Stability of peakons

Stability of Peakons

ADRIAN CONSTANTINUniversity of Zurich

AND

WALTER A. STRAUSSBrown University

Abstract

The peakons are peaked solitary wave solutions of a certain nonlinear dispersiveequation that is a model in shallow water theory and the theory of hyperelasticrods. We give a very simple proof of the orbital stability of the peakons in theH1 norm. c© 2000 John Wiley & Sons, Inc.

1 Introduction

The nonlinear dispersive equation

ut −utxx+3uux = 2uxuxx+uuxxx(1.1)

is an integrable equation whose solitary waves are peaked solitons. Its solutionseither exist globally in time or are breaking waves. The solitary waves (peakons)are

u(x,t) = cϕ(x−ct) = ce−|x−ct| , c∈ R .(1.2)

Equation (1.1) can be rewritten in conservation form as

ut +12

(u2 +ϕ∗

[u2 +

12

u2x

])x= 0.(1.3)

This is the precise sense in which the peakons can be understood as solutions.It turns out that the peakons are solitons and therefore their sizes and velocities

do not change as a result of collision, so that it is reasonable to expect that theyare stable. Because a small perturbation of a solitary wave can yield another onewith a different speed and phase shift, the appropriate notion of stability is orbitalstability. That is, a wave starting close to a solitary wave remains close to sometranslate of it at all later times. Thus the shape of the wave remains approximatelythe same for all times.

Communications on Pure and Applied Mathematics, Vol. LIII, 0603–0610 (2000)c© 2000 John Wiley & Sons, Inc.

604 A. CONSTANTIN AND W. A. STRAUSS

THEOREM If u ∈C([0,T);H1(R)) is a solution to(1.1)with

‖u(0,·)−cϕ‖H1 <

(ε

3c

)4

, 0< ε < c,

then

‖u(t, ·)−cϕ(·− ξ(t))‖H1 < ε for t ∈ (0,T) ,

whereξ(t) ∈ R is any point where the function u(t, ·) attains its maximum.

For the precise definition of a solution, see Section 2. The theorem gives aquantitative estimate of how closely the wave must approximate the peakon ini-tially in order to beε-close to some translate of the peakon at all later times. Thattranslate is located at a point where the wave is tallest. Even if a wave that is near apeakon breaks, its shape remains approximately the same up to the breaking time.For waves that approximate the peakons in a special way, a stability result wasproved in [9] by a different method (see Section 3).

It was first noticed by Fuchssteiner and Fokas [11] that equation (1.1) has abi-Hamiltonian structure and therefore an infinite number of conservation laws.Camassa and Holm [3] independently derived (1.1) as a model for shallow waterwaves whereu(t,x) represents the free surface above a flat bottom. They foundthe peakons, showed their soliton interaction, and constructed a Lax pair for theequation. Reference [4] also contains numerical calculations.

Equation (1.1) was also found independently by Dai [10] as a model for nonlin-ear waves in cylindrical hyperelastic rods withu(t,x) representing the radial stretchrelative to a prestressed state. Solitary waves in solids are very interesting from thepoint of view of applications, as they are easy to detect bcause they do not changetheir shapes during propagation and can be used to determine material propertiesand to detect flaws [17].

Section 2 is devoted to the proof of the theorem. Because the existence ofsolutions is implicitly assumed in our theorem, we include in Section 3 a series ofremarks about the current state of the theory of existence and wave breaking forequation (1.1).

2 Proof of Stability

Equation (1.1) has the following conservation laws:

E(u) :=∫

R

(u2 +u2x)dx, F(u) :=

∫

R

(u3 +uu2x)dx.(2.1)

We call u ∈ C([0,T);H1(R)) a solution to (1.1) ifu is a solution of (1.3) in thesense of distributions andE andF are conserved.

We proceed toward the proof of the theorem in several steps. For simplicity, wetakec = 1, the casec = 0 being trivial asE is conserved.

STABILITY OF PEAKONS 605

LEMMA 1 For every u∈ H1(R) andξ ∈ R,

E(u)−E(ϕ) = ‖u−ϕ(·− ξ)‖2H1 +4u(ξ)−4.

PROOF: We calculate

‖u−ϕ(·− ξ)‖2H1 = E(u)+E(ϕ(·− ξ))−2

∫

R

ux(x)ϕx(x− ξ)dx

−2∫

R

u(x)ϕ(x− ξ)dx

= E(u)+E(ϕ)−2∫ ξ

−∞ux(x)ϕ(x− ξ)dx+2

∫ ∞

ξux(x)ϕ(x− ξ)dx

−2∫

R

u(x)ϕ(x− ξ)dx

= E(u)+E(ϕ)−4u(ξ) = E(u)−E(ϕ)−4u(ξ)+4

using integration by parts andE(ϕ) = 2.

LEMMA 2 For u∈ H1(R), let M = maxx∈R{u(x)}. Then

F(u) ≤ ME(u)− 23

M3 .

PROOF: Let M be taken atx = ξ and define

g(x) :=

{u(x)−ux(x) , x < ξ ,

u(x)+ux(x) , x > ξ .

We calculate∫

R

g2(x)dx=∫ ξ

−∞[u(x)−ux(x)]2dx+

∫ ∞

ξ[u(x)+ux(x)]2dx

=∫

R

[u2(x)+u2x(x)]dx−u2(x)

∣∣∣ξ−∞

+u2(x)∣∣∣∞

ξ

= E(u)−2u2(ξ) = E(u)−2M2 .

(2.2)

Similarly,∫

R

u(x)g2(x)dx=∫ ξ

−∞u(x)[u(x)−ux(x)]2dx+

∫ ∞

ξu(x)[u(x)+ux(x)]2dx

=∫

R

u(x)[u2(x)+u2x(x)]dx− 2

3u3(x)

∣∣∣ξ−∞

+23

u3(x)∣∣∣∞

ξ

= F(u)− 43

u3(ξ) = F(u)− 43

M3 .


Thus

F(u)− 43

M3 =∫

R

u(x)g2(x)dx≤ M∫

R

g2(x)dx= ME(u)−2M3 .

LEMMA 3 For u∈ H1(R), if ‖u−ϕ‖H1 < δ, then

|E(u)−E(ϕ)| ≤ δ(2√

2+ δ)

and |F(u)−F(ϕ)| ≤ δ

(3√

2+3δ +1√2δ2

).

PROOF: Identity (2.2) shows that for allv ∈ H1(R),

supx∈R

|v(x)| ≤ 1√2

E(v)1/2 =1√2‖v‖H1 .(2.3)

Equality holds if and only ifv is proportional to a translate ofϕ. Note that

|E(u)−2| = |E(u)−E(ϕ)| = ∣∣(‖u‖H1 −‖ϕ‖H1

)(‖u‖H1 +‖ϕ‖H1

)∣∣≤ δ(2‖ϕ‖H1 + δ) = δ(2

√2+ δ) .

(2.4)

Similarly,∣∣∣∣F(u)− 43

∣∣∣∣ = |F(u)−F(ϕ)|

=∣∣∣∣∫

R

(u−ϕ)(u2 +u2x)dx+

∫

R

ϕ(u2 +u2x −ϕ2−ϕ2

x)dx

∣∣∣∣=

∣∣∣∣∫

R

(u−ϕ)(u2 +u2x)dx+

∫

R

ϕ[(u−ϕ)2 +(ux−ϕx)2]dx

+∫

R

ϕ[2(u−ϕ)ϕ+2(ux−ϕx)ϕx]dx

∣∣∣∣≤ ‖u−ϕ‖L∞E(u)+‖ϕ‖L∞‖u−ϕ‖2

H1 +2‖ϕ‖L∞‖u−ϕ‖H1‖ϕ‖H1

≤ 1√2

δ(2+2

√2δ + δ2)+ δ2 +2

√2δ

= δ

(3√

2+3δ +1√2

δ2)

by (2.3) and (2.4).

LEMMA 4 For u∈ H1(R), let M = maxx∈R{u(x)}. If

|E(u)−E(ϕ)| ≤ δ(2√

2+ δ)

and |F(u)−F(ϕ)| ≤ δ

(3√

2+3δ +1√2

δ2)

for someδ < 120, then

|M−1| ≤ 2√

δ .


PROOF: Note that, sinceE is near 2 andF is near43,

M ≥ FE

>12

.(2.5)

By Lemma 2,

M3− 32

ME(u)+32

F(u) ≤ 0.(2.6)

Consider the cubic polynomialP(y) = y3 − 32yE(u) + 3

2F(u). In caseE(u) =E(ϕ) = 2 andF(u) = F(ϕ) = 4

3, it takes the form

P0(y) = y3−3y+2 = (y−1)2(y+2) .

Because of the estimates onE(u) andF(u), there is a root ofP neary = −2, andthere may be two roots neary = 1. But (2.5) and (2.6) show that there must betwo roots neary = 1 andM must lie between these two roots. These two roots arecloser toy = 1 than the roots of the cubic

P1(y) = y3− 32

y(2+2

√2δ + δ2)+

32

(43−3

√2δ−3δ2− 1√

2δ3

),

whose graph onR+ lies below the graph ofP. Notice thatP1(1) < 0 andP1(1±2√

δ) > 0. Hence we reach the stated conclusion.

PROOF OF THETHEOREM: SinceE andF are both conserved by the evolutionequation (1.1), we have

E(u(t, ·)) = E(u0) and F(u(t, ·)) = F(u0) , t ∈ (0,T) .(2.7)

We apply Lemma 3 tou0 and toδ = 181ε

4. By assumption,ε < 1 so thatδ < 120. By

(2.7) the hypotheses of Lemma 4 are satisfied foru(t, ·). Hence

|u(t,ξ(t))−1| ≤ 29ε2 , t ∈ (0,T) .(2.8)

Combining (2.7) with Lemma 1, we find

‖u(t, ·)−ϕ(·− ξ(t))‖2H1 = E(u0)−E(ϕ)+4−4u(t,ξ(t)) , t ∈ (0,T) .

Therefore, by (2.8) and Lemma 3,

‖u(t, ·)−ϕ(·− ξ(t))‖2H1 ≤ 1

81ε4

(2√

2+181

ε4)

+89

ε2 ≤ ε2 , t ∈ (0,T) .


3 Comments

The only way that a classical solution of equation (1.1) may fail to exist for alltime is that the wave may break [5]. This means that the solution remains boundedwhile its slope becomes unbounded at a finite time while theH1(R) norm remainsconstant.

We now discuss the issue of well-posedness. It is known [6] that ifu0 ∈ H3(R),then there exists a maximal timeT = T(u0) > 0 such that (1.1) has a unique solu-tion u∈C([0,T);H3(R))∩C1([0,T);H2(R)) with E andF conserved.

Under some conditions the solution is global. Associate to each initial profileu0 the expressiony0 := u0−u0,xx. If y0 does not change sign, the solution is global[6]. The solution is also global if(x−x0)y0(x) ≥ 0 for somex0 ∈ R [5].

Wave breaking occurs ifu0 ∈ H3(R) and eitheru0 is odd andu′0(0) < 0 [6], orthe derivative ofu0 at some point is less than− 1√

2‖u0‖H1(R) [7]. If y0 ∈ H1(R) has

finitely many proper changes of sign and the condition(x−x0)y0(x) ≥ 0 for somex0 ∈ R fails, then breakdown must occur [14].

For initial profiles that are less regular,u0 ∈ Hs(R) with s > 32, it is known

[13, 16], that (1.1) has a unique solution inC([0,T);Hs(R)) for someT > 0 withE andF conserved.

These remarks point out the fact that our theorem can be applied to every initialprofileu0 ∈ Hs(R) with s> 3

2 that is close to some peakon. Note that foru0 = cϕ,y0 = 2cδ, whereδ is the Dirac distribution. The results formulated above in termsof y0 show that, arbitrarily close to a peakoncϕ, there exist waves that break andalso waves that do not break. Our theorem is applicable in both cases up to thebreaking time.

In [9], a variational approach characterizes the solitary waves of equation (1.1)as minima of constrained energy. Combining concentration compactness withcompensated compactness, the orbital stability is proved providedu0 ∈ H3(R) issuch thaty0 is nonnegative and integrable.

It is interesting to observe that a functional for which the peakonϕ is a mini-mizer has the formL = G+aE+bF, where

G(u) :=∫

R

u2(

12

u2 +u2x

)dx+

∫

R

(u2 +

12

u2x

)ϕ∗

(u2 +

12

u2x

)dx

is another invariant for (1.1). We calculate thatL′(ϕ) = 0 if and only ifa+b+2=0. Furthermore, with this choice

〈L′′(ϕ)v,v〉 = 4(b+4)v2(0)−2(b+2)∫

R

(1−ϕ(x))[v(x)+sgn(x)vx(x)]2dx

is positive for−4 < b < −2 andv 6≡ 0.


The conservation form (1.3) is reminiscent of the three-dimensional Euler equa-tion

vt +(v ·grad)v +gradp = 0, ∇ ·v = 0,(3.1)

of which (1.1) is an approximation in the shallow water regime [4]. Equation (1.1)is a re-expression of the geodesic flow in the group of compressible diffeomor-phisms of the line [8, 12, 15]. This resembles the fact that the Euler equation is anexpression of the geodesic flow in the group of incompressible diffeomorphisms ofR

3 [1].The Euler equation (3.1) has many approximations in the shallow water regime,

including (1.1), KdV, BBM, and Whitham’s equation. These four equations reflectsome of the interesting properties of the Euler equation. In particular, among theseequations, only KdV and (1.1) are integrable and have solitons, while only (1.1)and Whitham’s equation have waves that break as well as peaked solitary waves.On the other hand, all classical solutions of KdV and BBM are global in time. TheEuler equation has breaking waves [2] and a traveling wave solution, the wave ofgreatest height, which has a corner at its crest [18].

The single equation (1.1) possesses all of these interesting properties simulta-neously. Indeed, inequality (2.3) means that, among all waves of fixed energyE,the peakon is the tallest. Hence the tallest wave has a peak at its crest.

Acknowledgment.We thank the referee for some helpful comments.

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ADRIAN CONSTANTIN WALTER A. STRAUSS

University of Zurich Brown UniversityInstitute for Mathematics Department of Mathematics, Box 1917Winterthurerstrasse 190 Lefschetz Center for Dynamical SystemsCH-8057 Zurich Providence, RI 02912SWITZERLAND E-mail:[email protected]: [email protected]

Received March 1999.

Stability of peakons

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