New travelling wave solutions to the Fornberg-Whitham equation ut−uxxt+ux+uux=uuxxx+3uxuxx 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:
ut-uxxt+∂x(κu+αu2+βu3)=νuxuxx+γuuxxx,
where and α,β,γ,κ,ν are real constants.

The important cases of (1.1) are as follows. The hyperelastic-rod wave equation
ut-uxxt+3uux=γ(2uxuxx+uuxxx)
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
ut-uxxt+3uux=2uxuxx+uuxxx
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
ut-uxxt+4uux=3uxuxx+uuxxx,
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
ut-uxxt+ux+uux=uuxxx+3uxuxx
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
ut-uxxt+ux+uux=0
and the modified BBM equation
ut-uxxt+ux+3u2ux=0
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
(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 [40] 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 [41]. Its parametric form is
φ=φ2-φ1nsn2(w∣m)1-nsn2(w∣m),ξ=1εp(wφ1+(φ2-φ1)Π(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 [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
λ=2εp|φ1K(m)+(φ2-φ1)Π(n∣m)|,
where Π(n∣m) is the complete elliptic integral of the third kind.

When φ3=φ4, m=1, (2.5) becomes
φ=φ2-φ1ntanh2w1-ntanh2w,ξ=1ε(wφ3p-2tanh-1(ntanhw)).

The second form of solution of (2.5) is found using result 255.00 in [41]. Its parametric form is
φ=φ3-φ4nsn2(w∣m)1-nsn2(w∣m),ξ=1εp(wφ4-(φ4-φ3)Π(n;w∣m)),
where m,p,w are as in (2.6), and
n=φ3-φ2φ4-φ2.

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
λ=2εp|φ4K(m)-(φ4-φ3)Π(n∣m)|.

When φ1=φ2, m=1, (2.10) becomes
φ=φ3-φ4ntanh2w1-ntanh2w,ξ=1ε(wφ2p+2tanh-1(ntanhw)).

3. Periodic and Solitary Wave Solutions to Equaion (<xref ref-type="disp-formula" rid="Eq1.5">1.5</xref>)

Equation (1.5) can also be written in the form
(ut+uux)xx=ut+uux+ux.
Let u=φ(ξ)+c with ξ=x-ct be a travelling wave solution to (3.1), then it follows that
(φφξ)ξξ=φφξ+φξ.
Integrating (3.2) twice with respect to ξ, we have
(φφξ)2=14(φ4+83φ3+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+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,
and define φL, φR, bL, and bR by
φL=-12(2+4-2a),φR=-12(2-4-2a),bL=-φL2g(φL)=a24-2a+83+23(2-a)4-2a,bR=-φR2g(φR)=a24-2a+83-23(2-a)4-2a.
Obviously, φL, φR 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: φL, φR, 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 φL<0<φR and f(φL)<f(φR). For each value a<0 and 0<b<bR (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.

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=bR (a corresponding curve of f(φ) is shown in Figure 1(b)) corresponds to the limit φ3=φ4=φR so that m=1, and then the solution is an inverted loop-like solitary wave given by (2.9) with φ2≤φ<φR and
φ1=-16(2+34-2a+24+64-2a),φ2=-16(2+34-2a-24+64-2a);
see Figure 3(a), for an example.

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 φL<φR<0 and f(φL)<f(0). For each value 0<a<16/9 and bR<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
φ1=-43-1316-9a,φ2=-43+1316-9a.
In this case we obtain a weak solution, namely, the periodic upward-cusp wave
φ=φ(ξ-2jξm),(2j-1)ξm<ξ<(2j+1)ξm,j=0,±1,±2,…,
where
φ(ξ)=(φ2-φ1tanh2(ξ4))cosh2(ξ4),ξm=4tanh-1φ2φ1,
see Figure 3(b), for an example.

(3) a=16/9

In this case φL<φR<0 and f(φL)=f(0). For a=16/9 and each value bR<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=φL=-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
φ=43exp(-12|ξ|)-43,
that is, a single peakon solution with amplitude 4/3, see Figure 3(c).

(4) 16/9<a<2

In this case φL<φR<0 and f(φL)>f(0). For each value 16/9<a<2 and bR<b<bL (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=bL (a corresponding curve of f(φ) is shown in Figure 1(h)) corresponds to the limit φ1=φ2=φL so that m=1, and then the solution is a hump-like solitary wave given by (2.13) with φL<φ≤φ3 and
φ3=16(-2+34-2a-24-64-2a),φ4=16(-2+34-2a+24-64-2a),
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).

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