A wave breaking criterion for a modified periodic two component Camassa Holm system

R E S E A R C H

Open Access

A wave breaking criterion for a modified

periodic two-component Camassa-Holm

system

Ying Wang

*_{Correspondence:}

[email protected] School of Science, Sichuan University of Science and Engineering, Zigong, 643000, China

Abstract

In this paper, a wave-breaking criterion of strong solutions is acquired in the Soblev spaceHs_(S)×Hs–1_{(S) withs>}3

2 by employing the localization analysis in the transport equation theory, which is different from that of the two-component Camassa-Holm system.

MSC: 35D05; 35G25; 35L05; 35Q35

Keywords: a modified periodic two-component Camassa-Holm system; wave-breaking criterion; localization analysis

1 Introduction

The classical two-component Camassa-Holm system takes the form

( –∂_x)ut+u( –∂x)ux+ ux( –∂x)u+ρρx= , t> ,x∈R,

ρt+ (uρ)x= , t> ,x∈R,

()

where the variableu(t,x) represents the horizontal velocity of the fluid, andρ(t,x) is re-lated to the free surface elevation from equilibrium with the boundary assumptionsu→ andρ→ as|x| → ∞. System () was found originally in [], but it was firstly derived rigorously by Constantin and Ivanov []. The system has bi-Hamiltonian structure and is completely integrable. Since the birth of the system, a large number of literature was devoted to the study of the two-component Camassa-Holm system. Some mathematical and physical properties of the system have been obtained. Chenet al.[] established a reciprocal transformation between the two-component Camassa-Holm system and the first negative flow of the AKNS hierarchy. Escheret al.[] used Kato’s theory to establish local well-posedness for the two-component system and presented some precise blow-up scenarios for strong solutions of the system. In [], Constantin and Ivanov described suf-ficient conditions for wave-breaking and global solution to the system. Dynamics in the periodic case for system () were considered in []. It is worth mentioning that the wave-breaking criteria of strong solutions is determined in the lowest Soblev spaceHs_withs>



by applying the localization analysis in the transport equation theory []. The other results related to the system can be found in [–].

Inspired by the works mentioned, in this article, we consider a modified periodic two-component Camassa-Holm system on the circleSwithS=R/Z(the circle of unit lengh):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

mt+umx+ uxm+ρρx= , t> ,x∈R,

ρt+ (uρ)x= , t> ,x∈R, m(,x) =m(x), x∈R,

ρ(,x) =ρ(x), x∈R,

m(t,x+ ) =m(t,x), t> ,x∈R,

ρ(t,x+ ) =ρ(t,x), t> ,x∈R,

()

wherem= ( –∂

x)u, andRis the real number set. In fact, system () is a two-component

generalization of the equation (ifρ=  in system ())

mt+umx+ uxm= , m=

 –∂_xu. ()

Equation () was first derived as the Euler-Poincaré differential equation on the Bott-Virasoro group with respect to theHmetric [], and it is known as a modified Camassa-Holm equation and also viewed as a geodesic equation on some diffeomorphism group []. It is shown in [] that the well-posedness and dynamics of Eq. () on the unit circle

Sare significantly different from that of the Camassa-Holm equation. For example, Eq. () does not conform with blow-up solution in finite time.

As we know, differently from the Camassa-Holm equation, Eq. () has not blow-up so-lution. The motivation of the present paper is to find out whether or not system () has some similar dynamics as the classical two-component Camassa-Holm equation and Eq. () mathematically, for example, wave-breaking and global solution. One of the difficul-ties is the acquisition of the a priori estimates ofuxxL∞anduxxxL∞. This difficulty has

been overcome by Lemmas . and .. We mainly use the ideas of [] to derive a wave-breaking criterion (see Theorem ) of strong solutions for system () in the low Sobolev spacesHs(S)×Hs–(S) withs> _, where a new conservation law is necessary. We need to point out that in the Sobolev spacesHs_(R)×Hs–_{(R) withs>} 

, the wave-breaking

of the solution for system () only depends on the slope of the componentuof the solu-tion []. However, since the slope of the componentuof the solution is bounded by the Sobolev imbedding theoremH→L∞, the wave-breaking of the solution for system () is determined only by the slope of the componentρof solution definitely in the low Sobolev spacesHs_(S)×Hs–_{(S) withs>}

(see Theorem ). This implies that there exists some

dif-ference between system () and the two-component Camassa-Holm equation. Moreover, this is quite different from Eq. () because Eq. () does not admit a blow-up solution in infinite time.

2 The main results

We denote by∗the convolution. Note that ifg(x) :=  +  ∞_{n=+n}_+ncos(nx), then ( –

system () in the form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ut+uux+∂xg∗[u+ux–u 

xx– uxuxxx+_ρ], t> ,x∈R,

ρt+ (uρ)x= , t> ,x∈R, u(,x) =u(x), x∈R,

ρ(,x) =ρ(x), x∈R,

u(t,x+ ) =u(t,x), t> ,x∈R,

ρ(t,x+ ) =ρ(t,x), t> ,x∈R.

()

The main result of the present paper is as follows.

Theorem  Let z= (u,ρ)∈Hs(S)×Hs–(S),s>_,and T be the maximal existence time of the solution z= (u,ρ)to system().Assume that m∈L(S)and T<∞.Then

T



∂xρ(τ)_L∞_(S)dτ=∞.

3 Preliminaries

In order to prove Theorem , we first give some lemmas.

Lemma .([, ]) (-D Moser-type estimates) The following estimates hold:

(i) Fors≥,

fgHs≤Cf_Hsg_L∞+fL∞gHs. ()

(ii) Fors> ,

f∂xgHs≤Cf_Hs+g_L∞+f_L∞∂_xg_Hs. ()

(iii) Fors≤_,s>_,ands+s> ,

f∂xgHs≤CfHsgHs, ()

where C is a constant independent of f and g.

Lemma .([, ]) Suppose that s> –d_.Let v be a vector field such that∇v belongs to L([,T];Hs–)if s>  +d_ or to L([,T];Hd ∩_L∞_{)otherwise.Suppose also that f}∈_Hs_,

F∈L([,T];Hs),and that f ∈L∞([,T];Hs)∩C([,T];S)solves the d-dimensional linear transport equation

ft+v· ∇f=F, f|t==f.

()

Then f ∈C([,T];Hs).More precisely,there exists a constant C depending only s,p,and d,

and such that the following statements hold: ()If s=  +d_,then

fHs≤ f_Hs+C t



F(τ)_Hsdτ+C t



or

fHs≤eCV(t)

fHs+ t



e–CV(t)F(τ)_Hsdτ

()

with V(t) =_t∇v(τ)

Hd_∩L∞dτ if s<  + d

 and V(t) =

t

∇v(τ)Hs–dτelse.

()If f =v,then for all s> ,estimates()and()hold with V(t) =_t∂xu(τ)L∞dτ.

Lemma .([]) Let <σ< .Suppose that f∈Hσ,g∈L([,T];Hσ),ν,∂xν∈L([,T]; L∞),and f ∈L∞([,T];Hσ_{)∩C([,T];S)solves the-dimensional linear transport} equa-tion

ft+ν∂xf =g, f|t==f.

()

Then f ∈C([,T];Hσ_{).More precisely,there exists a constant C depending onlyσand such} that the following statement holds:

fHσ≤ f_Hσ+C

t



g(τ)_Hσdτ+C

t



V(τ)f(τ)_Hσdτ, ()

or

fHσ≤eCV(t)

fHσ+

t



Cg(τ)_Hσdτ

()

with V(t) =_t(ν(τ)L∞+∂xν(τ)L∞)dτ.

Lemma . For all x∈R,the following statements hold: (i) ∂_xg_L∞_(R)≤ +

π

 ()

and

(ii) ∂_xg_L∞_(R)≤ +ln +π. ()

Proof Letg(x) be the Green function for the operator ( –∂

x). Then from

 – ∂_x+∂_xg(x) =δ(x) =

∞

n=–∞

einx

we get

g(x) =

∞

n=–∞

  + n_+ne

inx_{=  + }

∞

n=



 + n_+ncos(nx).

Hence,

gxx(x) = –

∞

n= n

which results in

gxx(x)≤

∞

n= n

 + n_+ncos(nx)≤

∞

n= n

 + n_+n.

From Cauchy integral test we have

∞

n= n

 + n_+n≤_nlim_→∞

n

 x

( +x₎dx=

 +

π

.

It follows that

|gxx| ≤

∞

n= n

 + n_+n≤ +

π

.

Now, we prove (ii). From the Fourier series we have

h(x) =

∞

n=

sin(nx)

n =

π



 – x

π

for  <x< π,

from which we get

h(x) –gxxx=

 ∞ n=  n– n

 + n_+n

sin(nx)

≤ ∞ n=  n– n

 + n_+n

=  ∞ n= 

n( +n₎+ n

( +n₎

.

On the other hand,

∞

n=



n( +n₎+ n

( +n₎

≤ lim n→∞ n   x– x

 +x + x

( +x₎

dx

=  ln +

 . Hence, we have

∂_xg_L∞≤ +ln +π.

Lemma . Let z= (u,ρ)∈Hs(S)×Hs–(S)with s>_.Suppose that T is the maximal existence time of solution z= (u,ρ)of system()with the initial data z.Then,for all t∈

[,T),the following conservation law holds:

H=

S

u+ u_x+u_xx+ρdx=

S

Moreover,assume that m∈L.Then

uxxL∞(S)≤

 +π 

m_L+zH_×L

t



ρxL∞dτ

 

×exp



zH×L

t



 +ρxL∞

dτ

L(t) ()

and

uxxxL∞(S)≤( +ln +π)

m_L+zH_×L

t



ρxL∞dτ

 

×exp



zH×L

t



 +ρxL∞

dτ

M(t). ()

Proof Multiplying the first equation of system () byuand integrating by parts, we get 



d dt

S

u+ u_x+u_xxdx+

Suρρxdx= . ()

Multiplying the second equation of system () byρand integrating by parts, we get 



d dt

Sρ

_dx–

Suρρxdx= , ()

which, together with (), yields 



d dt

S

u+ u_x+u_xx+ρdx= , ()

which implies ().

Next, we prove (). Multiplying the first equation of system () bymand integrating by parts, we have

 

d dt

Sm

_{dx= –}

Summxdx– 

Suxm

_dx–

S

mρρxdx, ()

which results in

d dt

Sm

_{dx= –}

Suxm

_{dx– }

Sm

ρρxdx. ()

By the Hölder inequality we get from () that

d dtm



L_(S)≤uxL∞m_L+ mLρ_Lρ_xL∞ ≤uxL∞m_L+

 +m_L

ρ_Lρ_xL∞ ≤ m_L

uxL∞+ρLρ_xL∞

Applying Gronwall’s inequality, we obtain

m_L≤

m_L+

t



ρ_Lρ_xL∞dτ

exp

t



uxL∞+ρLρ_xL∞

dτ

,

which, together with (), yields

m_L≤

m_L+zH_×L

t



ρxL∞dτ

×exp

zH_×L

t



 +ρxL∞

dτ

. ()

On the other hand, from Lemma . we deduce

uxxL∞=gxx∗mL∞≤ gxxL∞mL≤

 +π 

mL. ()

It follows from () that

uxxL∞≤

 +π 

m_L+zH_×L

t



ρxL∞dτ

 

×exp



zH×L

t



 +ρxL∞

dτ

. ()

Similarly, we can obtain ().

This completes the proof of Lemma ..

4 Proof of main theorem

Proof of Theorem Using the maximal principle to the transport equation aboutρ,

ρt+uρx= –uxρ,

we have

ρ(t)_L∞_(S)≤ ρL∞(S)+C

t



∂xu(τ)_L∞ρ(τ)_L∞dτ.

Applying Gronwall’s inequality yields

ρ(t)_L∞_(S)≤ ρL∞exp

C

t



∂xu(τ)_L∞dτ

.

Using the Sobolev embedding theoremHs_→L∞_(s> 

), we get from Lemma . that uxL∞(S)≤C

uH+ρ_L

. Therefore, we have

ρ(t)_L∞_(S)≤ ρL∞eCt(uH+ρL)=ρL∞eCTzH×L. ()

Step. Fors∈(_, ), applying Lemma . to the second equation, we have

ρ_Hs–_(S)≤ ρ_Hs–+C

t



uxρHs–dτ

+C

t



ρ_Hs–u_L∞+∂_xu_L∞dτ.

From Lemma . () we get

ρuxHs–_(S)≤C

uxHs–ρ_L∞+ρ_Hs–u_xL∞

. ()

From () we obtain

ρ_Hs–_(S)≤ ρ_Hs–+C

t



uHsρ_L∞dτ

+C

t



ρ_Hs–

uL∞+∂xuL∞

dτ. ()

On the other hand, using Lemma ., we get from the first equation of system () that

u(t)_Hs_(S)≤C t



∂xg∗

u+u_x– u



xx– uxuxxx+

 ρ



Hs

dτ

+uHs+C t



u(t)_Hs∂xu(τ)_L∞dτ.

From Lemma .(b) of [], we have

∂xg∗

u+u_x– u



xx– uxuxxx+

 ρ



Hs

≤Cu+u_x– u



xx– uxuxxx+

 ρ



Hs–

≤CuHs–u_L∞+u_xHs–u_xL∞+u_xxHs–u_xxL∞

+uxxxHs–u_xL∞+ρ_Hs–ρ_L∞.

Hence, we get

u(t)_Hs_(S)≤ uHs_(S)+C t



ρ(τ)_Hs–ρ(τ)_L∞dτ

+C

t



uHsu_L∞+uxL∞+uxxL∞

dτ, ()

which, together with (), ensures that

u(t)_Hs_(S)+ρ(t)_Hs–_(S)

≤ uHs_(S)+ρ_Hs–_(S)+C

t



uHs+ρ(t)

Hs–

×uL∞+uxL∞+uxxL∞+ρL∞

Using Gronwall’s inequality, we have

u(t)_Hs_(S)+ρ(t)_Hs–_(S) ≤uHs_(S)+ρ_Hs–_(S)

×exp

C

t



uL∞+uxL∞+uxxL∞+ρL∞

dτ

. ()

From () and Lemma . we get

u(t)_Hs_(S)+ρ(t)_Hs–_(S) ≤uHs_(S)+ρ_Hs–_(S)

×exp

C

t



L(t) +zH_×L+ρ_L∞eCTzH×L

dτ

. ()

Therefore, if the maximal existence timeT<∞satisfies_tρxL∞dτ<∞, then we get

from () that

lim sup

t→T

u(t)_Hs_(S)+ρ(t)_Hs–_(S)

<∞, ()

which completes the proof of Theorem  fors∈(_, ).

Step. Fors∈[,_), applying Lemma . to the second equation of system (), we get

ρ_Hs–_(S)≤ ρ_Hs–+C

t



uxρHs–dτ

+C

t



ρ_Hs–∂_xu

L∞∩Hdτ.

Using () results in

ρHs–_(S)≤ ρHs–+C

t



uxHs–ρL∞dτ+C

t



ρHs–∂xu

L∞∩Hdτ,

which, together with (), yields

u(t)_Hs+ρ(t)_Hs–

≤ uHs+ρ_Hs–+C

t



uHs+ρ(t)

Hs–

×(uL∞+u

H +ε+uxxL∞+ρL∞)dτ, ()

whereε∈(,_), and we used the fact thatH+ε→L∞∩H.

Using Gronwall’s inequality, we have

u(t)_Hs+ρ(t)_Hs–

≤uHs+ρ_Hs–

exp

C

t



uL∞+u

H +ε+uxxL∞+ρL∞

dτ

From () and Lemma . we get

u(t)_Hs+ρ(t)_Hs– ≤uHs+ρ_Hs–

×exp

C

t



L(t) +zH_×L+ρ_L∞eCTzH×L

dτ

. ()

Applying the argument as in step , we complete the proof of Theorem  fors∈[,_).

Step. Fors∈(, ), differentiating once the second equation of system () with respect tox, we have

∂tρx+u∂xρx+ uxρx+uxxρ= . ()

Using Lemma ., we get

ρxHs–_(S)≤ ρ_xHs–+C

t



uHsρ_L∞dτ

+C

t



ρ_Hs–

uL∞+∂xuL∞

dτ, ()

where we used the estimates

uxρxHs–≤C

uxHs–ρ_L∞+ρ_xHs–u_xL∞

and

ρuxxHs–≤Cρ_Hs–u_xL∞+u_xxHs–ρ_L∞,

where Lemma . () was used.

Using (), (), and () (wheres–  is replaced bys– ) yields

u(t)_Hs+ρ(t)_Hs–≤ uHs+ρ_Hs–+C

t



uHs+ρ(t)

Hs–

×uL∞+uxL∞+uxxL∞+ρL∞

dτ. ()

Applying Gronwall’s inequality, we have

u(t)_Hs+ρ(t)_Hs–

≤uHs+ρ_Hs–

exp

C

t



uL∞+uxL∞+uxxL∞+ρL∞

dτ

. ()

From () and Lemma . we get

u(t)_Hs+ρ(t)_Hs– ≤uHs+ρ_Hs–

×exp

C

t



L(t) +zH_×L+ρ_L∞eCTzH×L

dτ

Using the argument as in step , we complete the proof of Theorem  fors∈(, ).

Step. Fors=k∈N,k≥, differentiatingk–  times the second equation of system () with respect tox, we obtain

(∂t+u∂x)∂xk–ρ+

l+l=k–,l,l≥

Cl,l∂ l+

x u∂xl+ρ+ρ∂x

∂_xk–u= . ()

Using Lemma ., we get from () that

∂_xk–ρ

H≤∂xk–ρ_H+C

t



∂_xk–ρ

H∂xu_H_∩L∞dτ

+C

t



l+l=k–,l,l≥

Cl,l∂ l+

x u∂xl+ρ+ρ∂xk–u

H

dτ. ()

SinceH_{is an algebra, we have}

ρ∂_xk–u_H≤CρH∂_xk–u

H≤CρHu_Hs

and

l+l=k–,l,l≥

Cl,l∂ l+ x u∂xl+ρ

H≤

Cρ_Hs–u_Hs–.

It follows that

∂_xk–ρ_H≤∂_xk–ρ_H+C

t



uHs+ρ_Hs–

×uHs–+ρ_H

dτ. ()

From the Gagliardo-Nirenberg inequality we have that, forσ∈(, ),

ρ_Hs–≤C

ρHσ+∂k–

x ρH

. ()

On the other hand, forσ∈(, ), rewrite () as

ρHσ_(S)≤ ρ_Hσ+C

t



uHσ+ρ_L∞dτ

+C

t



ρHσu_L∞+∂_xu_L∞dτ, ()

which, together with (), yields

ρ_Hs–≤Cρ_Hs–+C

t



uHs+ρ_Hs–

×uHs–+ρ_H

dτ, ()

Using Lemma . (), we get

u(t)_Hs_(S)≤ uHs+C t



uHsu_L∞+uxL∞+uxxL∞

+uxxxL∞

dτ+C

t



ρ(τ)_Hs–ρ(τ)_L∞dτ, ()

which, together with (), results in

u(t)_Hs+ρ(t)_Hs–

≤CuHs+ρ_Hs–+C

t



uHs+ρ(t)

Hs–

×uHs–+ρ_H+u_xxL∞+u_xxxL∞

dτ. ()

Using Gronwall’s inequality, we get

u(t)_Hs+ρ(t)_Hs– ≤CuHs+ρ_Hs–

×exp

C

t



uHs–+ρ_H+u_xxL∞+u_xxxL∞

dτ

. ()

IfT <∞satisfies_TρxL∞dτ <∞, applying step  and the induction assumption, we

obtain from Lemma . thatuHs–+ρ_H+u_xxL∞+u_xxxL∞is uniformly bounded.

From () we get

lim sup

t→T

_u(t)

Hs+ρ(t)_Hs–

<∞,

which contradicts the assumption thatT <∞is the maximal existence time. This com-pletes the proof of Theorem  fors=k∈Nandk≥.

Step. Fors∈(k,k+ ),k∈N, andk≥, differentiatingk–  times the second equation of system () with respect tox, we obtain

(∂t+u∂x)∂xk–ρ+

l+l=k–,l,l≥

Cl,l∂ l+

x u∂xl+ρ+ρ∂x

∂_xk–u= . ()

Using Lemma . withs–k∈(, ), we get from () that

∂_xk–ρ

Hs–k≤∂xk–ρ_Hs–k+C t



∂_xk–ρ

Hs–k

uL∞+∂xuL∞

dτ

+C

t



l+l=k–,l,l≥

Cl,l∂ l+

x u∂xl+ρ+ρ∂xku

Hs–k

dτ. ()

For eachε∈(,_), using Lemma . () and the fact thatH+ε→_L∞_{, we have}

ρ∂_xku_Hs–k≤C

ρ_Hs–k+∂_xk–u L∞+∂

k

xuHs–kρL∞

≤Cρ_Hs–k+u

Hk–  +ε+uHs–ρL∞

and

l+l=k–,l,l≥

Cl,l∂ l+ x u∂xl+ρ

Hs–k

≤C

l+l=k–,l,l≥

Cl,l∂ l+

x uHs–k+∂_xlρ_L∞

+∂l+ x uL∞∂

l+ x ρHs–k

≤CuHsρ

Hk–  +ε+uHk–  +ερHs–

. ()

Therefore, from (), (), and () we get

∂_xk–ρ_Hs–k ≤∂xk–ρ_Hs–k+C t



uHs+ρ_Hs–

×u

Hk–  +ε+ρ_Hk–  +ε

dτ. ()

Applying Lemma . to the first equation of system () fors∈(k,k+ ) withk≥, we obtain

u(t)_Hs_(S)≤ uHs+C t



ρ(τ)_Hs–ρ(τ)_L∞dτ

+C

t



uHsu_L∞+uxL∞+uxxL∞

dτ, ()

which, together with () and () (wheres–  is replaced bys–k), gives

u(t)_Hs+ρ(t)_Hs–

≤CuHs+ρ_Hs–

+C

t



uHs+ρ(t)

Hs–

×u

Hk–  +ε+ρHk–  +ε

dτ. ()

Using Gronwall’s inequality, we get

u(t)_Hs+ρ(t)_Hs–

≤CuHs+ρ_Hs–

exp

C

t



u

Hk–  +ε+ρHk–  +ε

dτ

. ()

Noting that k– _ +ε<k,k–_ +ε<k– , and k≥ and applying step , we obtain that u

Hk–  +ε+ρHk–  +ε is uniformly bounded. Therefore, we complete the proof of

Theorem  fors∈(k,k+ ),k∈N, andk≥.

So, the proof of Theorem  is completed.

Competing interests

Acknowledgements

The author thanks the referees for their valuable comments and suggestions. This work was supported by Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing [grant number 2013QZJ02], [grant number 2014QYJ03], Scientific Research Foundation of the Education Department of Sichuan province Project [grant number 16ZA0265], SUSER [grant number 2014RC03].

Received: 6 October 2015 Accepted: 18 February 2016

References

1. Olver, P, Rosenau, P: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E53, 1900-1906 (1996)

2. Constantin, A, Ivanov, R: On an integrable two-component Camassa-Holm shallow water system. Phys. Lett. A372, 7129-7132 (2008)

3. Chen, M, Liu, S, Zhang, Y: A 2-component generalization of the Camassa-Holm equation and its solutions. Lett. Math. Phys.75, 1-15 (2006)

4. Escher, J, Lechtenfeld, O, Yin, Z: Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete Contin. Dyn. Syst.19, 493-513 (2007)

5. Hu, Q, Yin, Z: Well-posedness and blow-up phenomena for the two-component periodic Camassa-Holm equation. Proc. R. Soc. Edinb., Sect. A141(1), 93-107 (2011)

6. Gui, G, Liu, Y: On the global existence and wave-breaking criteria for the two-component Camassa-Holm system. J. Funct. Anal.258, 4251-4278 (2010)

7. Falqui, G: On the Camassa-Holm type equation with two dependent variable. J. Phys. A, Math. Gen.39, 327-342 (2006)

8. Guan, C, Yin, Z: Global weak solutions for a two-component Camassa-Holm shallow water system. J. Funct. Anal.260, 1132-1154 (2011)

9. Henry, D: Infinite propagation speed for a two-component Camassa-Holm equation. Discrete Contin. Dyn. Syst., Ser. B12, 597-606 (2009)

10. Holm, D, Nraigh, L, Tronci, C: Singular solutions of modified two-component Camassa-Holm equation. Phys. Rev. E79, 016601 (2009)

11. Ivanov, R: Two-component integrable systems modeling shallow water waves: the constant vorticity case. Wave Motion46, 389-396 (2009)

12. Kuzmin, P: On two-component generalizations of the Camassa-Holm equation. Mat. Zametki81, 149-152 (2007) (in Russian); translation in Math. Notes81, 130-134 (2007)

13. Popowicz, Z: A 2-component orN= 2 supersymmetric Camassa-Holm equation. Phys. Lett. A354, 110-114 (2006) 14. Yan, K, Yin, Z: Analytic solutions of the Cauchy problem for two-component Camassa-Holm shallow water systems.

Math. Z.269, 1113-1127 (2011)

15. Zhang, P, Liu, Y: Stability of solitary waves and wave-breaking phenomena for the two-component CH system. Int. Math. Res. Not.2010(11), 1981-2021 (2010)

16. Mclachlan, R, Zhang, X: Well-posedness of modified Camassa-Holm equations. J. Differ. Equ.246, 3241-3259 (2009) 17. Danchin, R: Fourier Analysis Methods for PDEs. Lecture Notes (2003)

18. Danchin, R: A few remarks on the Camassa-Holm equation. Differ. Integral Equ.14, 953-988 (2001)