# Periodic and Solitary Wave Solutions to  the Fornberg-Whitham Equation

Authors: Jiangbo Zhou; Lixin Tian
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

New travelling wave solutions to the Fornberg-Whitham equationut−uxxt+ux+uux=uuxxx+3uxuxx are investigated. They are characterized by two parameters. The expresssions for the
periodic and solitary wave solutions are obtained.


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## Body

## 1. Introduction

Recently, Ivanov [1] investigated the integrability of a class of nonlinear dispersive wave equations:
(1.1)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
(1.2)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(1.3)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(1.4)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(1.5)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(1.6)ut-uxxt+ux+uux=0
and the modified BBM equation
(1.7)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 Section2, 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(2.1)(φφξ)2=ε2f(φ),
where
(2.2)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
(2.3)dξdζ=φε,
so that (2.1) becomes
(2.4)(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
(2.5)φ=φ2-φ1nsn2(w∣m)1-nsn2(w∣m),ξ=1εp(wφ1+(φ2-φ1)Π(n;w∣m)),
with w as the parameter, where
(2.6)m=(φ3-φ2)(φ4-φ1)(φ4-φ2)(φ3-φ1),p=12(φ4-φ2)(φ3-φ1),w=pζ,(2.7)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
(2.8)λ=2εp|φ1K(m)+(φ2-φ1)Π(n∣m)|,
where  Π(n∣m)  is the complete elliptic integral of the third kind.Whenφ3=φ4, m=1, (2.5) becomes
(2.9)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
(2.10)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
(2.11)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
(2.12)λ=2εp|φ4K(m)-(φ4-φ3)Π(n∣m)|.Whenφ1=φ2, m=1, (2.10) becomes
(2.13)φ=φ3-φ4ntanh2w1-ntanh2w,ξ=1ε(wφ2p+2tanh-1(ntanhw)).


## 3. Periodic and Solitary Wave Solutions to Equaion (1.5)

Equation (1.5) can also be written in the form
(3.1)(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
(3.2)(φφξ)ξξ=φφξ+φξ.
Integrating (3.2) twice with respect to ξ, we have
(3.3)(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
(3.4)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
(3.5)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 thata<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<0In this caseφL<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.The curve off(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)(e)(f)(g)(h)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.
(a)(b)(c)(d)The casea<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
(3.6)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.
(a)(b)(c)(d)(2)0<a<16/9In this caseφL<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 case0<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
(3.7)φ1=-43-1316-9a,φ2=-43+1316-9a.
In this case we obtain a weak solution, namely, the periodic upward-cusp wave
(3.8)φ=φ(ξ-2jξm),(2j-1)ξm<ξ<(2j+1)ξm,j=0,±1,±2,…,
where
(3.9)φ(ξ)=(φ2-φ1tanh2(ξ4))cosh2(ξ4),(3.10)ξm=4tanh-1φ2φ1,
see Figure 3(b), for an example.(3)a=16/9In this caseφL<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 casea=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
(3.11)φ=43exp(-12|ξ|)-43,
that is, a single peakon solution with amplitude 4/3, see Figure 3(c).(4)16/9<a<2In this caseφL<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 case16/9<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
(3.12)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 solutionsφ(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 parametersa 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.



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*Source: 507815-2009-06-02.xml*