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Phil. Trans. R. Soc. A (2007) 365, 2291–2298
doi:10.1098/rsta.2007.2009
Published online 14 March 2007
Classification of all travelling-wave solutions
for some nonlinear dispersive equations
B Y J ONATAN L ENELLS *
Department of Mathematics, Lund University, PO Box 118, 22100 Lund, Sweden
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 fractallike wave profiles.
Keywords: travelling waves; Camassa–Holm equation; Degasperis–Procesi equation
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
ut K6uux C uxxx Z 0; x 2R; tO 0;
ð1:1Þ
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
ut K utxx C 3uux Z 2ux uxx C uuxxx ; x 2R; tO 0
ð1:2Þ
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)).
*[email protected]
One contribution of 13 to a Theme Issue ‘Water waves’.
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J. Lenells
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
ut K utxx C 4uux Z 3ux uxx C uuxxx ;
x 2R; tO 0:
ð1:3Þ
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.
(i) While all solutions with initial data uð0; $Þ 2H 3 ðRÞ 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.
(ii) 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).
(iii) 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 1i ). An
interesting class of waves—called stumpons owing to their shape—is obtained by
Phil. Trans. R. Soc. A (2007)
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Classification of travelling-wave solutions
(a)
2293
(b)
j=c
(c)
(d)
j=c
(e)
(f)
j=c
(g)
(h)
j=c
(i)
( j)
j=c
Figure 1. The different kinds of travelling wave solutions of equations (1.2) and (1.3). (a) Smooth
periodic, (b) smooth with decay, (c) peaked periodic, (d ) peaked with decay, (e) cusped periodic,
( f ) cusped with decay, ( g) anticusped periodic, (h) anticusped with decay, (i ) composite waves
and ( j ) stumpons.
j=c
j=c
Figure 2. Two composite travelling waves with a fractal appearance.
inserting intervals where the solution equals a constant at the crests of suitable
cusped waves (figure 1j ). 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 2000a, 2002;
Constantin & Molinet 2001; Lenells 2004a,b). Furthermore, a numerical scheme
implemented by Kalisch & Lenells (2005) suggests that even some cusped and
composite travelling waves could be stable.
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J. Lenells
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 2006a,b).
(i) The class of equations
ut K utxx C 3uux Z gð2ux uxx C uuxxx Þ;
x 2R; t O 0;
ð1:4Þ
arising in elasticity with the physical parameter g ranging from K29.4760
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 nondimensional variables (Dai 1998). The only integrable equation in this
family indexed by g is the Camassa–Holm equation (gZ1; cf. Ivanov
2005), and for g!1 the solitary waves of equation (1.4) are smooth and
stable (Constantin & Strauss 2000b).
(ii) The class of nonlinear partial differential equations
ut Ka2 utxx C ð1 C 2bÞuux Z a2 ð2ux uxx C uuxxx Þ;
x 2R; tO 0;
2
where a R0 is a constant and bR0 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 2005b).
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)Z4(xKct) travelling with speed c, equation (1.3) takes the form
Kc4x C c4xxx C 344x Z 24x 4xx C 44xxx :
ð2:1Þ
We integrate and rewrite to get, for some constant a 2R,
ð2:2Þ
42x C 342 K2c4 Z ðð4KcÞ2 Þxx C a:
1
Since equation (2.2) makes sense whenever 4 2Hloc ðRÞ, the following definition
is natural.
1
Definition. A function 4 2Hloc
ðRÞ is a travelling wave of the Camassa–Holm
equation if there exists an a 2R such that 4 satisfies (2.2) in distributional sense.
(b ) Step 2. Smoothness away from {4Zc}
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 x 2R
where the crest is located satisfies 4(x)Zc, and 4 is smooth on the set Rnfxg.
Lemma 2.1 shows that this is not a coincidence.
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Lemma 2.1 (Lenells 2005a). Let p(v) be a polynomial with real coefficients.
1
Assume that v 2Hloc
ðRÞ satisfies
ðv 2 Þxx Z vx2 C pðvÞ
in distributional sense:
ð2:3Þ
Then
v k 2C j ðRÞ for k R 2 j :
ð2:4Þ
Applying lemma 2.1 with vZ4Kc, we get ð4KcÞk 2C j ðRÞ for kR2j. Taking
the k th square root, we conclude that 4 is smooth except possibly at points in the
boundary of the set 4K1(c).
(c ) Step 3. Characterization of travelling waves
Since 4 is continuous, Cd4K1(c) is a closed set. We deduce the existence of
disjoint open intervals Ei , iR1, such that RnC ZgN
iZ1 Ei . Within each interval Ei
where 4 is smooth, equation (2.2) may be integrated further to yield a first-order
ODE in 4 (equations (2.5) and (2.6)). Using this observation, lemma 2.2, which
characterizes the travelling waves, may be inferred. Observe the condition
(TW2)-(ii ) that the measure of the set of points where 4Zc can be strictly
positive only when aZc2.
1
Lemma 2.2. A function 4 2Hloc
ðRÞ 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 , iR1, and a closed set C such that
RnC ZgN
iZ1 Ei , 4 is smooth and non-constant on each Ei , 4(x)sc for
x 2gN
iZ1 Ei , and 4(x)Zc for x2C.
(TW2) There is an a 2R such that
(i ) for each iR1, there exists bi 2R such that
42x Z Fð4Þ for x 2Ei and 4/ c at any finite endpoint of Ei ;
ð2:5Þ
where
2
Fð4Þ Z
4 ðcK4Þ C a4 C bi
;
cK4
ð2:6Þ
(ii ) if meas(C )O0, then aZc2.
(TW3) ð4KcÞ2 2W 2;1
loc ðRÞ.
(d ) Step 4. Classification
The last step consists of determining the set of bounded functions satisfying
(TW1)–(TW3). Suppose we could find all solutions 4 of (2.5) and (2.6) for different
intervals Ei and different values of a and bi. Then we can join solutions defined on
intervals Ei whose union is RnC for some closed set C of measure 0. The function,
defined on R, that we get, will satisfy (TW1) and (TW2) if and only if all wave
segments satisfy (2.5) with the same a. Also, if we, for aZc2, allow meas(C )O0,
this procedure will give us all functions satisfying (TW1) and (TW2). The
following technical result (see Lenells (2005a) for a proof ) then shows that these
solutions automatically satisfy the regularity condition (TW3) also.
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2296
J. Lenells
Lemma 2.3. Any bounded function 4 satisfying (TW1) and (TW2) belongs to
1
Hloc
ðRÞ and satisfies (TW3).
The analysis of equation (2.5) is based on the following observations.
(i) Assume F(4) has a simple 0 at 4Zm so that F 0 (m)s0. Then a solution 4 of
equation (2.5) satisfies 42x Z ð4KmÞF 0 ðmÞC Oðð4KmÞ2 Þ as 4Ym. Hence
1
4ðxÞ Z m C ðx K x 0 Þ2 F 0 ðmÞ C Oððx K x 0 Þ4 Þ as x / x 0 ;
ð2:7Þ
4
where 4(x 0)Zm.
(ii) If F(4) instead has a double zero at m, so that F 0 (m)Z0, F 00 (m)s0, we
obtain 42x Z ð4KmÞ2 F 00 ðmÞC Oðð4KmÞ3 Þ as 4Ym, so that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4ðxÞKm wa expðKx jF 00 ðmÞjÞ as x /N;
ð2:8Þ
for some constant a. Thus, 4/m exponentially as x/N.
(iii) Suppose 4 approaches a simple pole 4Zc of F. Then, if 4(x 0)Zc,
4ðxÞKc Z ajx K x 0 j2=3 C Oððx K x 0 Þ4=3 Þ
as x / x 0 ;
ð2:9Þ
for some constant a. In particular, whenever F has a simple pole, the
solution 4 has a cusp.
(iv) Peakons occur when the evolution of 4 according to 42x Z Fð4Þ suddenly
changes direction: 4x 1K4x .
Writing F(4) in equation (2.5) as
ðM K4Þð4KmÞð4KzÞ
;
Fð4Þ Z
cK4
where M, m and z are the three zeros of the third-order polynomial 42(cK4)C
a4Cbi , 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.
Theorem 2.1. Let zZcKMKm. Any bounded travelling wave of equation (1.2)
falls into one of the following categories.
(i ) (Smooth periodic). If z!m!M!c, there is a smooth periodic travelling
wave 4(xKct) of equation (1.2) with mZ minx2R 4ðxÞ a nd
M Z maxx2R 4ðxÞ.
(ii ) (Smooth with decay). If zZm!M!c, there is a smooth travelling wave
4(xKct) of equation (1.2) with mZ inf x2R 4ðxÞ, M Z maxx2R 4ðxÞ and
4Ym exponentially as x/GN.
(iii ) (Periodic peakons). If z!m!MZc, there is a periodic peaked travelling
wave 4(xKct) of equation (1.2) with mZ minx2R 4ðxÞ a nd
M Z maxx2R 4ðxÞ.
(iv) (Peakons with decay). If zZm!MZc, there is a peaked travelling wave
4(xKct) of equation (1.2) with mZ inf x2R 4ðxÞ, M Z maxx2R 4ðxÞ and
4Ym exponentially as x/GN.
(v) (Periodic cuspons). If z!m!c!M, there is a periodic cusped travelling
wave 4(xKct) of equation (1.2) with mZ minx2R 4ðxÞ a nd
cZ maxx2R 4ðxÞ.
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Classification of travelling-wave solutions
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(vi ) (Cuspons with decay). If zZm!c!M there is a cusped travelling wave
4(xKct) of equation (1.2) with mZ inf x2R 4ðxÞ, cZ maxx2R 4ðxÞ and
4Ym exponentially as x/GN.
Replacing (4(x),c, m, M ) with (K4(Kx), Kc, Km, KM ) in (i)–(vi) yields dual
classes of waves travelling in the opposite direction. In particular, anticuspons
arise ( figure 1g,h).
(vii ) (Composite waves). For any fixed c 2R and aOKðc2 =3Þ, let
a ZKMmKðM C mÞðcKM KmÞ:
ð2:10Þ
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 4 ( figure 1i ). If meas(4K1(c))Z0, then 4 is a travelling wave of
equation (1.2).
(viii ) (Stumpons). If aZc2 intervals where 4hc are also allowed in the
composite waves ( figure 1j ).
The author thanks the Mittag-Leffler Institute for providing excellent working conditions.
References
Beals, R., Sattinger, D. & Szmigielski, J. 1998 Acoustic scattering and the extended Korteweg–de
Vries hierarchy. Adv. Math. 40, 190–206. (doi:10.1006/aima.1998.1768)
Camassa, R. & Holm, D. 1993 An integrable shallow water equation with peaked solitons. Phys.
Rev. Lett. 71, 1661–1664. (doi:10.1103/PhysRevLett.71.1661)
Colliander, J., Keel, M., Staffilani, G., Takaoka, H. & Tao, T. 2003 Sharp global well-posedness for
KdV and modified KdV on R and T. J. Am. Math. Soc. 16, 705–749. (doi:10.1090/S0894-034703-00421-1)
Constantin, A. 1997 On the Cauchy problem for the periodic Camassa–Holm equation. J. Differ.
Equations 141, 218–235. (doi:10.1006/jdeq.1997.3333)
Constantin, A. 1998 On the inverse spectral problem for the Camassa–Holm equation. J. Funct.
Anal. 155, 352–363. (doi:10.1006/jfan.1997.3231)
Constantin, A. 2000 Existence of permanent and breaking waves for a shallow water equation: a
geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362.
Constantin, A. 2001 On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. A
457, 953–970. (doi:10.1098/rspa.2000.0701)
Constantin, A. & Escher, J. 1998a Wave breaking for nonlinear nonlocal shallow water equations.
Acta Math. 181, 229–243. (doi:10.1007/BF02392586)
Constantin, A. & Escher, J. 1998b Well-posedness, global existence, and blowup phenomena for a
periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51, 475–504. (doi:10.
1002/(SICI)1097-0312(199805)51:5%3C475::AID-CPA2%3E3.0.CO;2-5)
Constantin, A. & Escher, J. 2000 On the blow-up rate and the blow-up set of breaking waves for a
shallow water equation. Math. Z. 233, 75–91. (doi:10.1007/PL00004793)
Constantin, A. & Ivanov, R. 2006 Poisson structure and action-angle variables for the Camassa–
Holm equation. Lett. Math. Phys. 76, 93–108. (doi:10.1007/s11005-006-0063-9)
Constantin, A. & Kolev, B. 2003 Geodesic flow on the diffeomorphism group of the circle.
Comment. Math. Helv. 78, 787–804. (doi:10.1007/s00014-003-0785-6)
Constantin, A. & McKean, H. P. 1999 A shallow water equation on the circle. Commun. Pure Appl.
Math. 52, 949–982. (doi:10.1002/(SICI)1097-0312(199908)52:8!949::AID-CPA3O3.0.CO;2-D)
Constantin, A. & Molinet, L. 2001 Orbital stability of solitary waves for a shallow water equation.
Physica D 157, 75–89. (doi:10.1016/S0167-2789(01)00298-6)
Constantin, A. & Strauss, W. 2000a Stability of peakons. Commun. Pure Appl. Math. 53, 603–610.
(doi:10.1002/(SICI)1097-0312(200005)53:5!603::AID-CPA3O3.0.CO;2-L)
Phil. Trans. R. Soc. A (2007)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 18, 2017
2298
J. Lenells
Constantin, A. & Strauss, W. 2000b Stability of a class of solitary waves in compressible elastic
rods. Phys. Lett. A 270, 140–148. (doi:10.1016/S0375-9601(00)00255-3)
Constantin, A. & Strauss, W. 2002 Stability of the Camassa–Holm solitons. J. Nonlin. Sci. 12,
415–422. (doi:10.1007/s00332-002-0517-x)
Dai, H.-H. 1998 Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin
rod. Acta Mech. 127, 193–207. (doi:10.1007/BF01170373)
Degasperis, A. & Procesi, M. 1999 Asymptotic integrability. In Symmetry and perturbation theory
(eds A. Degasperis & G. Gaeta), pp. 23–37. Singapore: World Scientific.
Degasperis, A., Holm, D. D. & Hone, A. N. W. 2002 A new integrable equation with peakon
solutions. Theor. Math. Phys. 133, 1463–1474. (doi:10.1023/A:1021186408422)
Drazin, P. G. & Johnson, R. S. 1989 Solitons: an introduction. Cambridge, UK: Cambridge
University Press.
Dullin, H., Gottwald, G. & Holm, D. 2001 An integrable shallow water equation with linear and
nonlinear dispersion. Phys. Rev. Lett. 87, 4501–4504. (doi:10.1103/PhysRevLett.87.194501)
Ivanov, R. 2005 On the integrability of a class of nonlinear dispersive wave equations. J. Nonlin.
Math. Phys. 12, 462–468. (doi:10.2991/jnmp.2005.12.4.2)
Kalisch, H. & Lenells, J. 2005 Numerical study of traveling-wave solutions for the Camassa–Holm
equation. Chaos Solitons Fractals 25, 287–298. (doi:10.1016/j.chaos.2004.11.024)
Kolev, B. 2004 Lie groups and mechanics: an introduction. J. Nonlin. Math. Phys. 11, 480–498.
(doi:10.2991/jnmp.2004.11.4.5)
Lenells, J. 2002 The scattering approach for the Camassa–Holm equation. J. Nonlin. Math. Phys.
9, 389–393. (doi:10.2991/jnmp.2002.9.4.2)
Lenells, J. 2004a Stability of periodic peakons. Int. Math. Res. Not. 10, 485–499. (doi:10.1155/
S1073792804132431)
Lenells, J. 2004b A variational approach to the stability of periodic peakons. J. Nonlin. Math. Phys.
11, 151–163. (doi:10.2991/jnmp.2004.11.2.2)
Lenells, J. 2005a Traveling wave solutions of the Camassa–Holm equation. J. Differ. Equations
217, 393–430. (doi:10.1016/j.jde.2004.09.007)
Lenells, J. 2005b Traveling wave solutions of the Degasperis–Procesi equation. J. Math. Anal. Appl.
306, 72–82. (doi:10.1016/j.jmaa.2004.11.038)
Lenells, J. 2006a Traveling waves in compressible elastic rods. Disc. Cont. Dyn. Sys. B 6, 151–168.
Lenells, J. 2006b Classification of traveling waves for a class of nonlinear wave equations. J. Dyn.
Differ. Equations 18, 381–391. (doi:10.1007/s10884-006-9009-2)
Li, Y. & Olver, P. 1997 Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian
dynamical system. I. Compactons and peakons. Discrete Cont. Dyn. Syst. 3, 419–432.
McKean, H. P. 1979 Integrable systems and algebraic curves. In Global analysis, vol. 755 (eds
M. Grmela & J. E. Marsden). Springer lecture notes in mathematics, pp. 83–200. Berlin,
Germany: Springer.
Misiolek, G. 1998 A shallow water equation as a geodesic flow on the Bott–Virasoro group.
J. Geom. Phys. 24, 203–208. (doi:10.1016/S0393-0440(97)00010-7)
Yin, Z. 2003 On the Cauchy problem for an integrable equation with peakon solutions. Illinois
J. Math. 47, 649–666.
Zhou, Y. 2004 Blow-up phenomenon for the integrable Degasperis–Procesi equation. Phys. Lett. A
328, 157–162. (doi:10.1016/j.physleta.2004.06.027)
Phil. Trans. R. Soc. A (2007)