑_𝜉+2𝜑𝜑_𝜉−6𝜑_𝜉𝜑_𝜉𝜉−2𝜑𝜑_𝜉𝜉𝜉=0, (3.2) where 𝜑_𝜉 is the derivative of function 𝜑 with respect to 𝜉.

Integrating (3.2) twice with respect to 𝜉 yields

(𝜑𝜑_𝜉)²=14(𝜑⁴−4𝑐3𝜑³+𝑎𝜑²+𝑏), (3.3) 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)=𝜑²−4𝑐3𝑓𝜑+𝑎,_′(𝜑)=2𝜑ℎ(𝜑),whereℎ(𝜑)=2𝜑²−2𝑐𝜑+𝑎, (3.4) and define 𝜑_𝐿, 𝜑_𝑅, 𝑏_𝐿, and 𝑏_𝑅 by

𝜑_𝐿=12(𝑐−√𝑐²−2𝑎),𝜑_𝑅=12(𝑐+√𝑐²𝑏−2𝑎),_𝐿=−𝜑²_𝐿𝑔(𝜑_𝐿𝑎)=²4−12𝑐²𝑐𝑎+⁴6−16(𝑐³−2𝑎𝑐)√𝑐²𝑏−2𝑎,_𝑅=−𝜑²_𝐿𝑔(𝜑_𝐿𝑎)=²4−12𝑐²𝑐𝑎+⁴6+16(𝑐³−2𝑎𝑐)√𝑐²−2𝑎. (3.5) Obviously, 𝜑_𝐿, 𝜑_𝑅 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: 𝜑_𝐿, 𝜑_𝑅, 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.

(1)𝑎<0

In this case 𝜑_𝐿<0<𝜑_𝑅 and 𝑓(𝜑_𝐿)>𝑓(𝜑_𝑅). For each value 𝑎<0 and 0<𝑏<𝑏_𝐿 (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.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1 

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.

(a)

(b)

(c)

(d)

Figure 2 

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

𝜑₃=12√𝑐²−2𝑎+𝑐6−13𝑐²+3𝑐√𝜑4−2𝑎,₄=12√𝑐²−2𝑎+𝑐6+13𝑐²+3𝑐√4−2𝑎. (3.6) See Figure 3(a) for an example.

(a)

(b)

(c)

(d)

Figure 3 

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.

(2)0<𝑎<4𝑐²/9

In this case 0<𝜑_𝐿<𝜑_𝑅 and 𝑓(𝜑_𝑅)<𝑓(0). For each value 0<𝑎<4𝑐²/9 and 𝑏_𝐿<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. (3.7) In this case we obtain a weak solution, namely, the periodic downward-cusp wave

𝜑=𝜑(𝜉−2𝑗𝜉_𝑚),(2𝑗−1)𝜉_𝑚<𝜉<(2𝑗+1)𝜉_𝑚,𝑗=0,±1,±2,…, (3.8) where

𝜑(𝜉)=(𝜑₃−𝜑₄tanh²(𝜉4))cosh²(𝜉4𝜉),_𝑚=4tanh⁻¹𝜑₃𝜑₄. (3.9) See Figure 3(b) for an example.

(3)𝑎=4𝑐²/9

In this case 0<𝜑_𝐿<𝜑_𝑅 and 𝑓(𝜑_𝑅)=𝑓(0). For 𝑎=4𝑐²/9 and each value 𝑏_𝐿<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=𝜑_𝑅=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=−2𝑐31exp(−2|𝜉|)+2𝑐3, (3.10) that is, a single valley-like peaked solution with amplitude 2𝑐/3. See Figure 3(c) for an example.

(4)4𝑐²/9<𝑎<𝑐²/2

In this case 0<𝜑_𝐿<𝜑_𝑅 and 𝑓(𝜑_𝑅)>𝑓(0). For each value 4𝑐²/9<𝑎<𝑐²/2 and 𝑏_𝐿<𝑏<𝑏_𝑅 (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=𝑏_𝑅 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(h)) corresponds to the limit 𝜑₃=𝜑₄=𝜑_𝑅 so that 𝑚=1, and then the solution is a velley-like solitary wave given by (2.10) with 𝜑_𝐿<𝜑≤𝜑₃ and

𝜑₁=𝑐6−12√𝑐²−2𝑎−13𝑐²−3𝑐√𝑐²𝜑−2𝑎,₂=𝑐6−12√𝑐²−2𝑎+13𝑐²−3𝑐√𝑐²−2𝑎. (3.11) 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).