## Abstract

We present a method for the classification of all weak travelling-wave solutions for some dispersive nonlinear wave equations. When applied to the Camassa–Holm or the Degasperis–Procesi equation, the approach shows the existence of not only smooth, peaked and cusped travelling-wave solutions, but also more exotic solutions with fractal-like wave profiles.

## 1. Introduction

Of interest when considering equations modelling wave phenomena is the existence of the so-called travelling-wave solutions. These are waves whose shape does not change as the wave travels along at some constant speed. An understanding of the class of travelling waves for an equation can be a first step towards the exploration of deeper properties of more intricate solutions. In this note, we will explain a simple method that determines all travelling-wave solutions for some nonlinear dispersive wave equations.

One of the most well-known models for the evolution of water waves is the Korteweg–de Vries equation(1.1)where *u*(*t*, *x*) represents the water's free surface in non-dimensional variables. It is the simplest equation embodying both nonlinearity and dispersion, and has served as the model equation for the development of soliton theory (Drazin & Johnson 1989). More recently, the Camassa–Holm equation(1.2)was discovered as another model for the propagation of water waves in shallow water (Camassa & Holm 1993). Equations (1.1) and (1.2) have plenty of structures tied into them, e.g. each of the equations is completely integrable: by means of an isospectral problem the equation can be converted into an infinite sequence of linear ordinary differential equations which can be integrated trivially (Drazin & Johnson (1989) and McKean (1979) for the case of equation (1.1), and Beals *et al.* (1998), Constantin (1998, 2001), Constantin & McKean (1999), Lenells (2002) and Constantin & Ivanov (2006) for the case of equation (1.2)).

A few years ago, it was discovered (Degasperis & Procesi 1999) that within a certain family of third-order nonlinear dispersive PDEs, there is in addition to equations (1.1) and (1.2) a third equation with the property of being formally integrable, namely(1.3)This equation has come to be known as the Degasperis–Procesi equation. Just like (1.1) and (1.2), equation (1.3) has a Lax pair formulation and a bi-Hamiltonian structure leading to an infinite number of conservation laws (Degasperis *et al.* 2002). However, despite their similarity, equations (1.1)–(1.3) exhibit many different properties. Some examples are the following.

While all solutions with initial data exist globally for equation (1.1) (cf. Colliander

*et al.*2003), both (1.2) and (1.3) have global as well as smooth solutions that blow up in finite time (Constantin 1997, 2000, 2001; Constantin & Escher 1998, 2000; Yin 2003; Zhou 2004). Equations (1.2) and (1.3) therefore provide interesting models for the study of wave-breaking phenomena.Whereas (1.1) and (1.2) are known to be integrable for a large class of initial data via the inverse scattering procedure, for equation (1.3) only the existence of an isospectral problem (very different from that known for the Korteweg–de Vries and Camassa–Holm equations) has been established (cf. Degasperis & Procesi 1999; Degasperis

*et al.*2002).Equations (1.1) and (1.2) are both re-expressions of geodesic flow (Misiolek 1998; Constantin 2000; Constantin & Kolev 2003; Kolev 2004), but no such geometric interpretation is valid for equation (1.3).

However, the difference of greatest concern to us is the following: while all travelling-wave solutions of equation (1.1) are smooth, it was noticed (Camassa & Holm 1993; Degasperis *et al.* 2002) that both (1.2) and (1.3) admit travelling waves with peaks at their crests. Furthermore, it was observed through phase-plane analysis (Li & Olver 1997) that cusped solutions of equation (1.2) also exist. Hence, in contrast to equation (1.1) for which it is straightforward to determine the rather limited set of travelling-wave solutions, it is clear that equations (1.2) and (1.3) allow for a wider class of travelling waves. Two natural questions to ask are therefore as follows.

Exactly in what sense are these peaked and cusped waves solutions?

Are there more travelling-wave solutions than the peaked and cusped ones?

Our objective in this note is to present an approach that gives an answer to both these questions for the Camassa–Holm and Degasperis–Procesi equations. In fact, using natural weak formulations, we can classify all weak travelling-wave solutions of equations (1.2) and (1.3). Both equations will be shown to admit qualitatively the same classes of travelling waves, including both peakons and cuspons. But in addition to the peaked and cusped solutions, there turns out to exist for each equation a multitude of peculiar waves obtained by combining peaked and cusped wave segments into new travelling waves (figure 1*i*). An interesting class of waves—called stumpons owing to their shape—is obtained by inserting intervals where the solution equals a constant at the crests of suitable cusped waves (figure 1*j*). Since a countable number of wave segments are permitted in these composite waves, the wave profiles can get highly intricate—figure 2 shows two fractal-like travelling wave solutions.

We expect some of these travelling waves to be unstable and therefore hard to detect physically. Nevertheless, for the Camassa–Holm equation, the peakons and the smooth solitary waves are stable (Constantin & Strauss 2000*a*, 2002; Constantin & Molinet 2001; Lenells 2004*a*,*b*). Furthermore, a numerical scheme implemented by Kalisch & Lenells (2005) suggests that even some cusped and composite travelling waves could be stable.

The suggested method for classifying travelling waves is also applicable to other dispersive nonlinear wave equations. For example, all travelling-wave solutions of the following two classes of equations can be determined by means of the same approach (cf. Lenells 2006*a*,*b*).

The class of equations(1.4)arising in elasticity with the physical parameter

*γ*ranging from −29.4760 to 3.1474. These equations are models for small-amplitude axial–radial deformation waves in compressible isotropic hyperelastic rods,*u*(*t*,*x*) representing the radial stretch relative to a prestressed state in non-dimensional variables (Dai 1998). The only integrable equation in this family indexed by*γ*is the Camassa–Holm equation (*γ*=1; cf. Ivanov 2005), and for*γ*<1 the solitary waves of equation (1.4) are smooth and stable (Constantin & Strauss 2000*b*).The class of nonlinear partial differential equationswhere

*α*^{2}≥0 is a constant and*β*≥0 is a bifurcation parameter. It originally arose as a model for the propagation of shallow water waves (Camassa & Holm 1993; Dullin*et al.*2001).

For the sake of definiteness, we will present our method for the case of the Camassa–Holm equation, the case of equation (1.3) being analogous (cf. Lenells 2005*b*).

## 2. Method

### (a) Step 1. Weak formulation

We first need to find a natural definition of a weak travelling-wave solution. For a solution *u*(*t*, *x*)=*φ*(*x*−*ct*) travelling with speed *c*, equation (1.3) takes the form(2.1)We integrate and rewrite to get, for some constant ,(2.2)Since equation (2.2) makes sense whenever , the following definition is natural.

A function is a *travelling wave* of the Camassa–Holm equation if there exists an such that *φ* satisfies (2.2) in distributional sense.

### (b) Step 2. Smoothness away from {*φ*=*c*}

It is easily observed that all peaked travelling-wave solutions of equations (1.2) and (1.3) have height equal to their speed. In particular, the point where the crest is located satisfies *φ*(*x*)=*c*, and *φ* is smooth on the set . Lemma 2.1 shows that this is not a coincidence.

**(****Lenells 2005 a**

**).**

*Let p*(

*v*)

*be a polynomial with real coefficients*.

*Assume that*

*satisfies*(2.3)

*Then*(2.4)

Applying lemma 2.1 with *v*=*φ*−*c*, we get for *k*≥2^{j}. Taking the *k*th square root, we conclude that *φ* is smooth except possibly at points in the boundary of the set *φ*^{−1}(*c*).

### (c) Step 3. Characterization of travelling waves

Since *φ* is continuous, *C*≔*φ*^{−1}(*c*) is a closed set. We deduce the existence of disjoint open intervals *E*_{i}, *i*≥1, such that . Within each interval *E*_{i} where *φ* is smooth, equation (2.2) may be integrated further to yield a first-order ODE in *φ* (equations (2.5) and (2.6)). Using this observation, lemma 2.2, which characterizes the travelling waves, may be inferred. Observe the condition (*TW*2)-(*ii*) that the measure of the set of points where *φ*=*c* can be strictly positive only when *a*=*c*^{2}.

*A function* *is a travelling wave of equation* *(1.2)* *with speed c if and only if the following three statements hold*.

- (TW1)
- There are disjoint open intervals Ei, i≥1, and a closed set C such that , φ is smooth and non-constant on each Ei, φ(x)≠c for , and φ(x)=c for x∈C.
- (TW2)
- There is an such that(i)for each i≥1, there exists such that(2.5)where(2.6)(ii)if meas(C)>0, then a=c2.
- (TW3)
- .

### (d) Step 4. Classification

The last step consists of determining the set of bounded functions satisfying (*TW*1)–(*TW*3). Suppose we could find all solutions *φ* of (2.5) and (2.6) for different intervals *E*_{i} and different values of *a* and *b*_{i}. Then we can join solutions defined on intervals *E*_{i} whose union is for some closed set *C* of measure 0. The function, defined on , that we get, will satisfy (*TW*1) and (*TW*2) if and only if all wave segments satisfy (2.5) with the same *a*. Also, if we, for *a*=*c*^{2}, allow meas(*C*)>0, this procedure will give us all functions satisfying (*TW*1) and (*TW*2). The following technical result (see Lenells (2005*a*) for a proof) then shows that these solutions automatically satisfy the regularity condition (*TW*3) also.

*Any bounded function φ satisfying* (*TW*1) *and* (*TW*2) *belongs to* *and satisfies* (*TW*3).

The analysis of equation (2.5) is based on the following observations.

Assume

*F*(*φ*) has a simple 0 at*φ*=*m*so that*F*′(*m*)≠0. Then a solution*φ*of equation (2.5) satisfies as*φ*↓*m*. Hence(2.7)where*φ*(*x*_{0})=*m*.If

*F*(*φ*) instead has a double zero at*m*, so that*F*′(*m*)=0,*F*″(*m*)≠0, we obtain as*φ*↓*m*, so that(2.8)for some constant*α*. Thus,*φ*→*m*exponentially as*x*→∞.Suppose

*φ*approaches a simple pole*φ*=*c*of*F*. Then, if*φ*(*x*_{0})=*c*,(2.9)for some constant*α*. In particular, whenever*F*has a simple pole, the solution*φ*has a cusp.Peakons occur when the evolution of

*φ*according to suddenly changes direction: .

Writing *F*(*φ*) in equation (2.5) aswhere *M*, *m* and *z* are the three zeros of the third-order polynomial *φ*^{2}(*c*−*φ*)+*aφ*+*b*_{i}, an investigation of all possible distributions of *M*, *m* and *z* using the above observations leads to the following final result. Note that the travelling waves are parameterized by their maximum, minimum and speed.

*Let z*=*c*−*M*−*m*. *Any bounded travelling wave of equation* *(1.2)* *falls into one of the following categories*.

*(Smooth periodic)*.*If z*<*m*<*M*<*c*,*there is a smooth periodic travelling wave φ*(*x*−*ct*)*of equation**(1.2)**with**and*.*(Smooth with decay)*.*If z*=*m*<*M*<*c*,*there is a smooth travelling wave φ*(*x*−*ct*)*of equation**(1.2)**with*,*and φ*↓*m exponentially as x*→±∞.*(Periodic peakons)*.*If z*<*m*<*M*=*c*,*there is a periodic peaked travelling wave φ*(*x*−*ct*)*of equation**(1.2)**with**and*.*(Peakons with decay)*.*If z*=*m*<*M*=*c*,*there is a peaked travelling wave φ*(*x*−*ct*)*of equation**(1.2)**with*,*and φ*↓*m exponentially as x*→±∞.*(Periodic cuspons)*.*If z*<*m*<*c*<*M*,*there is a periodic cusped travelling wave φ*(*x*−*ct*)*of equation**(1.2)**with**and*.*(Cuspons with decay)*.*If z*=*m*<*c*<*M there is a cusped travelling wave φ*(*x*−*ct*)*of equation**(1.2)**with*,*and φ*↓*m exponentially as x*→±∞.*Replacing*(*φ*(*x*),*c*,*m*,*M*)*with*(−*φ*(−*x*), −*c*, −*m*, −*M*)*in (i)–(vi) yields dual classes of waves travelling in the opposite direction*.*In particular, anticuspons arise**(**figure*1*g*,*h**)*.*(Composite waves)*.*For any fixed**and*,*let*(2.10)*A countable number of cuspons and peakons corresponding to points*(*m*,*M*)*with the same value of a*,*may be joined at their crests to form a composite wave φ**(**figure*1*i**)*.*If meas*(*φ*^{−1}(*c*))=0,*then φ is a travelling wave of equation**(1.2)*.*(Stumpons)*.*If a*=*c*^{2}*intervals where φ*≡*c are also allowed in the composite waves**(**figure*1*j**)*.

## Acknowledgments

The author thanks the Mittag-Leffler Institute for providing excellent working conditions.

## Footnotes

One contribution of 13 to a Theme Issue ‘Water waves’.

- © 2007 The Royal Society