Periodic and Solitary Wave Solutions to the Fornberg-Whitham Equation

Abstract

New travelling wave solutions to the Fornberg-Whitham equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢_𝑥+𝑢𝑢_𝑥=𝑢𝑢_𝑥𝑥𝑥+3𝑢_𝑥𝑢_𝑥𝑥 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: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝜕_𝑥(𝜅𝑢+𝛼𝑢²+𝛽𝑢³)=𝜈𝑢_𝑥𝑢_𝑥𝑥+𝛾𝑢𝑢_𝑥𝑥𝑥, (1.1) where and 𝛼,𝛽,𝛾,𝜅,𝜈 are real constants.

The important cases of (1.1) are as follows. The hyperelastic-rod wave equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥=𝛾(2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥) (1.2) has been recently studied as a model, describing nonlinear dispersive waves in cylindrical compressible hyperelastic rods [2–7]. The physical parameters of various compressible materials put 𝛾 in the range from –29.4760 to 3.4174 [2, 4].

The Camassa-Holm equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥 (1.3) 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 [9–12]. The solitary waves of (1.2) are smooth if 𝜅>0 and peaked if 𝜅=0 [9, 10]. Its solitary waves are stable solitons [13, 14], retaining their shape and form after interactions [15]. It models wave breaking [16–18].

The Degasperis-Procesi equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+4𝑢𝑢_𝑥=3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (1.4) 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 [19–26].

The Fornberg-Whitham equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢_𝑥+𝑢𝑢_𝑥=𝑢𝑢_𝑥𝑥𝑥+3𝑢_𝑥𝑢_𝑥𝑥 (1.5) 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], 𝑢(𝑥,𝑡)=𝐴exp(−1/2|𝑥−4/3𝑡|), where 𝐴 is an arbitrary constant. It is not completely integrable [1].

The regularized long-wave or BBM equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢_𝑥+𝑢𝑢_𝑥=0 (1.6) and the modified BBM equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢_𝑥+3𝑢²𝑢_𝑥=0 (1.7) have also been investigated by many authors [29–37].

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 𝑢(𝜉)(𝜉=𝑥−𝑐𝑡) 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 𝑢(𝜉) 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 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 Appendix A in [39]). For completeness and readability, we repeat 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 [40] 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 [41]. Its parametric form is 𝜑𝜑=²−𝜑₁𝑛sn²(𝑤∣𝑚)1−𝑛sn²,1(𝑤∣𝑚)𝜉=𝜀𝑝(𝑤𝜑₁+(𝜑₂−𝜑₁)Π(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−𝜑₁𝜑),𝑤=𝑝𝜁,𝑛=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 [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 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)Π(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 solution of (2.5) is found using result 255.00 in [41]. Its parametric form is 𝜑𝜑=³−𝜑₄𝑛sn²(𝑤∣𝑚)1−𝑛sn²,1(𝑤∣𝑚)𝜉=𝜀𝑝(𝑤𝜑₄−(𝜑₄−𝜑₃)Π(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)Π(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. Periodic and Solitary Wave Solutions to Equaion (1.5)

Equation (1.5) can also be written in the form (𝑢_𝑡+𝑢𝑢_𝑥)_𝑥𝑥=𝑢_𝑡+𝑢𝑢_𝑥+𝑢_𝑥. (3.1) Let 𝑢=𝜑(𝜉)+𝑐 with 𝜉=𝑥−𝑐𝑡 be a travelling wave solution to (3.1), then it follows that (𝜑𝜑_𝜉)_𝜉𝜉=𝜑𝜑_𝜉+𝜑_𝜉. (3.2) Integrating (3.2) twice with respect to 𝜉, we have (𝜑𝜑_𝜉)²=14(𝜑⁴+83𝜑³+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+8/3𝜑³+𝑎𝜑²+𝑏). For convenience we define 𝑔(𝜑) and ℎ(𝜑) by 𝑓(𝜑)=𝜑²𝑔(𝜑)+𝑏,where𝑔(𝜑)=𝜑²+83𝜑+𝑎,𝑓'(𝜑)=2𝜑ℎ(𝜑),whereℎ(𝜑)=2𝜑²+4𝜑+𝑎, (3.4) and define 𝜑_𝐿, 𝜑_𝑅, 𝑏_𝐿, and 𝑏_𝑅 by 𝜑_𝐿1=−2(2+√4−2𝑎),𝜑_𝑅=−12(2−√𝑏4−2𝑎),_𝐿=−𝜑²_𝐿𝑔(𝜑_𝐿𝑎)=²48−2𝑎+3+23(2−𝑎)√𝑏4−2𝑎,_𝑅=−𝜑²_𝑅𝑔(𝜑_𝑅𝑎)=²48−2𝑎+3−23(2−𝑎)√4−2𝑎. (3.5) Obviously, 𝜑_𝐿, 𝜑_𝑅 are the roots of ℎ(𝜑)=0.

In the following, suppose that 𝑎<2 and 𝑎≠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, (3.3) 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_𝑅 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(a)), there are periodic inverted loop-like solutions to (3.3) given by (2.5) so that 0<𝑚<1, and with wavelength given by (2.8); see Figure 2(a), for an example.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1 

The curve of 𝑓(𝜑). (a) 𝑎=−50, 𝑏=233; (b) 𝑎=−50, 𝑏=374.1346; (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=1.9, 𝑏=−0.24; (h) 𝑎=1.9, 𝑏=−0.2010.

(a)

(b)

(c)

(d)

Figure 2 

Periodic solutions to (3.3) with 0<𝑚<1. (a) 𝑎=−50, 𝑏=233 so 𝑚=0.7885; (b) 𝑎=1.5, 𝑏=−0.05 so 𝑚=0.6893; (c) 𝑎=16/9, 𝑏=−0.1 so 𝑚=0.8254; (d) 𝑎=1.9, 𝑏=−0.24 so 𝑚=0.6121.

The case 𝑎<0 and 𝑏=𝑏_𝑅 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(b)) corresponds to the limit 𝜑₃=𝜑₄=𝜑_𝑅 so that 𝑚=1, and then the solution is an inverted loop-like solitary wave given by (2.9) with 𝜑₂≤𝜑<𝜑_𝑅 and 𝜑₁1=−6(2+3√4−2𝑎+24+6√𝜑4−2𝑎),₂1=−6(2+3√4−2𝑎−24+6√4−2𝑎); (3.6) see Figure 3(a), for an example.

(a)

(b)

(c)

(d)

Figure 3 

Solutions to (3.3) with 𝑚=1. (a) 𝑎=−50, 𝑏=374.1346; (b) 𝑎=1.5, 𝑏=0; (c) 𝑎=16/9, 𝑏=0; (d) 𝑎=1.9, 𝑏=−0.2010.

(2) 0<𝑎<16/9

In this case 𝜑_𝐿<𝜑_𝑅<0 and 𝑓(𝜑_𝐿)<𝑓(0). For each value 0<𝑎<16/9 and 𝑏_𝑅<𝑏<0 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(c)), there are periodic hump-like solutions to (3.3) given by (2.5) so that 0<𝑚<1, and with wavelength given by (2.8); see Figure 2(b), for an example.

The case 0<𝑎<16/9 and 𝑏=0 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(d)) corresponds to the limit 𝜑₃=𝜑₄=0 so that 𝑚=1, and then the solution can be given by (2.9) with 𝜑₁ and 𝜑₂ given by the roots of 𝑔(𝜑)=0, namely 𝜑₁4=−3−13√16−9𝑎,𝜑₂=−43+13√16−9𝑎. (3.7) In this case we obtain a weak solution, namely, the periodic upward-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)=(𝜑₂−𝜑₁tanh²(𝜉4))cosh²(𝜉4𝜉),_𝑚=4tanh⁻¹𝜑₂𝜑₁, (3.9)(3.10) see Figure 3(b), for an example.

(3) 𝑎=16/9

In this case 𝜑_𝐿<𝜑_𝑅<0 and 𝑓(𝜑_𝐿)=𝑓(0). For 𝑎=16/9 and each value 𝑏_𝑅<𝑏<0 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(e)), there are periodic hump-like solutions to (3.3) given by (2.10) so that 0<𝑚<1, and with wavelength given by (2.12); see Figure 2(c), for an example.

The case 𝑎=16/9 and 𝑏=0 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(f)) corresponds to the limit 𝜑₁=𝜑₂=𝜑_𝐿=−4/3 and 𝜑₃=𝜑₄=0 so that 𝑚=1. In this case neither (2.9) nor (2.13) is appropriate. Instead we consider (3.3) with f(𝜑)=1/4𝜑²(𝜑+4/3)² and note that the bound solution has −4/3<𝜑≤0. On integrating (3.3) and setting 𝜑=0 at 𝜉=0 we obtain a weak solution 4𝜑=31exp(−24|𝜉|)−3, (3.11) that is, a single peakon solution with amplitude 4/3, see Figure 3(c).

(4) 16/9<𝑎<2

In this case 𝜑_𝐿<𝜑_𝑅<0 and 𝑓(𝜑_𝐿)>𝑓(0). For each value 16/9<𝑎<2 and 𝑏_𝑅<𝑏<𝑏_𝐿 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(g)), there are periodic hump-like solutions to (3.3) given by (2.10) so that 0<𝑚<1, and with wavelength given by (2.12); see Figure 2(d), for an example.

The case 16/9<𝑎<2 and 𝑏=𝑏_𝐿 (a corresponding curve of 𝑓(𝜑) is shown in Figure 1(h)) corresponds to the limit 𝜑₁=𝜑₂=𝜑_𝐿 so that 𝑚=1, and then the solution is a hump-like solitary wave given by (2.13) with 𝜑_𝐿<𝜑≤𝜑₃ and 𝜑₃=16(−2+3√4−2𝑎−24−6√𝜑4−2𝑎),₄=16(−2+3√4−2𝑎+24−6√4−2𝑎), (3.12) see Figure 3(d), for an example.

On the above, we have obtained expressions of parametric form for periodic and solitary wave solutions 𝜑(𝜉) to (3.3). So in terms of 𝑢=𝜑(𝜉)+𝑐, we can get expressions for the periodic and solitary wave solutions 𝑢(𝜉) 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 𝑎 and 𝑚. For 𝑚=1, there are inverted loop-like (𝑎<0), single peaked (𝑎=16/9), and hump-like (16/9<𝑎<2) solitary wave solutions. For 𝑚=1,0<𝑎<16/9 or 0<𝑚<1,𝑎<2, and 𝑎≠0, 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).