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Advanced Nonlinear Studies

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Volume 16, Issue 1

Issues

Traveling Wave Solutions to the Burgers-αβ Equations

Byungsoo Moon
Published Online: 2015-12-04 | DOI: https://doi.org/10.1515/ans-2015-5001

Abstract

The Burgers-αβ equation, which was first introduced by Holm and Staley [4], is considered in the special case where ν=0 and b=3. Traveling wave solutions are classified to the Burgers-αβ equation containing four parameters b,α,ν, and β, which is a nonintegrable nonlinear partial differential equation that coincides with the usual Burgers equation and viscous b-family of peakon equation, respectively, for two specific choices of the parameter β=0 and β=1. Under the decay condition, it is shown that there are smooth, peaked and cusped traveling wave solutions of the Burgers-αβ equation with ν=0 and b=3 depending on the parameter β. Moreover, all traveling wave solutions without the decay condition are parametrized by the integration constant k1. In an appropriate limit β=1, the previously known traveling wave solutions of the Degasperis–Procesi equation are recovered.

Keywords: Traveling Wave Solutions; Burgers-

MSC 2010: 35Q35; 35Q53; 35B65; 37K45

1 Introduction

Consider the Burgers-αβ equation [4] with ν=0 and b=3:

ut-α2utxx+(1+3β)uux=α2(3uxuxx+uuxxx),x,t>0,(1.1)

where the subscripts denote the partial derivatives with respect to the spatial coordinate x and temporal coordinate t, and α,β are real parameters. Indeed, using the transformations t1αt and x1αx, we can rewrite (1.1) as

ut-utxx+(1+3β)uux=3uxuxx+uuxxx,x,t>0.(1.2)

It is easy to see that (1.2) is the more general case compared to the Degasperis–Procesi equation. The Degasperis–Procesi equation is a special case of (1.2) with β=1. The formal integrability of the Degasperis–Procesi equation was obtained in [2] by constructing a Lax pair. It has a bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to the Camassa–Holm peakons [2]. The Degasperis–Procesi equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the Camassa–Holm shallow water equation [1, 3]. An inverse scattering approach for computing N-peakon solutions of the Degasperis–Procesi equation was presented in [8]. Its traveling wave solutions were investigated in [6, 9].

Note that if p(x):=12e-|x|, x, then u=(1-x2)-1m=p*m, where m:=u-uxx and * denotes the convolution product on , given by

(f*g)(x):=f(y)g(x-y)𝑑y.

This formulation allows us to define a weak form of (1.2) as follows:

ut+x(12u2+p*(3β2u2))=0,x,t>0.(1.3)

We also note that the peaked solitons are not classical solutions of (1.2) with β=1. They satisfy the Degasperis–Procesi equation in the weak form (1.3) with β=1.

Recently, Holm and Staley in [4] studied traveling wave solutions of the Burgers-αβ equation with (3-b)β=1 and ν=0. The aim of the present paper is to classify all weak traveling wave solutions of the Burgers-αβ equation with b=3 and ν=0 by the idea used in [5, 7].

Our main results of this paper are Theorem 2.3 (traveling wave solution with decay; Figures 1 and 2) and Theorems 3.13.4 (traveling wave solution without decay; Figures 3, 4 and 5)

This paper is organized as follows. In Section 2, we classify the traveling wave solutions of (1.2) under the decay condition. In particular, we show the existence of peaked traveling wave solutions for β=1. In Section 3, we categorize the traveling wave solutions of (1.2) without the decay condition, by using an analogous analysis in [6]. Finally, we give our concluding remarks in Section 4.

2 Traveling Wave Solutions With the Decay Condition

In this section, we study all weak traveling wave solutions of (1.2), i.e. solutions of the form

u(t,x)=φ(x-ct),c(2.1)

for some φ: such that φ0 as |x|. Note that if φ(x-ct) is a traveling wave solution of (1.2), then -φ(x+ct) is also a traveling wave solution of (1.2). Substituting (2.1) into (1.2) and integrating it, we have

(c-φ)φxx-φx2=cφ-1+3β2φ2,(2.2)

We rewrite (2.2) as

12[(φ-c)2]xx=-cφ+1+3β2φ2.

Now we give the definition of a traveling wave solution of (1.2).

A function φ(x-ct)H1() is a nontrivial traveling wave solution of (1.2) with c and φ0 as |x|.

The following lemma deals with the regularity of the traveling wave solutions. The idea is inspired by the study of the traveling waves of the Camassa–Holm equation [5].

Let φ(x-ct) be a traveling wave solutions of (1.2). Then

(φ-c)kCj(φ-1(c)),k2j.(2.3)

Therefore

φC(φ-1(c)).

Proof.

Let ν=φ-c and denote P(ν)=-2c(ν+c)+(1+3β)(ν+c)2. So P(ν) is a polynomial in ν. Then ν satisfies

(ν2)xx=P(ν).

Since νH1(), we know that (ν2)xxLloc1(). Hence (ν2)x is absolutely continuous and ν2C1(). Then νC1(ν-1(0)). Moreover, we have

(νk)xx=(kνk-1νx)x=k2(νk-2(ν2)x)x=k(k-2)νk-2νx2+k2νk-2(ν2)xx=k(k-2)νk-2νx2+k2νk-2P(ν).(2.4)

For k=3, the right-hand side of (2.4) is in Lloc1(). Thus we conclude that ν3C1(). For k22=4 we know that (2.4) implies

(νk)xx=k4(k-2)νk-4[(ν2)x]2+k2νk-2P(ν)C1().

Therefore νkC1() for k22=4.

For k23=8 we see from the above that ν4,νk-4,νk-2,νk-2P(ν)C2(). Moreover, we have

νk-2νx2=14(ν4)x1k-4(νk-4)xC1().

Hence from (2.4) we deduce that

νkC3(ν-1(0)),k23=8.

Applying the same argument to higher values of k we prove that νkCj(ν-1(0)) for k2j, and hence (2.3). This completes the proof of Lemma 2.2. ∎

Let φx=G. Then (2.2) becomes

(G2)φ-2c-φG2=2(cφ-1+3β2φ2)c-φ.(2.5)

Solving the first-order ordinary differential equation (2.5), we have

φx2=φ2[1+3β4φ2-(1+β)cφ+c2](φ-c)2:=F(φ).(2.6)

Consider the polynomial

P(φ)=φ2[1+3β4φ2-(1+β)cφ+c2](2.7)

with a double root at φ=0. Then we can classify all traveling wave solutions of (1.2) depending on the different behaviors of this polynomial.

If β=-13, then we know that P(φ) is the third-degree polynomial with a double zero at φ=0 and a simple zero at φ=32c such that P(φ)=-23cφ2(φ-32c).

If β-13, then P(φ) is the fourth-degree polynomial with a double zero at φ=0 and there are the three cases

  • (i)

    Δ<0,

  • (ii)

    Δ=0,

  • (iii)

    Δ>0,

where Δ:=c2β(β-1) is the determinant of 1+3β4φ2-(1+β)cφ+c2.

(i) Δ<0: For 0<β<1, we have that P(φ) is the fourth-degree polynomial with a double zero at φ=0.

(ii) Δ=0:

  • For β=0, P(φ)=φ2(12φ-c)2 is the fourth-degree polynomial with a double zero at φ=0 and φ=2c.

  • For β=1, P(φ)=φ2(φ-c)2 is the fourth-degree polynomial with a double zero at φ=0 and φ=c.

(iii) Δ>0: For β<0 or β>1, P(φ)=1+3β4φ2(c-l1-φ)(c-l2-φ) is the fourth-degree polynomial with a double zero at φ=0 and a simple zero at φ=c-l1 or φ=c-l2, where

l1=c[β-11+3β+(β-11+3β)2+β-11+3β],l2=c[β-11+3β-(β-11+3β)2+β-11+3β]

for c>0 and

l1=c[β-11+3β-(β-11+3β)2+β-11+3β],l2=c[β-11+3β+(β-11+3β)2+β-11+3β]

for c<0 are two roots of the equation

y2-2(β-1)1+3βcy-β-11+3βc2=0.

If β<-13 then

l2<0<c<l1orl2<c<0<l1.(2.8)

If -13<β<0 then

l2<l1<0<corc<0<l2<l1.(2.9)

If β>1 then

l2<0<l1<corc<l2<0<l1.

Using the idea as introduced in [5] gives us the following conclusions:

  • (a)

    Assume F(φ) has a simple zero at φ=m so that F(m)=0 and F(m)0. The solution φ of (2.6) satisfies

    φx2=(φ-m)F(m)+O((φ-m)2)as xm,

    where f=O(g) as xa means that |f(x)g(x)| is bounded in some interval [a-ϵ,a+ϵ] with ϵ>0. Therefore

    φ(x)=m+14(x-ξ)2F(m)+O((x-ξ)4)as xξ,(2.10)

    where φ(ξ)=m.

  • (b)

    If F(φ) has a double zero at m, so that F(m)=0,F′′(m)>0, then

    φx2=(φ-m)2F′′(m)+O((φ-m)3)as φm.

    We obtain

    φ(x)-mαexp(-xF′′(m))as x(2.11)

    for some constant α. Thus φm exponentially as x.

  • (c)

    If φ approaches a double pole φ(x0)=c of F(φ). Then

    φ(x)-c=α|x-x0|1/2+O((x-x0)3/2)as xx0,(2.12)φx={12α|x-x0|-1/2+O((x-x0)1/2)as xx0,-12α|x-x0|-1/2+O((x-x0)1/2)as xx0(2.13)

    for some constant α. In particular, when F has a double pole, the solution φ has a cusp.

  • (d)

    If the evolution of φ according to (2.6) suddenly changes direction φx-φx, then peaked solitary waves occur.

In view of (a)–(d), we give the following theorem on all bounded traveling wave solutions of (1.2) with decay.

Any bounded traveling wave of (1.2) with decay belongs to one of the following categories.

  • (1)

    For β<-13 :

    • If c>0 , then there is a smooth traveling wave with decay φ<0 with minxφ(x)=c-l1 and a cusped traveling wave with decay φ>0 with maxxφ(x)=c.

    • If c<0 , then there is a smooth traveling wave with decay φ>0 with maxxφ(x)=c-l2 and an anticusped traveling wave with decay φ<0 with minxφ(x)=c.

  • (2)

    For β=-13 :

    • If c>0 , then there is a cusped traveling wave with decay φ>0 with maxxφ(x)=c.

    • If c<0 , then there is an anticusped traveling wave with decay φ<0 with minxφ(x)=c.

  • (3)

    For -13<β<0 :

    • If c>0 , then there is a cusped traveling wave with decay φ>0 with maxxφ(x)=c.

    • If c<0 , then there is an anticusped traveling wave with decay φ<0 with minxφ(x)=c.

  • (4)

    For β=0 :

    • If c>0 , then there is a cusped traveling wave with decay φ>0 with maxxφ(x)=c.

    • If c<0 , then there is an anticusped traveling wave with decay φ<0 with minxφ(x)=c.

  • (5)

    For 0<β<1 :

    • If c>0 , then there is a cusped traveling wave with decay φ>0 with maxxφ(x)=c.

    • If c<0 , then there is an anticusped traveling wave with decay φ<0 with minxφ(x)=c.

  • (6)

    For β=1 :

    • If c>0 , then there is a peaked traveling wave with decay φ>0 with maxxφ(x)=c.

    • If c<0 , then there is an antipeaked traveling wave with decay φ<0 with minxφ(x)=c.

  • (7)

    For β>1 :

    • If c>0 , then there is a smooth traveling wave with decay φ>0 with maxxφ(x)=c-l1.

    • If c<0 , then there is a smooth traveling wave with decay φ<0 with minxφ(x)=c-l2.

Proof.

First, we consider c>0. The other case c<0 can be handled in a very similar way.

If β=-13, then (2.6) becomes

φx2=-23cφ2(φ-32c)(φ-c)2:=F1(φ).

When c>0, φ=c is a double pole of F1(φ). Hence from (2.12) and (2.13) we see that we obtain a traveling wave with cusp at φ=c, which decays exponentially.

If 0<β<1 and β=0, then φ=c is a double pole of F(φ) and φ=0 is a double zero of F(φ) in (2.6). Therefore, in the same manner as above, we obtain a cusped traveling wave with maxxφ(x)=c for c>0. If β=1, then (2.6) becomes

φx2=φ2(φ-c)2(φ-c)2:=F2(φ).

When c>0, the smooth solution can be constructed until φ=c. But it can make a sudden turn at φ=c and so give rise to a peak. Since φ=0 is still a double zero of F2(φ), we still have the exponential decay.

There are three different kinds of traveling wave solutions with decay for c>0${c>0}$ in Theorem 2.3.(i) Cusped traveling waves with maxx∈ℝ⁡φ⁢(x)=c${\max_{x\in\mathbb{R}}\varphi(x)=c}$ for β<1${\beta<1}$.(ii) Peaked traveling waves with maxx∈ℝ⁡φ⁢(x)=c${\max_{x\in\mathbb{R}}\varphi(x)=c}$ for β=1${\beta=1}$.(iii) Smooth traveling waves with maxx∈ℝ⁡φ⁢(x)=c-l1${\max_{x\in\mathbb{R}}\varphi(x)=c-l_{1}}$ for β>1${\beta>1}$.There are three different kinds of traveling wave solutions with decay for c>0${c>0}$ in Theorem 2.3.(i) Cusped traveling waves with maxx∈ℝ⁡φ⁢(x)=c${\max_{x\in\mathbb{R}}\varphi(x)=c}$ for β<1${\beta<1}$.(ii) Peaked traveling waves with maxx∈ℝ⁡φ⁢(x)=c${\max_{x\in\mathbb{R}}\varphi(x)=c}$ for β=1${\beta=1}$.(iii) Smooth traveling waves with maxx∈ℝ⁡φ⁢(x)=c-l1${\max_{x\in\mathbb{R}}\varphi(x)=c-l_{1}}$ for β>1${\beta>1}$.There are three different kinds of traveling wave solutions with decay for c>0${c>0}$ in Theorem 2.3.(i) Cusped traveling waves with maxx∈ℝ⁡φ⁢(x)=c${\max_{x\in\mathbb{R}}\varphi(x)=c}$ for β<1${\beta<1}$.(ii) Peaked traveling waves with maxx∈ℝ⁡φ⁢(x)=c${\max_{x\in\mathbb{R}}\varphi(x)=c}$ for β=1${\beta=1}$.(iii) Smooth traveling waves with maxx∈ℝ⁡φ⁢(x)=c-l1${\max_{x\in\mathbb{R}}\varphi(x)=c-l_{1}}$ for β>1${\beta>1}$.
Figure 1

There are three different kinds of traveling wave solutions with decay for c>0 in Theorem 2.3.(i) Cusped traveling waves with maxxφ(x)=c for β<1.(ii) Peaked traveling waves with maxxφ(x)=c for β=1.(iii) Smooth traveling waves with maxxφ(x)=c-l1 for β>1.

The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.
Figure 2

The graph of the polynomial (2.7) displayed for different values of β. The seven cases give rise to the categories(1)–(7) in Theorem 2.3.

If β<-13, -13<β<0, and β>1, then (2.6) becomes

φx2=(1+3β)φ2(c-φ-l1)(c-φ-l2)4(φ-c)2:=F3(φ).

When β<-13, we know that l2<0<c<l1 from (2.8). F3(φ) has a simple zero at φ=c-l1<0 and a double zero at φ=0. Therefore from (2.10) and (2.11) we see that in this case we can obtain a smooth traveling wave with minxφ(x)=c-l1 and an exponential decay to zero at infinity. Moreover, since F3(φ) has a double pole at φ=c, we can also obtain a cusped traveling wave with maxxφ(x)=c, which decays exponentially.

If -13<β<0, we know that l2<l1<0<c from (2.9). In this case F3(φ) has a double pole at φ=c and a double zero at φ=0. Hence we obtain a cusped traveling wave with maxxφ(x)=c, which decays exponentially.

If β>1, we see that l2<0<l1<c from (2.9). F3(φ) has a simple zero at φ=c-l1>0 and a double zero at φ=0. Therefore from (2.10) and (2.11) we see that in this case we can obtain a smooth traveling wave with maxxφ(x)=c-l1 and an exponential decay to zero at infinity. This completes the proof of Theorem 2.3. ∎

3 Traveling Wave Solutions Without the Decay Condition

In this section, we consider all weak traveling wave solutions of (1.2), i.e. solutions of the form

u(t,x)=φ(x-ct),c(3.1)

without the decay condition at infinity. Note that if φ(x-ct) is a traveling wave solution of (1.2), then -φ(x+ct) is also a traveling wave solution of (1.2). Thus we only consider traveling wave with a positive speed c>0. Substituting (3.1) into (1.2) and integrating it, we have

(c-φ)φxx-φx2=cφ-1+3β2φ2+k1,(3.2)

where k1 is an integration constant. Let φx=G. Then (3.2) becomes

(G2)φ-2c-φG2=2(cφ-1+3β2φ2+k1)c-φ.(3.3)

Solving the first-order ordinary differential equation (3.3), we have

φx2=(φ-c)2[1+3β4φ2+β-12cφ+β-14c2-k1]+k2(φ-c)2:=H(φ),

where k2 is an integration constant.

Consider the polynomial

P1(φ)=(φ-c)2[1+3β4φ2+β-12cφ+β-14c2-k1]

with a double root at φ=c. Then we can classify all traveling wave solutions of (1.2) depending on the different behaviors of this polynomial. Once k1 is fixed, a change in k2 will shift the graph vertically up or down. Hence we can easily determine which k2 yield bounded traveling waves. There are qualitatively different cases.

Case 1. We consider β=-13. Then P1(φ)=(φ-c)2(-23cφ-13c2-k1) becomes a third-degree polynomial with a double zero at φ=c.

Arguments similar to the ones in [5] and (2.10)–(2.13) give us the following theorem for β=-13.

Let β=-13 and c>0. Any bounded traveling wave of (1.2) belongs to one of the following categories. The waves are parametrized by k1 as follows:

  • (1)

    If k1-c2 , then there are no bounded solutions of ( 1.2 ).

  • (2)

    If k1>-c2 , then there exist a one-parameter group of cusped periodic traveling waves and one cusped traveling wave with decay.

Proof.

If β=-13, we have

P1(φ)=-23c(φ-c)2(φ+12c+3k12c)=-23c(φ-c)3-(c2+k1)(φ-c)2.(3.4)

Since c>0 and k1-c2, we see that

P1(φ)=-2c(φ-c)2-2(φ-c)(c2+k1)<0for φ<c.

Therefore, P1(φ) is decreasing for φ<c. There are no bounded solutions for any k2.

If -c2<k1<-c23, k1=-c23, and k1>-c23, then P1(φ) has a double zero at φ=c and a simple zero at φ=-c2+3k12c>0, φ=-c2+3k12c=0, and φ=-c2+3k12c<0, respectively. Hence there are cusped traveling waves for some k2>0. This completes the proof of Theorem 3.1. ∎

The graph of the polynomial (3.4) displayed for different values of k1${k_{1}}$. The four cases give rise to the categories(1)–(2) in Theorem 3.1.The graph of the polynomial (3.4) displayed for different values of k1${k_{1}}$. The four cases give rise to the categories(1)–(2) in Theorem 3.1.The graph of the polynomial (3.4) displayed for different values of k1${k_{1}}$. The four cases give rise to the categories(1)–(2) in Theorem 3.1.The graph of the polynomial (3.4) displayed for different values of k1${k_{1}}$. The four cases give rise to the categories(1)–(2) in Theorem 3.1.
Figure 3

The graph of the polynomial (3.4) displayed for different values of k1. The four cases give rise to the categories(1)–(2) in Theorem 3.1.

Case 2. Consider β-13. We know that

P1(φ)=(φ-c)2[1+3β4φ2+β-12cφ+β-14c2-k1](3.5)

is a fourth-degree polynomial. We distinguish two cases:

  • (i)

    β>-13,

  • (ii)

    β<-13.

We need the determinant Δ:=1-β22c2+(1+3β)k1 of the second-degree polynomial

1+3β4φ2+β-12cφ+β-14c2-k1.

(i) β>-13:

Δ>0:k1>β2-12(1+3β)c2,Δ=0:k1=β2-12(1+3β)c2,Δ<0:k1<β2-12(1+3β)c2.

(ii) β<-13:

Δ>0:k1<β2-12(1+3β)c2,Δ=0:k1=β2-12(1+3β)c2,Δ<0:k1>β2-12(1+3β)c2.

The idea as introduced in [5] and (2.10)–(2.13) give us the following theorems.

Let β>-13 and c>0. Any bounded traveling wave of (1.2) belongs to one of the following categories. The waves are parametrized by k1 as follows:

  • (1)

    If k1-c22(1+3β) , then there are no bounded solutions of ( 1.2 ).

  • (2)

    If -c22(1+3β)<k1<β2-12(1+3β)c2 , then there exist a one-parameter group of smooth periodic traveling waves and one smooth traveling wave with decay.

  • (3)

    Let k1=β2-12(1+3β)c2.

    • If β>0 , then there exist a one-parameter group of smooth periodic traveling waves and one peaked solitary wave.

    • If -13<β0 , then there are no bounded solutions of ( 1.2 ).

  • (4)

    If β2-12(1+3β)c2<k1<3β-12c2 , then there exist a one-parameter group of smooth periodic traveling waves, one peaked periodic traveling wave, a one-parameter group of cusped periodic traveling waves, and one cusped traveling wave with decay.

  • (5)

    Let k1=3β-12c2.

    • If β>0 , then there exist a one-parameter group of cusped periodic traveling waves and one cusped traveling wave with decay.

    • If -13<β0 , then there are no bounded solutions of ( 1.2 ).

  • (6)

    If k1>3β-12c2 , then there exist a one-parameter group of cusped periodic traveling waves, one cusped traveling with decay, a one-parameter group of anticusped periodic traveling waves, and one anticusped traveling wave with decay.

  • (C)

    (Composite waves) Any countable number of cuspons, anticuspons, and peakons from the categories (1)(6) corresponding to the same value of k1 may be joined at points where φ=c to form a composite wave φ. If the Lebesgue measure μ(φ-1(c)) equals 0 , then φ is a traveling wave of ( 1.2 ).

  • (S)

    (Stumpons) For k1=3β-12c2 the composite waves are traveling waves of ( 1.2 ) even if the Lebesgue measure μ(φ-1(c)) exceeds 0 . Consequently, these waves may contain intervals where φc.

Proof.

Let β>-13 and c>0.

(1) If k1-c22(1+3β), a direct computation gives us

P1(φ)=(1+3β)(φ-c)(φ-c1+3β)2-(φ-c)(c21+3β+2k1)<0for φ<c.

There are no bounded traveling wave solutions of (1.2) for any k2 since P1(φ) is decreasing for φ<c.

(2) If -c22(1+3β)<k1<β2-12(1+3β)c2, there are smooth traveling waves for some negative k2.

(3) Let k1=β2-12(1+3β)c2. If β>0, then there are a peaked solitary wave for k2=0 and smooth traveling waves for some negative k2. If -13<β0, there are no bounded solutions for any k2.

(4) If β2-12(1+3β)c2<k1<3β-12c2, then there are smooth traveling waves for some negative k2, a peaked periodic traveling wave for k2=0, and cusped traveling wave solutions for some k2>0.

(5) Let k1=3β-12c2. If β>0, there are cusped traveling waves for some positive k2 and the constant φc is a solution for k2=0. If -13<β0, there are no bounded solutions for any k2.

The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.The polynomial (3.5) displayed for different values of k1${k_{1}}$. All cases give rise to the categories (1)–(6) in Theorem 3.2.
Figure 4

The polynomial (3.5) displayed for different values of k1. All cases give rise to the categories (1)–(6) in Theorem 3.2.

The polynomial (3.5) displayed for different values of k1${k_{1}}$. The six cases give rise to the categories (1)–(6) inTheorem 3.4.The polynomial (3.5) displayed for different values of k1${k_{1}}$. The six cases give rise to the categories (1)–(6) inTheorem 3.4.The polynomial (3.5) displayed for different values of k1${k_{1}}$. The six cases give rise to the categories (1)–(6) inTheorem 3.4.The polynomial (3.5) displayed for different values of k1${k_{1}}$. The six cases give rise to the categories (1)–(6) inTheorem 3.4.The polynomial (3.5) displayed for different values of k1${k_{1}}$. The six cases give rise to the categories (1)–(6) inTheorem 3.4.The polynomial (3.5) displayed for different values of k1${k_{1}}$. The six cases give rise to the categories (1)–(6) inTheorem 3.4.
Figure 5

The polynomial (3.5) displayed for different values of k1. The six cases give rise to the categories (1)–(6) inTheorem 3.4.

(6) If k1>3β-12c2, then there are cusped and anticusped traveling waves for some positive k2. ∎

In [6], Lenells categorized traveling wave solutions of the Degasperis–Procesi equation. His categories (1)–(8) correspond to our categories (1)–(6), (C), and (S) for β=1 in Theorem 3.2.

Let β<-13 and c>0. Any bounded traveling wave of (1.2) belongs to one of the following categories. The waves are parametrized by k1 as follows:

  • (1)

    If k1-c22(1+3β) , then there are no bounded solutions of ( 1.2 ).

  • (2)

    If β2-12(1+3β)c2<k1<-c22(1+3β) , then there exist a one-parameter group of cusped periodic traveling waves and one cusped traveling wave with decay.

  • (3)

    If k1=β2-12(1+3β)c2 , then there exist a one-parameter group of smooth periodic traveling waves, one smooth traveling waves with decay, a one-parameter group of cusped periodic traveling waves, and one cusped traveling wave with decay.

  • (4)

    If 3β-12c2<k1<β2-12(1+3β)c2 , then there exist a one-parameter group of smooth periodic traveling waves, a one-parameter group of cusped periodic traveling waves, and one cusped traveling wave with decay.

  • (5)

    If k1=3β-12c2 , then there exist a one-parameter group of smooth periodic traveling waves and cusped periodic traveling waves.

  • (6)

    If k1>3β-12c2 , then there exist a one-parameter group of smooth periodic traveling waves, one peaked periodic traveling waves, and cusped periodic traveling waves.

  • (C)

    (Composite waves) Any countable number of cuspons, anticuspons, and peakons from the categories (1)(6) corresponding to the same value of k1 may be joined at points where φ=c to form a composite wave φ. If the Lebesgue measure μ(φ-1(c)) equals 0 , then φ is a traveling wave of ( 1.2 ).

  • (S)

    (Stumpons) For k1=3β-12c2 the composite waves are traveling waves of ( 1.2 ) even if the Lebesgue measure μ(φ-1(c)) exceeds 0 . Consequently, these waves may contain intervals where φc.

Proof.

Let β<-13 and c>0.

(1) If k1-c22(1+3β), a direct computation gives us

P1(φ)=(1+3β)(φ-c)(φ-c1+3β)2-(φ-c)(c21+3β+2k1)>0for φ<c.

There are no bounded traveling wave solutions of (1.2) for any k2 since P1(φ) is increasing for φ<c.

(2) If β2-12(1+3β)c2<k1<-c22(1+3β), there are cusped traveling waves for some positive k2.

(3) If k1=β2-12(1+3β)c2, there are smooth traveling waves and cusped traveling waves for some positive k2.

(4) If 3β-12c2<k1<β2-12(1+3β)c2, there are smooth periodic traveling waves for k2=0 and cusped traveling waves for some positive k2.

(5) If k1=3β-12c2, there are smooth periodic traveling waves for some negative k2, cusped periodic traveling waves for positive k2, and smooth periodic traveling waves and the constant φc for k2=0.

(6) If k1>3β-12c2, there are smooth periodic traveling waves for some negative k2, a peaked periodic traveling wave for k2=0, and cusped periodic traveling waves for some positive k2. ∎

4 Concluding Remarks

In this paper, we have investigated traveling wave solutions of the Burgers-αβ equation with ν=0 and b=3, including the well-studied integrable Degasperis–Procesi equation [2], β=1. Hence the present paper generalizes some priori traveling wave results from [6] of the Degasperis–Procesi equation. The free parameter β and the integration constant k1 play an important role in the type of traveling wave solutions of (1.2). Our study shows that there are three different kinds of traveling wave solutions with the decay condition to (1.2) such as cusped (β<1), peaked (β=1), and smooth (β>1). Traveling wave solutions without the decay condition to (1.2) are parametrized by the integration constant k1.

References

  • [1]

    Constantin A. and Lannes D., The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations, Arch. Rational Mech. Anal. 192 (2009), 165–186.  Google Scholar

  • [2]

    Degasperis A., Holm D. D. and Hone A. N. W., A new integral equation with peakon solutions, Theoret. Math. Phys. 133 (2002), 1463–1474.  Google Scholar

  • [3]

    Dullin H. R., Gottwald G. A. and Holm D. D., Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynam. Res. 33 (2003), 73–79.  Google Scholar

  • [4]

    Holm D. D. and Staley M. F., Nonlinear balances and exchange of stability in dynamics of solitons, peakons, ramp/cliffs and leftons in a 1+1 nonlinear evolutionary PDE, Phy. Lett. A 308 (2003), 437–444.  Google Scholar

  • [5]

    Lenells J., Traveling wave solutions of the Camassa–Holm equation, J. Differential Equations 217 (2005), 393–430.  Google Scholar

  • [6]

    Lenells J., Traveling wave solutions of the Degasperis–Procesi equation, J. Math. Anal. Appl. 306 (2005), 72–82.  Google Scholar

  • [7]

    Lenells J., Classification of travelling waves for a class of nonlinear wave equations, J. Dynam. Differential Equations 18 (2006), no. 2, 381–391.  Google Scholar

  • [8]

    Lundmark H. and Szmigielski J., Multi-peakon solutions of the Degasperis–Procesi equation, Inverse Problems 19 (2003), 1241–1245.  Google Scholar

  • [9]

    Vakhnenko V. O. and Parkes E. J., Periodic and solitary-wave solutions of the Degasperis–Procesi equation, Chaos Solitons Fractals 20 (2004), 1059–1073.  Google Scholar

About the article

Received: 2015-02-25

Accepted: 2015-08-16

Published Online: 2015-12-04

Published in Print: 2016-02-01


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 147–157, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5001.

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