orbital stability of periodic peakons to a generalized μ-camassa–holm equation

Digital Object Identifier (DOI) 10.1007/s00205-013-0672-2Arch. Rational Mech. Anal. 211 (2014) 593–617

Orbital Stability of Periodic Peakonsto a Generalized μ-Camassa–Holm Equation

Changzheng Qu, Ying Zhang, Xiaochuan Liu & Yue Liu

Communicated by F. Lin

Abstract

In this paper, we study the orbital stability of the periodic peaked solitonsof the generalized μ-Camassa–Holm equation with nonlocal cubic and quadraticnonlinearities. The equation is aμ-version of a linear combination of the Camassa–Holm equation and the modified Camassa–Holm equation. It is also integrable withthe Lax-pair and bi-Hamiltonian structure and admits the single peakons and multi-peakons. By constructing an inequality related to the maximum and minimumof solutions with the conservation laws, we prove that, even in the case that theCamassa–Holm energy counteracts in part the modified Camassa–Holm energy,the shapes of periodic peakons are still orbitally stable under small perturbationsin the energy space.

1. Introduction

We consider the following generalizedμ-Camassa–Holm (μ-CH) equation [35]

yt + k1

((2μ(u)u − u2

x )y)

x+ k2 (2yux + uyx ) = 0, t > 0, x ∈ S, (1.1)

where S = R/Z denotes the unit circle on R2, k1 and k2 are two constants, u(t, x)

is a real-valued spatially periodic function and y = μ(u) − uxx with the mean ofu, that is, μ(u) = ∫

Su(t, x) dx . It is observed that equation (1.1) reduces to the

μ-CH equation [24]

yt + 2yux + uyx = 0, y = μ(u)− uxx , (1.2)

if k1 = 0, k2 = 1, and the modified μ-CH equation [34]

yt +((2μ(u)u − u2

x )y)

x= 0, y = μ(u)− uxx , (1.3)

594 Changzheng Qu, Ying Zhang, Xiaochuan Liu & Yue Liu

if k1 = 1, k2 = 0, respectively. (1.1) was introduced in [35] as a μ-version of thegeneralized Camassa–Holm equation with quadratic and cubic nonlinearities

yt + k1

((u2 − u2

x ) y)

x+ k2 (2yux + uyx ) = 0, y = u − uxx , (1.4)

which was derived by Fokas [16] from the hydrodynamical wave, and can alsobe obtained using the approach of tri-Hamiltonian duality [18,32] to the bi-Hamiltonian Gardner equation

ut + uxxx + k1u2ux + k2uux = 0. (1.5)

Note that the Lax pair of (1.4) was obtained in [33].It was shown in [35] that a scale limit of equation (1.1) yields the following

integrable equation

vxt − k1v2xvxx + k2

(vvxx + 1

2v2

x

)= 0,

which describes asymptotic dynamics of a short capillary-gravity wave [15], wherev(t, x) denotes the fluid velocity on the surface.

The μ-CH equation (1.2), regarded as a μ-version of the Camassa–Holm equa-tion, was introduced first in [24] by Khesin, Lenells and Misiolek, and models theevolution of rotators in liquid crystals with an external magnetic field and self-interaction. It is interesting to note that this equation is integrable in the sense thatit admits the Lax-pair and bi-Hamiltonian structure, and also describes a geodesicflow on the diffeomorphism group of S with Hμ(S) metric (which is equivalentto H1(S)metric). Its integrability, well-posedness, blow-up and peakons were dis-cussed in [17,24].

The Camassa–Holm (CH) equation [1,19]

yt + uyx + 2ux y = 0, y = u − uxx (1.6)

was proposed as a model for the unidirectional propagation of the shallow waterwaves over a flat bottom, with u(t, x) representing the water’s free surface in non-dimensional variables [1] (for a discussion of the physical relevance of equation(1.6), see [10,23]). It was found by using the method of recursion operators due toFokas and Fuchssteiner [19], and it can also be derived by tri-Hamiltonian dualityfrom the KdV equation [18,32]. Such derivations reveal properties of (1.6) as anintegrable system. Interestingly, the CH equation (1.6) has several nice geometricformulations [4,9,25,30], which provide us with new insights to understand itsproperties. Well-posedness and wave breaking of the CH equation have been studiedextensively, and many interesting results have been obtained, see the references [5–8,28], for example. A scale limit equation to the CH equation is the integrableHunter-Saxton (HS) equation [22]

uxt + uuxx + 1

2u2

x = 0. (1.7)

A midway equation between the CH equation and HS equation is the so-calledμ-CH equation (1.2).

Orbital Stability of Peakons to a Generalized μ-CH Equation 595

The modified μ-CH equation (1.3) was introduced in [34] as a μ-version of themodified CH equation with cubic nonlinearity [16,18,32], that is

yt +((u2 − u2

x ) y)

x= 0, y = u − uxx . (1.8)

Equation (1.8) can be derived by using the tri-Hamiltonian duality approachto the bi-Hamiltonian representation of the modified KdV equation [18,32]. It wasshown that (1.3) and (1.8) arise from invariant non-stretching curve flows in theEuclidean geometry and two-dimensional sphere [20,34]. Indeed, the modified μ-CH equation (1.3) is equivalent to the following Euclidean invariant plane flow forthe curve C ∈ R

2 [34]

∂C

∂t= −2usn +

(2 − (2μ(u)u − u2

s ))

t,

where s is the arc-length parameter of the curve C,n and t denote the unit normaland tangent vector of the curve. It is worth mentioning that equation (1.3) andequation (1.8) have new features of wave breaking, blowup criteria and peakedsoliton structures [20,34], compared to the CH equation and the μ-CH equation. Itis also noticed that the short pulse equation

vxt = 1

6(v3)xx + γ v (1.9)

is a scaling limit equation of equation (1.3) and equation (1.8) with the first-orderterm ux , which was derived by Schäfer and Wayne [37] as a model for the propa-gation of ultra-short light pulses in silica optical fibers; it is also an approximationof nonlinear wave packets in dispersive media in the limit of few cycles on theultra-short pulse scale.

It has been known that the generalized μ-CH equation (1.1) is formally inte-grable. Indeed, it can be written in the following bi-Hamiltonian form [35]

yt = JδH1

δy= K

δH2

δy,

where

J = 1

2

(k1(−∂x y ∂−1

x y ∂x )+ k2(−y ∂x − ∂x y))

and K = ∂3x

are compatible Hamiltonian operators, while

H1[u] = μ2(u)+∫

S

u2x dx, (1.10)

and

H2[u] = k1

∫

S

(μ2(u)u2 + μ(u)uu2

x − 1

12u4

x

)dx

+ k2

∫

S

(μ(u)u2 + 1

2uu2

x

)dx (1.11)


are the corresponding Hamiltonian functionals. Note that H0[u] = μ(u) = ∫S

u(t, x) dx is another conservation law of (1.1). Equation (1.1) also admits the fol-lowing Lax formulation [35]

(ψ1ψ2

)

x= U (y, λ)

(ψ1ψ2

), U (y, λ) =

(0 λ y

k1λ y + k2λ 0

),

(ψ1ψ2

)

t= V (y, u, λ)

(ψ1ψ2

), V (y, u, λ) =

( 12 k2 ux P

Q − 12 k2 ux

),

with

P = −1

2λ−1μ(u)+ k1λ(2μ(u)u − u2

x )y + k2λ uy,

Q = −k2(1

2λ−1 − u)− 1

2λ−1k1μ(u)+ k2

1λ(2μ(u)u − u2x )y

+ k1k2λ((2μ(u)u − u2

x )+ uy)

− (k2 − k22λ)u.

One remarkable property of the nonlinear integrable equations with nonlineardispersion is the existence of (periodic) peaked solitons. Usually, the dual nonlinearintegrable systems, such as the CH equation and the modified CH equation, areendowed with nonlinear dispersion; they admit peaked solitons. It has been shownthat the two systems admit single peakons of the form

u(t, x) = ϕc(x − ct) = be−|x−ct |, (1.12)

where the amplitude b is given by the constants c and√

3c/2, for the CH equation[1] and the modified CH equation [20], respectively. Their corresponding periodicpeakons take the form

u(t, x) = ϕc(x − ct) = bcosh(x − ct − [x − ct] − 1

2 )

cosh( 12 )

, (1.13)

where the notation [ξ ] denotes the greatest integer part of ξ, and the amplitude bis also given by the wave speed c for the CH equation and√

3c cosh2( 1

2

)/(1 + 2 cosh2

( 12

))for the modified CH equation [36], respec-

tively. Both of equations also admit multi-peakon solutions

u(t, x) =N∑

i=1

pi (t) e−|x−qi (t)|,

where pi (t) and qi (t) satisfy the system, respectively, for the CH equation [1,13]

⎧⎪⎨⎪⎩

q i = ∑j

p j e−|qi −q j |,

pi = ∑j �=i

pi p j sign (qi − q j ) e−|qi −q j |,(1.14)


and for the modified CH equation [20]⎧⎪⎨⎪⎩

q i = 23 p2

i + 2N∑

j=1pi p j e−|qi −q j | + 4

∑1�k<i,i< j�N

pk p j e−|q j −qk |,

pi = 0,

Note that the rigorous analysis for the systems of pi and qi defined in (1.14)was verified in [21]. The solutions consisting of a train of infinitely many peakedsolitary waves were established in [2].

Recently, it was found that the μ-CH equation [24,27] and the modified μ-CHequation [34] admit periodic peakons of the form

u(t, x) = ϕc(x − ct) = aϕ(x − ct), (1.15)

where

ϕ(x) = 1

2

(x2 + 23

12

), x ∈ [− 1

2 ,12

], (1.16)

and ϕ is extended periodically to the real line, and the amplitude a takes values12c/13 and 2

√3c/5, respectively, for the μ-CH equation and the modified μ-CH

equation.More interestingly, in Section 2, we shall show that equation (1.1) admits the

periodic peakon, which is given by (1.15) with a replaced by

a =−13k2 ±

√169k2

2 + 1200ck1

50k1, (1.17)

where the wave speed c satisfies 169k22 + 1200ck1 � 0.

The aim of this paper is to investigate the stability of periodic peaked solitons ofthe generalized μ-CH equation (1.1). Localized, peaked traveling-wave solutionsof certain nonlinear, dispersive wave equations are known in many circumstancesto play a distinguished role in the long-time evolution of an initial disturbance.Therefore, because the issues are interesting in their own right, the dynamicalstability of these special solutions has been a central theme of development for manyyears. In an intriguing paper by Constantin and Strauss [12], it was verified that thesingle peakons of the CH equation (1.6) are orbitally stable. Their approach reliesheavily on the conservation laws of the CH equation and the features of the peakons.This allowed them to establish an inequality relating to the conservation laws withthe maximal value of approximate solutions. Such an approach can be extended tostudy the orbital stability of single peakons for the Degasperis–Procesi equation [29]and the modified CH equation [36]. A variational approach for proving the orbitalstability of the peakons was established by Constantin and Molinet [11]. For thetrains of peaked solitons to the CH equation, their orbital stability was established byDika and Molinet [14] using the approaches in [12] and [31]. Stability of the periodicpeaked solitons of the CH equation was proved by Lenells [26]. The approach in[26] was further extended in [3] to prove the orbital stability of the periodic peakonsfor the μ-CH equation [24]. In this paper, we shall establish the following stabilityresult on the single peakons of the generalized μ-CH equation (1.1).


Theorem 1.1. Assume that k1 > 0, k2 > 0, and c � −169k22/(1200k1) or k1 >

0, k2 � 0 and c > 49k22/(16k1). The periodic peakons ϕc of the generalized μ-CH

equation (1.1) are orbitally stable in the energy space H 1(S).

Due to the conservation law H1, it is expected that there will be orbital stabilityof periodic peakons for the generalized μ-CH equation (1.1) in the sense of theenergy space H1 norm. It is found that equation (1.1) consists of two parts: the cubicnonlinear term and the quadratic nonlinear term. Regarding the signs of k1 and k2,we shall consider two possibilities: (1) k1 > 0 and k2 > 0 (2) k1 > 0, k2 � 0. Forthe case of k1 > 0 and k2 > 0, our approach is inspired by [3,26], where the stabilityof periodic peakons for the μ-CH equation and the CH equation was established.The most difficult part is to establish a suitable inequality relating the maximumand minimum of solutions to conservation laws H0, H1 and H2. The case of k1 > 0and k2 � 0 is much more subtle to deal with because of the interaction between twoparts of the energy H2 with the different signs. In this case, however, it is of interestto find that the first part related to H2 with a cubic nonlinear part in (1.1), namely,

J1[u] =∫

S

(μ2(u)(u − m)2 + μ(u)(u − m)u2

x − 1

12u4

x

)dx

can be dominated by the second part of H2 with the quadratic nonlinear part in(1.1),

J2[u] =∫

S

(μ(u)(u − m)2 + 1

2(u − m)u2

x

)dx .

More precisely,

J1[u] � 4

3μ(u) J2[u].

Based on this observation, it is possible to establish the orbital stability ofpeakons for this case by deriving an inequality related to the maximum and mini-mum of solutions with the three conservation laws H0, H1 and H2.

The rest of the paper is outlined as follows. In Section 2, a brief review of thewell-posedness of the Cauchy problem of equation (1.1) is given and the existence ofperiodic peakons of (1.1) is then justified. The orbital stability of periodic peakonsof (1.1) in the energy space H1 is established in Section 3.

2. Preliminaries

Consider the initial-value problem of Eq. (1.1) on the unit circle S, that is⎧⎪⎪⎨⎪⎪⎩

yt + k1

((2μ(u)u − u2

x )y)

x+ k2 (2yux + uyx ) = 0, t > 0, x ∈ R,

u(0, x) = u0(x), y = μ(u)− uxx , x ∈ R,

u(t, x + 1) = u(t, x), t � 0, x ∈ R.

(2.1)

In the following, for a given Banach space Z , we denote its norm by ‖ · ‖Z .Since all spaces of functions are over S, for simplicity, we drop S in our notations


of function spaces if there is no ambiguity. For any s, we let Hs(S) denote theSobolev space in S with the norm given by

‖ f ‖2Hs (S) =

∞∑n=−∞

(1 + n2)s | fn|2,

where { fn} is the Fourier series of f ∈ L2(S).First, we give the notion of strong solutions as follows.

Definition 2.1. If u ∈ C([0, T ), Hs(S)) ∩ C1([0, T ), Hs−1(S)) with s > 5/2 andsome T > 0 satisfies (2.1), then u is called a strong solution on [0, T ). If u is astrong solution on [0, T ) for every T > 0, then it is called a global strong solution.

We also have the following local well-posedness result and properties for strongsolutions on the unit circle [35].

Proposition 2.1. [35] Let u0 ∈ Hs(S)with s > 5/2. Then there exists a time T > 0such that the Cauchy problem (2.1) has a unique strong solution u ∈ C([0, T ),Hs(S)) ∩ C1([0, T ), Hs−1(S)) and the map u0 �→ u is continuous from a neigh-borhood of u0 in Hs(S) into C([0, T ), Hs(S)) ∩ C1([0, T ), Hs−1(S)).

Proposition 2.2. [35] The Hamiltonian functionals (1.10)–(1.11) and H0[u] =μ(u) are conserved for the strong solution u in Proposition 2.1, that is, for allt ∈ [0, T ), there hold

d

dtH0[u] = 0,

d

dtH1[u] = 0,

and

d

dtH2[u] = 0.

Furthermore, if y0(x) = (μ− ∂2x )u0(x) does not change sign, then y(t, x) will not

change sign for any t ∈ [0, T ). It follows that if y0(x) � 0, then the correspondingsolution u(t, x) is positive for (t, x) ∈ [0, T )× S.

Plugging the formula for y in terms of u into equation (1.1) results in thefollowing equation

ut + k1

(2μ(u)u − 1

3u2

x

)ux + k1∂x A−1

(2μ2(u)u + μ(u)u2

x

)

+k1

3A−1(u3

x )+ k2uux + k2∂x A−1(

2μ(u)u + 1

2u2

x

)= 0. (2.2)

Recall that

u = A−1 y = g ∗ y, (2.3)

where g is the Green function of the operator A−1, given by [27]

g(x) = 1

2

(x − 1

2

)2

+ 23

24. (2.4)


Its derivative at x = 0 can be assigned to zero, so one has

gx (x)def=

{0, x = 0,

x − 12 , 0 < x < 1.

The above formulation (2.2) allows us to define a weak solution as follows.

Definition 2.2. Given initial data u0 ∈ W 1,3(S), the function u ∈ L∞([0, T ),W 1,3(S)) is said to be a weak solution to the initial-value problem (2.1) if it satisfiesthe following identity∫ T

0

∫S

(uϕt + k1μ(u)u2ϕx + k1

3 u3xϕ−k1gx ∗ (2μ2(u)u + μ(u)u2

x )ϕ− k13 g ∗ (u3

x )ϕ

−k2gx ∗ (2μ(u)u + 12 u2

x )ϕ + k22 u2ϕx

)dx dt + ∫

Su0(x)ϕ(0, x) dx = 0, (2.5)

for any smooth test function ϕ(t, x) ∈ C∞c ([0, T )× S). If u is a weak solution on

[0, T ) for every T > 0, then it is called a global weak solution.

We have the following existence result of single peakons for equation (1.1).

Theorem 2.1. Let the wave speed c satisfy ck1 � −169k22/1200. Then equa-

tion (1.1) admits the periodic peaked traveling-wave solutions of the form: u =ϕc(ξ), ξ = x − ct, with

ϕc(ξ) = a

(1

2

(ξ − 1

2

)2

+ 23

24

), ξ ∈

[−1

2,

1

2

], (2.6)

where the amplitude a = −13k2±√

169k22+1200ck1

50k1, and ϕc(ξ) is extended periodically

to the real line with period one.

Proof. Inspired by the forms of periodic peakons for the μ-CH equation [27], weassume that the periodic peakon of (1.1) is given by

uc(t, x) = a

(1

2

(ξ − [ξ ] − 1

2

)2

+ 23

24

), ξ = x − ct.

By Definition 2.2, it is found that uc satisfies the following equation

6∑j=1

I j :=∫ T

0

∫

S

uc,tϕ dx dt +∫ T

0

∫

S

k1

(2μ(uc)uc − 1

3u2

c,x

)uc,xϕ dx dt

+∫ T

0

∫

S

k1gx ∗(

2μ2(uc)uc + μ(uc)u2c,x

)ϕ dx dt

+∫ T

0

∫

S

k1

3μ(u3

c,x )ϕ dx dt

+∫ T

0

∫

S

k2gx ∗(

2μ(uc)uc+ 1

2u2

c,x

)ϕ dx dt+

∫ T

0

∫

S

k2ucuc,xϕ dx dt

= 0, (2.7)


for some T > 0 and every test function ϕ(t, x) ∈ C∞c ([0, T )× S), where g(x) =

12 (x − [x] − 1

2 )2 + 23

24 . For any x ∈ S, one finds that

μ(uc) = a∫ ct

0

(1

2

(x − ct + 1

2

)2

+ 23

24

)dx

+ a∫ 1

ct

(1

2

(x − ct − 1

2

)2

+ 23

24

)dx = a,

μ(u3c,x ) = a3

∫ ct

0

(x − ct + 1

2

)3

dx + a3∫ 1

ct

(x − ct − 1

2

)3

dx = 0.

To compute the I j , j = 1, . . . , 6, we need to consider two cases: (i) x > ct and(ii) x � ct .

For x > ct , it follows from the proof of Theorem 7.1 in [34] that

I1 + I2 + I3 + I4 =∫ T

0

∫

S

(25

12k1a3 − ca

) (ξ − 1

2

)ϕ(t, x) dx dt

for any ϕ(t, x) ∈ C∞c ([0, T )×S). Here we only need to compute I5 and I6. Firstly,

we have for x > ct ,

2μ(uc)uc + 1

2u2

c,x = 3

2a2

(ξ − 1

2

)2

+ 23

12a2.

Using the above relation, a simply computation then shows that

gx ∗(

2μ(uc)uc + 1

2u2

c,x

)

=∫

S

(x − y − [x − y] − 1

2

)(3

2a2(y − ct − [y − ct] − 1

2)2 + 23

12a2

)dy

=∫ ct

0

(x − y − 1

2

) (3

2a2(y − ct + 1

2)2 + 23

12a2

)dy

+∫ x

ct

(x − y − 1

2

) (3

2a2(y − ct − 1

2)2 + 23

12a2

)dy

+∫ 1

x

(x − y + 1

2

) (3

2a2(y − ct − 1

2)2 + 23

12a2

)dy

= 1

4a2ξ(ξ − 1)(1 − 2ξ).

It follows that

I5 =∫ T

0

∫

S

k2gx ∗(

2μ(uc)uc + 1

2u2

c,x

)ϕ(t, x) dx dt

= 1

4k2a2

∫ T

0

∫

S

(−2ξ3 + 3ξ2 − ξ

)ϕ(t, x) dx dt.


Moreover, it is easy to get

I6 =∫ T

0

∫

S

k2ucuc,xϕ(t, x) dx dt

=∫ T

0

∫

S

k2a2

(1

2

(ξ − [ξ ] − 1

2

)2

+ 23

24

) (ξ − [ξ ] − 1

2

)ϕ(t, x) dx dt

= k2a2∫ T

0

∫

S

(1

2

(ξ − 1

2

)3

+ 23

24

(ξ − 1

2

))ϕ(t, x) dx dt.

Plugging the above expressions into (2.7), we deduce that for any ϕ(t, x) ∈C∞

c ([0, T )× S),

6∑j=1

I j =∫ T

0

∫

S

((25

12k1a3 − ca

) (ξ − 1

2

)+ 13

12k2a2

(ξ − 1

2

))ϕ(t, x) dx dt

=∫ T

0

∫

S

a

(25

12k1a2 + 13

12k2a − c

) (ξ − 1

2

)ϕ(t, x) dx dt.

Similarly, for x � ct , we have

2μ(uc)uc + 1

2u2

c,x = 3

2a2

(ξ + 1

2

)2

+ 23

12a2,

and

gx ∗(

2μ(uc)uc + 1

2u2

c,x

)

=∫

S

(x − y − [x − y] − 1

2

)(3

2a2

(y − ct − [y − ct] − 1

2

)2

+ 23

12a2

)dy

=∫ x

0

(x − y − 1

2

) (3

2a2

(y − ct + 1

2

)2

+ 23

12a2

)dy

+∫ ct

x

(x − y + 1

2

) (3

2a2

(y − ct + 1

2

)2

+ 23

12a2

)dy

+∫ 1

ct

(x − y + 1

2

) (3

2a2

(y − ct − 1

2

)2

+ 23

12a2

)dy

= −1

4a2ξ(ξ + 1)(2ξ + 1).

This allows us to calculate

I1 + I2 + I3 + I4 =∫ T

0

∫

S

(25

12k1a3 − ca

) (ξ − 1

2

)ϕ(t, x) dx dt,

I5 = 1

4k2a2

∫ T

0

∫

S

(2ξ3 + 3ξ2 + ξ)ϕ(t, x) dx dt,


I6 = k2a2∫ T

0

∫

S

(1

2

(ξ + 1

2

)3

+ 23

24

(ξ + 1

2

))ϕ(t, x) dx dt.

Whence we arrive at

6∑j=1

I j =∫ T

0

∫

S

((25

12k1a3 − ca

) (ξ + 1

2

)+ 13

12k2a2

(ξ + 1

2

))ϕ(t, x) dx dt

=∫ T

0

∫

S

a

(25

12k1a2 + 13

12k2a − c

) (ξ + 1

2

)ϕ(t, x) dx dt.

Since ϕ is arbitrary, both cases imply that the constant a fulfills

25

12k1a2 + 13

12k2a − c = 0,

which gives

a =−13k2 ±

√169k2

2 + 1200ck1

50k1.

Thus the theorem is proved. ��Remark 2.1. Note that equation (1.1) is invariant under the transformation

u → −u, k2 → −k2,

so it suffices to consider the peakons in (2.6) with “+”.

3. Proof of Orbital Stability

Attention is now turned to proof of the stability of peakons for the generalizedμ-CH equation (1.1). We first present a precise reformulation of the theorem statedin the introduction.

Theorem 3.1. Assume that k1 > 0, k2 > 0 and c � −169k22/(1200k1) or k1 >

0, k2 � 0 and c > 49k22/(16k1). For every ε > 0, there is a δ > 0 such that if

u ∈ C([0, T ), Hs(S)), s > 5/2, is a solution to (2.1) with

‖u(0, ·)− ϕc‖H1(S) < δ,

then

‖u(t, ·)− ϕc (· − ξ(t))‖H1(S) < ε f or t ∈ [0, T ),

where ξ(t) ∈ R is any point where the function u(t, ξ(t)+ 12 ) attains its maximum.

Remark 3.1. When k1 > 0 and k2 � 0, the condition c > 49k22/(16k1) is equiva-

lent to 2k1μ(ϕc)+ 3k2 > 0.


Remark 3.2. If the initial value u(0, x) satisfies ‖u(0, ·)−ϕc‖H1 (S) < δ, then wecan ensure that u(0, x) > 0, for all x ∈ S, with a choice of small δ > 0. It turnsout that μ(u(t)) = μ(u(0, ·)) > 0, for all t ∈ [0, T ). In fact, by the estimate

|u(0, x)− ϕc(x)| � C‖u(0, ·)− ϕc‖H1(S) < Cδ,

it follows that u(0, x) > minx∈S{ϕc(x)} − Cδ = 2324 H0[ϕc] − Cδ > 0, where

H0[ϕc] is defined in Lemma 3.1.

The proof of Theorem 3.1 is approached via a series of lemmas. The followinglemma summarizes the properties of the single peakons.

Lemma 3.1. The periodic peakon ϕc(x) defined in (2.6) is continuous on S withpeak at x = ± 1

2 . The extrema of ϕc are

Mϕc = maxx∈S

{ϕc(x)} = ϕc

(1

2

)= 13

12H0[ϕc],

mϕc = minx∈S

{ϕc(x)} = ϕc(0) = 23

24H0[ϕc],

with

H0[ϕc] = μ(ϕc) =−13k2 +

√169k2

2 + 1200ck1

50k1= a > 0.

Moreover, we have

H1[ϕc] = 13

12H2

0 [ϕc], H2[ϕc] = 1043

960k1 H4

0 [ϕc] + 47

45k2 H3

0 [ϕc].

We also have

limx↑ 1

2

ϕc,x (x) = 1

2H0[ϕc], lim

x↓− 12

ϕc,x (x) = −1

2H0[ϕc].

Proof. The properties follow easily from the definition of ϕc(x) and the definitionof Hi , i = 0, 1, 2. For example, recalling that H0[ϕc] = μ(ϕc), we have

H0[ϕc] =∫ 1

2

− 12

−13k2 +√

169k22 + 1200ck1

50k1

(x2

2+ 23

24

)dx = a.

Thus we get

H1[ϕc] = H20 [ϕc] +

∫ 12

− 12

(ax)2 dx = 13

12H2

0 [ϕc].

��


We define the μ-inner product 〈·, ·〉μ and the associated μ-norm ‖ · ‖μ by

〈u, v〉μ = μ(u)μ(v)+∫

S

uxvx dx, ‖u‖2μ = 〈u, u〉μ = H1[u], u, v ∈ H1(S),

(3.1)

and consider the expansion of the conservation law H1[u] around the peakon ϕc inthe μ-norm. The following Lemma 3.2 tells us that the error term in this expansionis given by 2a times the difference between ϕc and the perturbed solution u at thepoint of the peak.

Lemma 3.2. For every u ∈ H1(S) and ξ ∈ R,

H1[u] − H1[ϕc] = ‖u − ϕc(· − ξ)‖2μ + 2a

(u

(ξ + 1

2

)− Mϕc

).

Proof. By Proposition 2.2 and relation (3.1), we compute by integration by parts,

‖u−ϕc(· − ξ)‖2μ = H1[u] + H1[ϕc(· − ξ)] − 2μ(u)μ(ϕc)−2

∫

S

ux (x)ϕc,x (x−ξ) dx

= H1[u] + H1[ϕc] − 2μ(u)μ(ϕc)+ 2∫

S

u(x + ξ)ϕc,xx (x) dx .

Using the expression

ϕc,xx = a

(1 − δ

(x − 1

2

)), (3.2)

we deduce that∫

S

u(x + ξ)ϕc,xx (x) dx = a∫

S

u(x + ξ)

(1 − δ

(x − 1

2

))dx

= a

(∫

S

u(x) dx − u

(ξ + 1

2

)).

Then in view of Lemma 3.1, we obtain

‖u − ϕc(· − ξ)‖2μ =H1[u] − H1[ϕc] + 2a

(Mϕc − u

(ξ + 1

2

)).

This completes the proof of the lemma.

Remark 3.3. For a wave profile u ∈ H1 (S), the functional H1[u] represents thekinetic energy. Lemma 3.2 tells us that if a wave u ∈ H1 (S) with energy H1[u]and height Mu is close to the peakon’s energy and height, then the whole shape ofu is close to that of the peakon. It also follows from Lemma 3.2 that the peakon hasmaximal height among all waves of fixed energy. In fact, if u ∈ H1 (S) ⊂ C (S)is such that H1[u] = H1[ϕc] and u(ξ) = max

x∈S

{u(x)}, then u(ξ) � Mϕc . The same

remarks also apply to the CH, the modified CH and the μ-CH equations [3,26,36].


The following lemmas are crucial to establish the result of the stability ofperiodic peakons.

Lemma 3.3. For any function u ∈ H1(S) with μ(u) > 0, ki > 0, i = 1, 2, definethe function

Fu : {(M,m) ∈ R2 : M � m} → R

by

Fu(M,m) = 4

3k1(2M + m)H0[u]H1[u] − 64

45k1(M − m) (2H0[u](M − m))

32

+ 4

3k1 (2M + m) H3

0 [u]−(

4

3k1m(4M − m)−2k2(2m + M)

)H2

0 [u]

+ 2k2 M(H1[u] − 2m H0[u])− 32

15k2(M − m)

52√

2H0[u] − 4H2[u].(3.3)

Then it satisfies

Fu(Mu,mu) � 0,

where Mu = maxx∈S

{u(x)} and mu = minx∈S

{u(x)}.

Remark 3.4. Note that the function Fu depends on u only through the three con-servation laws H0[u], H1[u] and H2[u].Proof of Lemma 3.3. First we observe that the peakon ϕc satisfies the followingequation

∂xϕc(x) =

⎧⎪⎨⎪⎩

−√2μ(ϕc)(ϕc − mϕc ), − 1

2< x � 0,

√2μ(ϕc)(ϕc − mϕc ), 0 � x <

1

2.

Let u ∈ H(S) ⊂ C(S) with μ(u) > 0. Write M = Mu = maxx∈S

{u(x)},m = mu = min

x∈S

{u(x)}. Let ξ and η be such that u(ξ) = M and u(η) = m. We

define the real-valued function g(x) by

g(x) ={

ux + √2μ(u)(u − m), ξ < x � η,

ux − √2μ(u)(u − m), η � x < ξ + 1,

(3.4)

and extend it periodically to the real line. A simple computation leads to

∫

S

g2(x) dx = H1[u] + H20 [u]−2m H0[u]− 4

3H0[u] (2μ(u)(M − m))3/2 . (3.5)


On the other hand, we define the real-valued function h(x) by

h(x)=

⎧⎪⎪⎨⎪⎪⎩

2k1

(μ(u)u + 1

3

√2μ(u)(u − m) ux − 1

6u2

x

)+ 2k2u, ξ < x � η,

2k1

(μ(u)u− 1

3

√2μ(u)(u − m) ux − 1

6u2

x

)+2k2u, η � x < ξ + 1,

and extend it periodically to the entire real line. It is then found that∫

S

h(x)g2(x) dx

=∫ η

ξ

(2k1

(μ(u)u + 1

3

√2μ(u)(u − m)ux − 1

6u2

x

)+ 2k2u

)

·(

ux + √2μ(u)(u − m)

)2dx

+∫ ξ+1

η

(2k1

(μ(u)u − 1

3

√2μ(u)(u − m)ux − 1

6u2

x

)+ 2k2u

)

·(

ux − √2μ(u)(u − m)

)2dx

:= I1 + I2. (3.6)

By a direct calculation, we obtain

I1 = 4k1

∫ η

ξ

(μ2(u)u2 + μ(u)uu2

x − 1

12u4

x

)dx + 4k2

∫ η

ξ

(μ(u)u2 + 1

2uu2

x

)dx

+ 16

3k1

∫ η

ξ

μ(u)uux

√2μ(u)(u − m) dx− 4

3k1

∫ η

ξ

μ(u)mux

√2μ(u)(u − m) dx

− 2k1

∫ η

ξ

μ(u)m(2μ(u)u + u2

x

)dx + 4k2

∫ η

ξ

uux

√2μ(u)(u − m) dx

− 4k2

∫ η

ξ

μ(u)mu dx .

Using the identities

d

dx

(1

5(2μ(u)(u − m))3/2

(2

3m + u

))= μ(u)uux

√2μ(u)(u − m)

andd

dx

(1

3m (2μ(u)(u − m))3/2

)= μ(u)mux

√2μ(u)(u − m),

we deduce that16

3k1

∫ η

ξ

μ(u) uux

√2μ(u)(u − m) dx =16

15k1 (2μ(u)(u − m))3/2

(2

3m + u

) ∣∣∣η

ξ

= − 16

15k1 (2μ(u)(M − m))3/2

(2

3m + M

),

4k2

∫ η

ξ

uux

√2μ(u)(u − m) dx = 4

5μ(u)k2 (2μ(u)(u − m))3/2

(2

3m + u

) ∣∣∣η

ξ

= − 4

5μ(u)k2 (2μ(u)(M−m))3/2

(2

3m+M

)


and

−4

3k1

∫ η

ξ

μ(u)mux

√2μ(u)(u − m) dx = − 4

9k1m (2μ(u)(u − m))3/2

∣∣∣η

ξ

= 4

9k1m (2μ(u)(M − m))3/2 .

It then follows that

I1 = 4k1

∫ η

ξ

(μ2(u)u2+μ(u)uu2

x − 1

12u4

x

)dx+4k2

∫ η

ξ

(μ(u)u2 + 1

2uu2

x

)dx

− 2k1μ(u)∫ η

ξ

(μ(u)u + u2

x

)dx − 2k1μ

2(u)m∫ η

ξ

u dx

− 16

15k1 (2μ(u)(M − m))3/2

(2

3m + M

)+ 4

9k1m (2μ(u)(M − m))3/2

− 4

5μ(u)k2 (2μ(u)(M − m))3/2

(2

3m + M

)− 4k2μ(u)m

∫ η

ξ

u dx .

In a similar way, we obtain

I2 = 4k1

∫ ξ+1

η

(μ2(u)u2+μ(u)uu2

x − 1

12u4

x

)dx+4k2

∫ η

ξ

(μ(u)u2+ 1

2uu2

x

)dx

− 2k1μ(u)m∫ η

ξ

(μ(u)u + u2

x

)dx − 2k1μ

2(u)m∫ ξ+1

η

u dx

− 16

15k1 (2μ(u)(M − m))3/2

(2

3m + M

)+ 4

9k1m (2μ(u)(M − m))3/2

− 4

5μ(u)k2 (2μ(u)(M − m))3/2

(2

3m + M

)− 4k2μ(u)m

∫ ξ+1

η

u dx .

Substituting the above two equalities into (3.6), and using (1.10) and (1.11), weobtain∫

S

h(x)g2(x) dx

= 4H2[u] − 2k1μ3(u)m − 2k1μ(u)m H1[u] + 8

9k1m (2μ(u)(M − m))3/2

−32

15k1 (2μ(u)(M − m))3/2

(2

3m + M

)

− 8

5μ(u)k2 (2μ(u)(M − m))3/2

(2

3m + M

)− 4k2μ

2(u)m. (3.7)

A direct use of the Cauchy inequality gives rise to

h(x) = 2k1

(μ(u)u + 1

3

√2μ(u)(u − m) ux − 1

6u2

x

)+ 2k2u

� 2k1

(μ(u)u + 1

6u2

x + 1

3μ(u)(u − m)− 1

6u2

x

)+ 2k2u

�2

3k1μ(u)(4M − m)+ 2k2 M.


Combining the above inequality with (3.7) yields

4H2[u] − 2k1μ(u)m H1[u] − 32

15k1 (2μ(u)(M − m))3/2

(2

3m + M

)

+ 8

9k1m (2μ(u)(M − m))3/2 − 2k1μ

3(u)m

− 8

5μ(u)k2 (2μ(u)(M − m))3/2

(2

3m + M

)− 4k2μ

2(u)m

�(

2

3k1μ(u)(4M − m)+ 2k2 M

)∫

S

g2(x) dx .

Using the equality (3.5) and noting that μ(u) = H0[u], it is inferred that

4

3k1(2M + m)H0[u]H1[u] − 64

45k1(M − m) (2H0[u](M − m))3/2

+ 4

3k1 (2M + m) H3

0 [u]− 4

3k1m (4M−m) H2

0 [u] + 2k2(2m + M)H20 [u]

+ 2k2 M (H1[u] − 2m H0[u])− 32

15k2(M − m)2

√2H0[u](M − m)

− 4H2[u] � 0.

(3.8)

This completes the proof of Lemma 3.3. ��For the case of k1 > 0, k2 � 0, one has to construct a different functional

Fu which is non-negative at the maximum and minimum values of the perturbedsolution u but is concave down at the peakon ϕc. It is noted that constructing thisnon-trivial functional Fu is crucial in the proof of the stability of peakons for theinteraction case with k1k2 < 0. The corresponding result can be enunciated in thefollowing lemma.

Lemma 3.4. For any function u ∈ H1(S) with μ(u) > 0, k1 > 0, k2 � 0, and4k1μ(u)+ 3k2 > 0, define the function

Fu : {(M,m) ∈ R2 : M � m} → R (3.9)

by

Fu(M,m) = 1

3k1(2M + m)H0[u]H1[u] − 16

45k1(M − m) (2H0[u](M − m))

32

+1

3k1 (2M + m) H3

0 [u] − 1

3k1m (4M − m) H2

0 [u]

+1

2k2(2m + M)H2

0 [u] + 1

2k2 M H1[u] − k2 Mm H0[u]

− 8

15k2(M − m)

52√

2H0[u] − H2[u]. (3.10)

Then it satisfies

Fu(Mu,mu) � 0,

where Mu = maxx∈S

{u(x)} and mu = minx∈S

{u(x)}.


Proof. Let u ∈ H1(S) ⊂ C(S) with μ(u) > 0. Denote M = Mu = maxx∈S

{u(x)},m = mu = min

x∈S

{u(x)}. Let ξ and η be such that u(ξ) = M and u(η) = m. Define

H2[u] = k1

∫

S

(μ2(u)(u − m)2 + μ(u)(u − m)u2

x − 1

12u4

x

)dx

+k2

∫

S

(μ(u)(u − m)2 + 1

2(u − m)u2

x

)dx

≡ k1 J1[u] + k2 J2[u], (3.11)

where

J1[u] =∫

S

(μ2(u)(u − m)2 + μ(u)(u − m)u2

x − 1

12u4

x

)dx, and

J2[u] =∫

S

(μ(u)(u − m)2 + 1

2(u − m)u2

x

)dx .

By the Cauchy inequality, we have the estimate

J1[u] � 4

3μ(u) J2[u]. (3.12)

The equality holds if and only if u is the peakon of equation (1.1). On the otherhand, a straightforward computation yields

J1[u] = J1[u] − m H30 [u] + m2 H0[u]2 − m H0[u]H1[u], and

H2[u] = H2[u] − k1m(

H30 [u] − m H2

0 [u] + H0[u]H1[u])

−k2m

(3

2H2

0 [u] − m H0[u] + 1

2H1[u]

), (3.13)

where

J1[u] =∫

S

(μ2(u)u2 + μ(u)uu2

x − 1

12u4

x

)dx .

We then deduce that∫

h(x)g2(x) dx = 4J1[u] − 2m H30 [u] − 2m H0[u]H1[u]

− 8

15(m + 4M) (2H0[u](M − m))

32 , (3.14)

where

h(x) =

⎧⎪⎨⎪⎩

2μ(u)u + 2

3

√2μ(u)(u − m) ux − 1

3u2

x , ξ < x � η,

2μ(u)u − 2

3

√2μ(u)(u − m) ux − 1

3u2

x , η � x < ξ + 1,


and g(x) is given by (3.4). Notice that

h(x) � 2M H0[u] + 2

3(M − m)H0[u] = 2

3(4M − m)H0[u]. (3.15)

It then follows from (3.12), (3.14) and (3.15) that

4J1[u] − 2m H30 [u] − 2m H0[u]H1[u] − 8

15(m + 4M) (2H0[u](M − m))

32

� 2

3(4M − m)H0[u]

(H1[u] + H2

0 [u] − 2m H0[u]

− 4

3H0[u] (2H0[u](M − m))32

).

Using this inequality and combining (3.11), (3.12) and (3.13), we are able toget (3.10). This completes the proof of the lemma. ��

The following lemmas reveal the properties of the functions Fu(M,m) in Lem-mas 3.3 and Lemma 3.4 associated to the peakon ϕc.

Lemma 3.5. For the peakon ϕc with ki > 0, i = 1, 2, let Fu(M,m) be given inLemma 3.3. Then it satisfies

Fϕc(Mϕc ,mϕc ) = 0,∂Fϕc

∂M(Mϕc ,mϕc ) = 0,

∂Fϕc

∂m(Mϕc ,mϕc ) = 0,

∂2 Fϕc

∂M∂m(Mϕc ,mϕc ) = 0,

∂2 Fϕc

∂M2 (Mϕc ,mϕc ) = −16

3k1 H2

0 [ϕc] − 4k2 H0[ϕc], and

∂2 Fϕc

∂m2 (Mϕc ,mϕc ) = −8

3k1 H2

0 [ϕc] − 4k2 H0[ϕc].Moreover, (Mϕc ,mϕc ) is the unique maximum of Fϕc .

Proof. Since g(x) corresponding to the peakon ϕc is identical to zero, it is easy tosee that Fϕc(Mϕc ,mϕc ) = 0.

On the other hand, it follows from Lemma 3.1 that

Mϕc − mϕc = 1

8H0[ϕc], 2H0[ϕc](Mϕc − mϕc) = 1

4H2

0 [ϕc],√

2H0[ϕc](Mϕc − mϕc) = 1

2H0[ϕc],

(2H0[ϕc](Mϕc − mϕc)

)3/2 = 1

8H3

0 [ϕc].

(3.16)

In view of equality (3.3), a differentiation then gives

∂Fu

∂M= 8

3k1 H0[u]H1[u] − 32

9k1 (2H0[u](M − m))3/2

+ 8

3k1 H3

0 [u] − 16

3k1m H2

0 [u] + 2k2 H20 [u] + 2k2 H1[u]

− 4k2m H0[u] − 16

3k2(M − m)

32√

2H0[u],


∂Fu

∂m= 4

3k1 H0[u]H1[u] + 32

9k1 (2H0[u](M − m))3/2

−8

3k1(2M − m)H2

0 [u] + 4

3k1 H3

0 [u] + 4k2 H20 [u] − 4k2 M H0[u]

+16

3k2(M − m)

32√

2H0[u]. (3.17)

Furthermore, we have

∂2 Fu

∂M2 = − 32

3k1 H0[u]√2H0[u](M − m)− 8k2

√2H0[u](M − m),

∂2 Fu

∂m2 = − 32

3k1 H0[u]√2H0[u](M − m)+ 8

3k1 H2

0 [u]− 8k2

√2H0[u](M − m),

∂2 Fu

∂M∂m= ∂2 Fu

∂m∂M= 32

3k1 H0[u]√2H0[u](M − m)− 16

3k1 H2

0 [u]− 4k2 H0[u] + 8k2

√2H0[u](M − m).

(3.18)

To complete the first part of the proof, we take Fu = Fϕc ,M = Mϕc andm = mϕc in the expressions (3.17)–(3.18) for the partial derivatives of F and use(3.16) and Lemma 3.1.

Next, we show that (Mϕc ,mϕc ) is the unique maximum of Fϕc . Firstly, we havethe following expression for Fϕc(M,m)

Fϕc(M,m) = 25

9k1(2M + m)H3

0 [ϕc] − 128√

2

45k1(M − m)5/2 (H0[ϕc])3/2

− 4

3k1m(4M − m)H2

0 [ϕc] + k2

(4m + 25

6M

)H2

0 [ϕc]− 4k2 Mm H0[ϕc]

− 32√

2

15k2(M − m)5/2

√H0[ϕc] − 4H2[ϕc],

which has a unique critical point (M,m) = (Mϕc ,mϕc ). Hence it suffices to showthat Fϕc < 0 on the boundary of its domain.

On {M = m > 0},

Fϕc(M,M) = −4k1 H20 [ϕc]

(M− 25

24H0[ϕc]

)2

−4k2 H0[ϕc](

M− 49

48H0[ϕc]

)2

− 1

180k1 H4

0 [ϕc] − 3

320k2 H3

0 [ϕc] < 0.

On {m = 0},

Fϕc(M, 0) = 50

9k1 M H3

0 [ϕc] − 128√

2

45k1 M5/2 (H0[ϕc])3/2 + 25

6k2 M H2

0 [ϕc]

− 32√

2

15k2 M5/2

√H0[ϕc] − 4H2[ϕc],


which has a maximum at M = 5 3√1016 H0[ϕc] with the value < −2.102k1 H4

0 [ϕc] −2.495k2 H3

0 [ϕc].When M → ∞, it is obvious that Fϕc(M,m) → −∞. Thus the lemma is

proved. ��For the case of k1 > 0, k2 � 0, we have the following similar result which can

be obtained by a direct computation.

Lemma 3.6. For the peakon ϕc with k1 > 0, k2 � 0, and c > 49k22/(16k1), the

functional Fu(M,m) given in Lemma 3.4 satisfies

Fϕc(Mϕc ,mϕc ) = 0,∂Fϕc

∂M(Mϕc ,mϕc ) = 0,

∂Fϕc

∂m(Mϕc ,mϕc ) = 0,

∂2 Fϕc

∂M∂m(Mϕc ,mϕc ) = 0,

∂2 Fϕc

∂M2 (Mϕc ,mϕc ) = −1

3H0[ϕc] (2k1 H0[ϕc] + 3k2) ,

∂2 Fϕc

∂m2 (Mϕc ,mϕc ) = −1

3H0[ϕc] (4k1 H0[ϕc] + 3k2) .

Moreover, (Mϕc ,mϕc ) is the unique maximum of Fϕc .

Lemma 3.7. [3] If f ∈ H1(S), then it satisfies

maxx∈S

| f (x)| �√

13

12‖ f ‖μ, (3.19)

where the μ-norm is defined in (3.1). Moreover,√

1312 is the best constant and

equality holds in (3.19) if and only if f = cϕc(· − ξ + 12 ) for some c, ξ ∈ R, i.e. if

and only if f has the shape of a peakon.

Note that equality in (3.19) also holds for the μ-CH peakons [3]. The nextlemma shows that the μ-norm is equivalent to the H1(S)-norm.

Lemma 3.8. [3] Every u ∈ H1(S) satisfies

‖u‖2μ � ‖u‖2

H1(S) � 3‖u‖2μ.

Lemma 3.9. [26] If u ∈ C([0, T ), H1(S)), then

Mu(t) = maxx∈S

{u(t, x)}, mu(t) = minx∈S

{u(t, x)}

are continuous functions of t ∈ [0, T ).

Lemma 3.10. Assume that k1 > 0, k2 > 0, and c � −169k22/(1200k1) or k1 >

0, k2 � 0, c > 49k22/(16k1). Let u ∈ C([0, T ), Hs(S)), s > 5/2, be a solution of

(2.1). Given a small neighborhood U of (Mϕc ,mϕc ) in R2, there exists a δ > 0such that

(Mu(t),mu(t)) ∈ U f or t ∈ [0, T ) i f ‖u(0, ·)− ϕc‖H1(S) < δ. (3.20)


Proof. We just prove this lemma for the case of k1 > 0 and k2 > 0. The case ofk1 > 0 and k2 � 0 can be dealt with similarly. Note that the function Fu(t)(M,m)depends on u only through the three conservation laws H0[u], H1[u] and H2[u].Hence, Fu(t)(M,m) = Fu is independent of time. Suppose Hi [u] = Hi [ϕc] +εi , i = 0, 1, 2. A direct computation gives

Fu(M,m) = 4

3k1(2M + m)

((H0[ϕc] + ε0) (H1[ϕc] + ε1)+ (H0[ϕc] + ε0)

3)

− 64

45k1(M − m) (2(H0[ϕc] + ε0)(M − m))3/2

− 4

3k1m (4M − m) (H0[ϕc] + ε0)

2 + 2k2(2m + M) (H0[ϕc] + ε0)2

+ 2k2 M (H1[ϕc] + ε1)− 4k2 Mm (H0[ϕc] + ε0)

− 32

15k2(M − m)2

√2(H0[ϕc] + ε0)(M − m)− 4H2[ϕc] − 4ε2

= Fϕc(M,m)+ 4

3k1(2M + m)ε3

0 + 2

3(6k1(2M + m)H0[ϕc]

− 2k1m(4M − m)

+3k2(2m + M)) ε20 + 4

3

(k1(2M + m)

(H1[ϕc] + 3H2

0 [ϕc])

−2k1m(4M − m)H0[ϕc] + 3k2(2m + M)H0[ϕc] − 3k2 Mm) ε0

+ 4

3k1(2M + m) (ε0ε1 + H0[ϕc]ε1)+ 2k2 Mε1 − 4ε2 + O(ε0).

Here O(ε0) is the term which can not be expressed explicitly. So Fu is a smallperturbation of Fϕc . The effect of the perturbation near the point (Mϕc ,mϕc ) canbe made arbitrarily small by choosing the ε′i s small. It follows from Lemma 3.5that Fϕc(Mϕc ,mϕc ) = 0 and that Fϕc has a critical point with a negative definitesecond derivative at (Mϕc ,mϕc ). By continuity of the second derivative, there isa neighborhood around (Mϕc ,mϕc ) where Fϕc is concave with curvature boundedaway from zero. Therefore, the set where Fu � 0 near (Mϕc ,mϕc )will be containedin a neighborhood of (Mϕc ,mϕc ).

Let U be given as the above statement. Shrinking U if necessary, we infer theexistence of a δ′ > 0 such that for all u ∈ C([0, T ), Hs(S)) (s > 5/2) with

|Hi [u] − Hi [ϕc]| < δ′, i = 0, 1, 2, (3.21)

then it follows that the set where Fu(t) � 0 near (Mϕc ,mϕc ) is contained in U forall t ∈ [0, T ). By Lemma 3.3 and Lemma 3.9, we see that Mu(t) and mu(t) arecontinuous functions of t ∈ [0, T ) and Fu(Mu(t),mu(t)) � 0 for t ∈ [0, T ). It isthen implied that for u satisfying (3.21), we have

(Mu(t),mu(t)) ∈ U f or t ∈ [0, T ) i f (Mu(0),mu(0)) ∈ U.

On the other hand, the continuity of the conserved functionals Hi : H1(S) →R, i = 0, 1, 2, also implies that there is a δ > 0 such that (3.21) holds for all u with

‖u(0, ·)− ϕc‖H1(S) < δ.


Moreover, by the inequality (3.19), taking a smaller δ if necessary, we may alsoassume that (Mu(0),mu(0)) ∈ U if ‖u(0, ·) − ϕc‖H1(S) < δ. This completes theproof of the lemma. ��

With all the preparations given above, we are now in a position to carry out theproof of Theorem 3.1.

Proof of Theorem 3.1. Assume that ki > 0, i = 1, 2 and c � −169k22/(1200k1)

or k1 > 0, k2 � 0 and c > 49k22/(16k1). Let u ∈ C([0, T ), Hs(S)) (s > 5/2) be a

solution of (2.1) and suppose ε > 0 is given. Pick a neighborhood U of (Mϕc ,mϕc )

small enough such that |M−Mϕc | < 25k1ε2

−78k2+6√

169k22+1200ck1

if (M,m) ∈ U . Choose

a δ > 0 as in Lemma 3.10 so that (3.20) holds. Taking a smaller δ if necessary, wemay assume that μ(u) > 0 and

|H1[u] − H1[ϕc]| < ε2

6if ‖u(0, ·)− ϕc‖H1(S) < δ.

Then, by Lemma 3.2 and Lemma 3.8, we get for t ∈ [0, T ),

‖u(t, ·)− ϕc(· − ξ(t))‖2H1(S)

� 3‖u(t, ·)− ϕc(· − ξ(t))‖2μ

� 3|H1[u] − H1[ϕc]| + 6a|Mϕc − Mu(t)|< ε2,

where ξ(t) ∈ R is any point such that u(t, ξ(t) + 12 ) = Mu(t). The stability

conclusion then follows. ��

Acknowledgments The work of Yue Liu is partially supported by the NSF Grant DMS-1207840, the NSF-China Grant-11271192, and the NHARP Grant-003599-0001-2009. Thework of Changzheng Qu is supported partially by NSF-China for Distinguished YoungScholars Grant-10925104 and Ph.D. Programs Foundation of Ministry of Education ofChina-20106101110008. The work of Xiaochuan Liu is supported by the NSF-ChinaGrant-11001220 and Grant-11001219.

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Department of Mathematics,Ningbo University,

Ningbo 315211,People’s Republic of China.

e-mail: [email protected]

and

Department of Mathematics,Tianshui Normal University,

Tianshui 741001,People’s Republic of China.

e-mail: [email protected]

and

Department of Mathematics,Northwest University,

Xi’an 710069,People’s Republic of China.e-mail: [email protected]

and

Department of Mathematics,University of Texas,

Arlington,TX 76019,

USA.e-mail: [email protected]

(Received May 21, 2013 / Accepted August 5, 2013)Published online September 4, 2013 – © Springer-Verlag Berlin Heidelberg (2013)

http://xxx.lanl.gov/abs/1205.2028

http://dx.doi.org/10.1007/s00220-013-1749-3

http://dx.doi.org/10.1007/s00220-013-1749-3

orbital stability of periodic peakons to a generalized μ-camassa–holm equation

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