analytic solutions of the cauchy problem for two-component shallow water systems

Math. Z. (2011) 269:1113–1127DOI 10.1007/s00209-010-0775-5 Mathematische Zeitschrift

Analytic solutions of the Cauchy problemfor two-component shallow water systems

Kai Yan · Zhaoyang Yin

Received: 27 June 2010 / Accepted: 26 August 2010 / Published online: 26 September 2010© Springer-Verlag 2010

Abstract This paper is mainly concerned with the Cauchy problem for a modified two-component Camassa–Holm shallow water system with analytic initial data. The analyticityof its solutions is proved in both variables, globally in space and locally in time. The obtainedresults can be also applied to the classical integrable two-component Camassa–Holm shallowwater system and the b-equation.

Keywords A modified two-component Camassa–Holm shallow water system · Analyticsolutions · Two-component Camassa–Holm shallow water system · The b-equation

Mathematics Subject Classification (2000) 35A10 · 35Q53

1 Introduction

In this paper we mainly consider the Cauchy problem of the following modified two-com-ponent Camassa–Holm shallow water system:

⎧⎪⎪⎨

⎪⎪⎩

mt + umx + 2mux = −σρρx , t ∈ R, x ∈ R,

ρt + (ρu)x = 0, t ∈ R, x ∈ R,

m(0, x) = m0(x), x ∈ R,

ρ(0, x) = ρ0(x), x ∈ R,

(1.1)

where m = u − uxx , ρ = (1 − ∂2x )(ρ − ρ0), ρ0 is a constant and σ = ±1.

The system (1.1) was recently introduced by Holm et al. [31]. The modified two-compo-nent Camassa–Holm equation (MCH2) is written in terms of velocity u and locally averageddensity ρ (or depth, in the shallow-water interpretation) and ρ0 is taken to be constant. MCH2is defined as geodesic motion on the semidirect product Lie group [30,39] with respect to a

K. Yan (B) · Z. YinDepartment of Mathematics, Sun Yat-sen University, Guangzhou 510275, Chinae-mail: [email protected]

Z. Yine-mail: [email protected]

123

1114 K. Yan, Z. Yin

certain metric and is given as a set of Euler-Poincaré equations on the dual of the correspond-ing Lie algebra.

For ρ ≡ 0, the system (1.1) becomes the Camassa–Holm equation, modeling the uni-directional propagation of shallow water waves over a flat bottom. Here u(t, x) stands forthe fluid velocity at time t in the spatial x direction [5,12,22,35–37]. A long-standing openproblem in hydrodynamics was the derivation of a model equation that can capture breakingwaves as well as peaked traveling waves, cf. the discussion in [49]. The quest for peakedtraveling waves is motivated by the desire to find waves replicating a feature that is char-acteristic for the waves of great height-waves of largest amplitude that are exact travelingsolutions of the governing equations for water waves, whether periodic or solitary, cf. thepapers [11,14,46], while by breaking waves one understands solutions that remain boundedbut their slope becomes unbounded in finite time—see the discussion in [13]. Both theseaspects are modeled by the Camassa–Holm equation. The Camassa–Holm equation is alsoa model for the propagation axially symmetric waves in hyperelastic rods [18,19]. It has abi-Hamiltonian structure [8,27] and is completely integrable [5,10]. Its solitary waves arepeaked [6], capturing thus the shape of solitary wave solutions to the governing equations forwater waves [14]. The orbital stability of the peaked solutions is proved in [17]. The explicitinteraction of the peaked solutions is given in [4].

The analyticity of solutions to Euler equations of hydrodynamics have been studied exten-sively (It was initiated by [43,44] and later further developed in [15,41,42,47] and in thepapers of [1,2] where the approach is based on a contraction type argument in a suitablescale of Banach spaces). In particular, we here concentrate on the analyticity of solutions fora class of shallow water systems, especially for a modified two-component Camassa–Holmsystem. It is worthy to point out that our approach is strongly motivated by the work [29]where the analyticity of periodic solutions to the Camassa–Holm equation in both space andtime variables is obtained, provided that the initial data is analytic.

In this paper we will prove the analyticity of solutions to the system (1.1) in both variables,with x on the line R (or on the circle T = R/Z) and t in an neighborhood of zero, providedthat the initial data is analytic on R (or on T). Note that the classical Cauchy-Kowalevskitheorem does not apply to (1.1) since the initial line t = 0 is characteristic. Therefore theresults above can be viewed as a Cauchy-Kowalevski theorem for the nonlinear case (1.1).

We now conclude this introduction by outlining the rest of the paper. In Sect. 2, we obtainthe analytic solutions to the system (1.1) on the line R. In Sect. 3, we get a similar analyticityresult of the system (1.1) on the circle T. In Sect. 4, we give two examples of the classicalwater wave system or equation on the line R and on the circle T, whose analyticity of thesolutions can be also obtained by our approach.

2 Analytic solutions on the line

In this section, we will show the existence and uniqueness of analytic solutions to the system(1.1) on the line.

First, we can rewrite problem (1.1) with m = u − uxx , ρ = γ − γxx and γ = ρ − ρ0 asfollows:

⎧⎪⎪⎨

⎪⎪⎩

mt + mx u + 2mux = −σργx , t ∈ R, x ∈ R,

ρt + (uρ)x = 0, t ∈ R, x ∈ R,

m(0, x) = u0(x) − u0,xx (x), x ∈ R,

ρ(0, x) = γ0 − γ0,xx , x ∈ R.

(2.1)

123

Analytic solutions of the Cauchy problem for two-component shallow water systems 1115

Note that if p(x) := 12 e−|x |, x ∈ R, then p ∗ m = u and p ∗ ρ = γ . Thus we can rewrite

(2.1) as follows:

⎧⎪⎪⎨

⎪⎪⎩

∂t u + 12∂x u2 + ∂x (1 − ∂2

x )−1(u2 + 12 (∂x u)2 + 1

2σγ 2 − 12σ(∂xγ )2) = 0,

∂tγ + u∂xγ + ∂x (1 − ∂2x )−1(∂x u · ∂xγ ) + (1 − ∂2

x )−1(γ ∂x u) = 0,

u(0, x) = u0(x),

γ (0, x) = γ0(x),

(2.2)

where t ∈ R, x ∈ R, u0, γ0 ∈ Cω(R).

Setting

f (x) = x2, P1 = −∂x , P2 = −∂x (1 − ∂2x )−1, and P3 = (1 − ∂2

x )−1,

we can rewrite (2.2) in the following form:

∂t u =(

1

2P1 + P2

)

f (u) + 1

2P2 f (P1u) + 1

2P2σ f (γ ) − 1

2P2σ f (P1γ ), (2.3)

∂tγ = u P1γ + P2(P1u · P1γ ) + P3(γ · P1u). (2.4)

Next, we transform (2.3) and (2.4) into a system. For this we let

u1 = u, u2 = P1u, u3 = γ, and u4 = P1γ.

Then

∂t u1 = 1

2P1 f (u1) + P2 f (u1) + 1

2P2 f (u2) + 1

2σ P2 f (u3) − 1

2P2 f (u4)

= P1

(1

2f (u1)

)

+ P2

(

f (u1) + 1

2f (u2) + 1

2σ f (u3) − 1

2σ f (u4)

)

≡ F1(u1, u2, u3, u4),

∂t u2 = P1(∂t u1)

= P21

(1

2f (u1)

)

+ P1 P2

(

f (u1) + 1

2f (u2) + 1

2σ f (u3) − 1

2σ f (u4)

)

= P1(u1u2) + P1 P2

(

f (u1) + 1

2f (u2) + 1

2σ f (u3) − 1

2σ f (u4)

)

≡ F2(u1, u2, u3, u4),

and

∂t u3 = u1u4 + P2(u2u4) − P3(u2u3)

≡ F3(u1, u2, u3, u4),

∂t u4 = P1(∂t u3)

= P1(u1u4) + P1 P2(u2u4) − P2(u2u3)

≡ F4(u1, u2, u3, u4).

123

1116 K. Yan, Z. Yin

Therefore our initial value problem takes the following form:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂t u1 = F1(u1, u2, u3, u4),

∂t u2 = F2(u1, u2, u3, u4),

∂t u3 = F3(u1, u2, u3, u4),

∂t u4 = F4(u1, u2, u3, u4),

u1(0, x) = u0(x),

u2(0, x) = −∂x u0(x),

u3(0, x) = γ0(x),

u4(0, x) = −∂xγ0(x).

(2.5)

Define

U ≡ (u1, u2, u3, u4)

and

F(U ) = F(u1, u2, u3, u4)

≡ (F1(u1, u2, u3, u4), F2(u1, u2, u3, u4), F3(u1, u2, u3, u4), F4(u1, u2, u3, u4)).

Then we have

{∂U (t)

∂t = F(t, U (t)),U (0) = (u0(x),−∂x u0(x), γ0(x),−∂xγ0(x)).

(2.6)

Now we have the following analyticity result:

Theorem 2.1 Let

(m0

ρ0

)

be real analytic on R. There exists an ε > 0 and a unique solution(

mρ

)

of the Cauchy problem (1.1) that is analytic on (−ε, ε) × R.

For the proof of Theorem 2.1, we will need a suitable scale of Banach spaces as follows.For any s > 0, we define

Es ≡{

u ∈ C∞(R) : |||u|||s ≡ supk∈N

sk‖∂ku‖Hr (R)

k!/(k + 1)2 < ∞}

,

where r > 12 is any fixed real number. It is not difficult to cheek that Es equipped the norm

||| · |||s is a Banach space by the completeness of Hr (R) and the closedness of the differentialoperator ∂x , and that, for any 0 < s′ < s, Es ⊂ Es′ with |||u|||s′ < |||u|||s .

Proposition 2.1 Es ⊂ Cω(R). That is, any u in Es is real analytic on R.

Proof For any u ∈ Es , we first formally write

u(x) =∞∑

k=0

∂ku(x0)

k! (x − x0)k .

123


For convenience, let x0 = 0. It follows from Hr (R) ↪→ L∞(R) that

∞∑

k=0

∂ku(0)

k! xk ≤ c∞∑

k=0

||∂ku||Hr (R)

k! xk

= c∞∑

k=0

sk ||∂ku||Hr (R)

k!/(k + 1)2 · xk

sk(k + 1)2

≤ c|||u|||s∞∑

k=0

(x

s)k .

Then the convergence radius of the power series∑∞

k=0∂k u(0)

k! xk is s, that is, u can be expandeda power series on (−s, s). Thus u ∈ Cω(R). ��

Now we set

|||(u1, u2, u3, u4)|||s ≡4∑

i=1

|||ui |||s

and

|||F(u1, u2, u3, u4)|||s ≡4∑

i=1

|||Fi (u1, u2, u3, u4)|||s .

Then Theorem 2.1 is a straightforward consequence of the following result.

Theorem 2.2 [1,29] Let (Xs, |||.|||s)0<s≤1 be a scale of decreasing Banach spaces, suchthat for any 0 < s′ < s we have Xs ⊂ Xs′ with |||.|||s′ ≤ |||.|||s . Consider the Cauchyproblem

{dudt = F(t, u(t)),u(0) = 0.

(2.7)

Let T, R and C be positive numbers and suppose that F satisfies the following conditions:

(a) If for any 0 < s′ < s < 1, the function t −→ u(t) is holomorphic on |t | < T andcontinuous on |t | ≤ T with values in Xs and

sup|t |≤T

|||u(t)|||s < R,

then t −→ F(t, u(t)) is a holomorphic function on |t | < T with values in Xs′ .(b) For any 0 < s′ < s ≤ 1 and any u, v ∈ B(0, R) ⊂ Xs, that is, |||u|||s < R, |||v|||s <

R, we have

sup|t |≤T

|||F(t, u) − F(t, v)|||s′ ≤ C

s − s′ |||u − v|||s .

(c) There exists a M > 0, such that for any 0 < s < 1,

sup|t |≤T

|||F(t, 0)|||s ≤ M

1 − s.

123

1118 K. Yan, Z. Yin

Then there exists a T0 ∈ (0, T ) and a unique function u(t), which is holomorphic in|t | < (1 − s)T0 with values in Xs for every s ∈ (0, 1) and is a solution to the initial valueproblem (2.7).

The conditions (a)–(c) above can be easily verified once our system (2.5) or (2.6) istransformed into a new system with zero initial data as in (2.7). To complete our proof ofTheorem 2.1, it suffices to show the following key result.

Theorem 2.3 Let R > 0. There exists a constant C = C(r, R) > 0, such that for any0 < s′ < s ≤ 1, we have

|||F(u1, u2, u3, u4)−F(v1, v2, v3, v4)|||s′ ≤ C

s − s′ |||(u1, u2, u3, u4)−(v1, v2, v3, v4)|||s,

for any u j and v j in the ball B(0, R) ⊂ Es, j = 1, 2, 3, 4.

The proof of Theorem 2.3 requires the following two useful lemmas.

Lemma 2.1 Let s > 0. There is a constant C > 0, independent of s, such that for any u andv in Es , we have

|||uv|||s ≤ C |||u|||s |||v|||s, (2.8)

where C = C(r) depends only on r.In particular, for any s > 0, we have

||| f (u) − f (v)|||s = |||u2 − v2|||s ≤ C |||u + v|||s |||u − v|||s,

for any u, v ∈ Es.

Proof From the algebra property of Hr (R) as r > 12 , we have

|| f g||Hr (R) ≤ c0|| f ||Hr (R)||g||Hr (R),

where c0 = c0(r).Using Leibniz rule, noting meanwhile that for any s > 0, || f ||Hr (R) ≤ ||| f |||s , we esti-

mate

||∂k(uv)||Hr (R) ≤k∑

l=0

(k

l

)

||∂k−lu · ∂ lv||Hr (R)

≤ c0

k∑

l=0

(k

l

)

||∂k−lu||Hr (R)||∂ lv||Hr (R)

= c0||∂ku||Hr (R)||v||Hr (R) + c0

k∑

l=1

(k

l

)

||∂k−lu||Hr (R)||∂ lv||Hr (R)

≤ c0|||v|||s ||∂ku||Hr (R) + c0

k∑

l=1

(k

l

)

||∂k−lu||Hr (R)||∂ lv||Hr (R).

123


Noting that∞∑

l=1

1l2 = π2

6 < 2, we have

|||uv|||s = supk∈N

sk‖∂k(uv)‖Hr (R)

k!/(k + 1)2

≤ c0 supk∈N

sk

k!/(k + 1)2

k∑

l=1

(k

l

)

||∂k−lu||Hr (R)||∂ lv||Hr (R)

+c0|||v|||s supk∈N

sk‖∂ku‖Hr (R)

k!/(k + 1)2

= c0 supk∈N

sk

k!/(k + 1)2

k∑

l=1

(k

l

)sk−l‖∂k−lu‖Hr (R)

(k − l)!/(k − l + 1)2 · sl‖∂ lv‖Hr (R)

l!/(l + 1)2 ·

· (k − l)!/(k − l + 1)2

sk−l· l!/(l + 1)2

sl+ c0|||u|||s |||v|||s

≤ c0|||u|||s |||v|||s supk∈N

k∑

l=1

(k

l

)sk

k!/(k + 1)2 · (k − l)!/(k − l + 1)2

sk−l·

· l!/(l + 1)2

sl+ c0|||u|||s |||v|||s

= c0|||u|||s |||v|||s(

1 + supk∈N

k∑

l=1

(k + 1)2

(l + 1)2(k − l + 1)2

)

≤ c0|||u|||s |||v|||s(

1 + supk∈N

k∑

l=1

(k + 1)2

l2(k − l + 1)2

)

= c0|||u|||s |||v|||s(

1 + supk∈N

k∑

l=1

(1

l+ 1

k − l + 1)2

)

≤ c0|||u|||s |||v|||s(

1 + 2 supk∈N

k∑

l=1

(1

l2 + 1

(k − l + 1)2

))

= c0|||u|||s |||v|||s(

1 + 4 supk∈N

k∑

l=1

1

l2

)

≤ C |||u|||s |||v|||s,where C = 9c0 depends only on r . ��

Lemma 2.2 For any s > s′ > 0, we have

|||P1u|||s′ ≤ 1

s − s′ |||u|||s, (2.9)

|||P2u|||s ≤ |||u|||s, (2.10)

and

|||P3u|||s ≤ |||u|||s . (2.11)

123

1120 K. Yan, Z. Yin

Proof We first prove (2.9). Note that s > s′ > 0 or 0 < s′s < 1. Then we have

|||P1u|||s′ = supk∈N

s′k‖∂k P1u‖Hr (R)

k!/(k + 1)2

= supk∈N

s′k‖∂k+1u‖Hr (R)

k!/(k + 1)2

= supk∈N

sk+1‖∂k+1u‖Hr (R)

(k + 1)!/(k + 2)2 · s′k

sk+1 · (k + 1)2

(k + 2)2 (k + 1)

≤ |||u|||s supk∈N

1

s

(s′

s

)k

(k + 1)


1

s

(

1 + s′

s+

(s′

s

)2

+ · · · +(

s′

s

)k)


1

s· 1

1 − s′s

= 1

s − s′ |||u|||s .

Next we prove (2.10). Since

‖∂k P2u‖2Hr (R) =

∫

R

(1 + ξ2)r∣∣∣ ∂k P2u(ξ)

∣∣∣2

dξ

=∫

R

(1 + ξ2)r∣∣∣∣

ξ kξ

1 + ξ2 u(ξ)

∣∣∣∣

2

dξ

≤∫

R

(1 + ξ2)r∣∣∣ξ

k u(ξ)

∣∣∣2

dξ

= ‖∂ku‖2Hr (R),

it follows that

|||P2u|||s = supk∈N

sk‖∂k P2u‖Hr (R)

k!/(k + 1)2

≤ supk∈N

sk‖∂ku‖Hr (R)

k!/(k + 1)2

= |||u|||s .For the proof of (2.11), it suffices to show that ‖∂k P3u‖Hr (R) ≤ ‖∂ku‖Hr (R).In fact,

‖∂k P3u‖2Hr (R) =

∫

R

(1 + ξ2)r∣∣∣ ∂k P3u(ξ)

∣∣∣2

dξ

=∫

R

(1 + ξ2)r∣∣∣∣

ξ k

1 + ξ2 u(ξ)

∣∣∣∣

2

dξ

123


≤∫

R

(1 + ξ2)r∣∣∣ξ

k u(ξ)

∣∣∣2

dξ

= ‖∂ku‖2Hr (R).

��Proof of Theorem 2.3 For any u j and v j ∈ B(0, R) ⊂ Es, ( j = 1, 2, 3, 4), we have

|||F(u1, u2, u3, u4) − F(v1, v2, v3, v4)|||s′

=4∑

i=1

|||Fi (u1, u2, u3, u4) − Fi (v1, v2, v3, v4)|||s′

≡ I1 + I2 + I3 + I4.

Next, we will estimate I1, I2, I3 and I4 respectively.

I1 ≤ 1

2|||P1( f (u1) − f (v1))|||s′ + |||P2( f (u1) − f (v1))|||s′

+1

2|||P2( f (u2) − f (v2))|||s′ + 1

2|||P2( f (u3) − f (v3))|||s′

+1

2|||P2( f (u4) − f (v4))|||s′

≤ 1

2

1

s − s′ ||| f (u1) − f (v1)|||s′ + ||| f (u1) − f (v1)|||s′

+1

2||| f (u2) − f (v2)|||s′ + 1

2||| f (u3) − f (v3)|||s′ + 1

2||| f (u4) − f (v4)|||s′

≤ 1

2

C

s − s′ |||u1 + v1|||s′ |||u1 − v1|||s′ + C |||u1 + v1|||s′ |||u1 − v1|||s′

+1

2C |||u2 + v2|||s′ |||u2 − v2|||s′ + 1

2C |||u3 + v3|||s′ |||u3 − v3|||s′

+1

2C |||u4 + v4|||s′ |||u4 − v4|||s′

≤ C(r, R)

s − s′ |||(u1, u2, u3, u4) − (v1, v2, v3, v4)|||s,I2 ≤ |||P1(u1u2 − v1v2)|||s′ + |||P1 P2( f (u1) − f (v1))|||s′

+1

2|||P1 P2( f (u2) − f (v2))|||s′ + 1

2|||P1 P2( f (u3) − f (v3))|||s′

+1

2|||P1 P2( f (u4) − f (v4))|||s′

≤ 1

s − s′ |||u1u2 − v1v2|||s′ + 1

s − s′ ||| f (u1) − f (v1)|||s′

+1

2

1

s − s′ ||| f (u2) − f (v2)|||s′ + 1

2

1

s − s′ ||| f (u3) − f (v3)|||s′

+1

2

1

s − s′ ||| f (u4) − f (v4)|||s′

≤ C

s − s′ (|||u2|||s′ |||u1 − v1|||s′ + |||v1|||s′ |||u2 − v2|||s′)

+ 1

s − s′ ||| f (u1) − f (v1)|||s′ + 1

2

1

s − s′ ||| f (u2) − f (v2)|||s′

123

1122 K. Yan, Z. Yin

+1

2

1

s − s′ ||| f (u3) − f (v3)|||s′ + 1

2

1

s − s′ ||| f (u4) − f (v4)|||s′

≤ C(r, R)

s − s′ |||(u1, u2, u3, u4) − (v1, v2, v3, v4)|||s,

I3 ≤ |||u1u4 − v1v4|||s′ + |||P2(u2u4 − v2v4)|||s′ + |||P3(u2u3 − v2v3)|||s′

≤ |||u1u4 − v1v4|||s′ + |||u2u4 − v2v4|||s′ + |||u2u3 − v2v3|||s′

≤ C(|||u1|||s′ |||u4 − v4|||s′ + |||v4|||s′ |||u1 − v1|||s′)

+C(|||u2|||s′ |||u4 − v4|||s′ + |||v4|||s′ |||u2 − v2|||s′)

+C(|||u2|||s′ |||u3 − v3|||s′ + |||v3|||s′ |||u2 − v2|||s′)

≤ C |||(u1, u2, u3, u4) − (v1, v2, v3, v4)|||s≤ C(r, R)

s − s′ |||(u1, u2, u3, u4) − (v1, v2, v3, v4)|||s,

I4 ≤ |||P1(u1u4 − v1v4)|||s′ + |||P1 P2(u2u4 − v2v4)|||s′ + |||P2(u2u3 − v2v3)|||s′

≤ 1

s − s′ |||u1u4 − v1v4|||s′ + 1

s − s′ |||u2u4 − v2v4|||s′ + |||u2u3 − v2v3|||s′

≤ C(r, R)

s − s′ |||(u1, u2, u3, u4) − (v1, v2, v3, v4)|||s .

This completes the proof of Theorem 2.3. ��

3 Analytic solutions on the circle

In this section, we study the analytic solutions to the following modified two-componentperiodic Camassa–Holm shallow water system:

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

mt + umx + 2mux = −σρρx , t ∈ R, x ∈ R,

ρt + (ρu)x = 0, t ∈ R, x ∈ R,

m(0, x) = m0(x), x ∈ R,

ρ(0, x) = ρ0(x), x ∈ R,

m(t, x) = m(t, x + 1), t ∈ R, x ∈ R,

ρ(t, x) = ρ(t, x + 1), t ∈ R, x ∈ R,

(3.1)

where m = u − uxx , ρ = (1 − ∂2x )(ρ − ρ0), ρ0 is a constant and σ = ±1.

Similar to Sect. 2, we can rewrite the system (3.1) as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂t u1 = F1(u1, u2, u3, u4),

∂t u2 = F2(u1, u2, u3, u4),

∂t u3 = F3(u1, u2, u3, u4),

∂t u4 = F4(u1, u2, u3, u4),

u1(0, x) = u0(x),

u2(0, x) = −∂x u0(x),

u3(0, x) = γ0(x),

u4(0, x) = −∂xγ0(x),

(3.2)

where t ∈ R, x ∈ T, u0, γ0 ∈ Cω(T).

123


Define

U ≡ (u1, u2, u3, u4)

and

F(U ) = F(u1, u2, u3, u4)

≡ (F1(u1, u2, u3, u4), F2(u1, u2, u3, u4), F3(u1, u2, u3, u4), F4(u1, u2, u3, u4)).

Then we have{

∂U (t)∂t = F(t, U (t)),

U (0) = (u0(x),−∂x u0(x), γ0(x),−∂xγ0(x)).(3.3)

where t ∈ R, x ∈ T.In this case, we also have the analyticity result as follows:

Theorem 3.1 Let

(m0

ρ0

)

be real analytic on T. There exists an ε > 0 and a unique solution(

mρ

)

of the Cauchy problem (3.1) that is analytic on (−ε, ε) × T.

For the proof of Theorem 3.1, we will need another Banach space similar to Es definedin Sect. 2.

For any s > 0, we define

E ′s ≡

{

u ∈ C∞(T) : |||u|||′s ≡ supk∈N

sk‖∂ku‖Hr (T)

k!/(k + 1)2 < ∞}

,

where r > 12 is any fixed real number. Obviously, E ′

s equipped the norm ||| · |||′s is a Banachspace, and that, for any 0 < s′ < s, E ′

s ⊂ E ′s′ with |||u|||′s′ < |||u|||′s . Another simple

consequence of the definition is that any u in E ′s is real analytic on the circle T.

Now we set

|||(u1, u2, u3, u4)|||′s ≡4∑

i=1

|||ui |||′s

and

|||F(u1, u2, u3, u4)|||′s ≡4∑

i=1

|||Fi (u1, u2, u3, u4)|||′s .

Using the algebra property and the Fourier series of Hr on the circle T, we can easily obtainthe following results similar to Lemma 2.1 and Lemma 2.2.

Lemma 3.1 Let s > 0. There is a constant C > 0, independent of s, such that for any u andv in E ′

s , we have

|||uv|||′s ≤ C |||u|||′s |||v|||′s, (3.4)

where C = C(r) depends only on r.

Lemma 3.2 For any s > s′ > 0, we have

|||P1u|||′s′ ≤ 1

s − s′ |||u|||′s, (3.5)

|||P2u|||′s ≤ |||u|||′s, (3.6)

123

1124 K. Yan, Z. Yin

and

|||P3u|||′s ≤ |||u|||′s . (3.7)

Then we can easily complete the proof of Theorem 3.1 by using the above key lemmas asin Sect. 2.

4 Some other examples

Example 4.1 (System)The classical integrable two-component Camassa–Holm shallow water system:

⎧⎪⎪⎨

⎪⎪⎩

ut − utxx + kux + 3uux = 2ux uxx + uuxxx + σρρx ,

ρt + (uρ)x = 0,

u(0, x) = u0(x),

ρ(0, x) = ρ0(x),

(4.1)

where k is a constant, σ = ±1, t ∈ R, x ∈ R(or T) and u0, ρ0 ∈ Cω.

The system (4.1) was recently derived by Constantin and Ivanov [16] in the context ofshallow water theory. The variable u(t, x) describes the horizontal velocity of the fluid andthe variable ρ(t, x) is in connection with the horizontal deviation of the surface from equilib-rium, all measured in dimensionless units [16]. The case σ = −1 corresponds to the situationin which the gravity acceleration points upwards [16]. The system (4.1) with σ = −1 wasoriginally proposed by Chen et al. in [7] and Falqui in [26]. The system (4.1) with σ = −1 isidentified with the first negative flow of the AKNS hierarchy and has peakon and multi-kinksolutions [7]. The extended N = 2 supersymmetric Camassa–Holm equation was presentedrecently by Popowicz in [45]. The Cauchy problem of the system (4.1) with initial datau0 ∈ Hs × Hs−1, s ≥ 2 was recently studied in [24,28,34].

For (4.1), we can also rewrite it in the following nonlocal form:⎧⎪⎪⎨

⎪⎪⎩

∂t u + ∂x (12 u2) + (1 − ∂2

x )−1∂x (u2 + 12 (∂x u)2 + ku − 1

2σρ2) = 0,

∂tρ + ∂x (uρ) = 0,

u(0, x) = u0(x),

ρ(0, x) = ρ0(x),

(4.2)

where t ∈ R, x ∈ R(or T) and u0, ρ0 ∈ Cω.Then we can obtain the analyticity results that resemble those in Sects. 2 and 3.

Example 4.2 (Equation)We study the following nonlinear dispersive equation:

{ut − α2utxx + c0ux + (b + 1)uux + γ uxxx = α2(bux uxx + uuxxx ),

u(0, x) = u0(x),(4.3)

where c0, b, γ, α are real constants, t ∈ R, x ∈ R(or T) and u0 ∈ Cω.

Using the notation y = u − α2uxx , we can rewrite Eq. (4.3) as follows:{

yt + c0ux + uyx + bux y + γ uxxx = 0,

u(0, x) = u0(x),(4.4)

where t ∈ R, x ∈ R (or T) and u0 ∈ Cω.

123


The b-equation (4.4) can be derived as the family of asymptotically equivalent shallowwater wave equations that emerges at quadratic order accuracy for any b �= −1 by an appro-priate Kodama transformation, cf. [22,23]. For the case b = −1, the corresponding Kodamatransformation is singular and the asymptotic ordering is violated, cf. [22,23]. The solutionsof the b-equation (4.4) with c0 = γ = 0 were studied numerically for various values of b in[32,33], where b was taken as a bifurcation parameter. The symmetry conditions necessaryfor integrability of the b-equation (4.4) was investigated in [40]. The KdV equation, theCamassa–Holm equation and the Degasperis–Procesi equation are the only three integrableequations in the b-equation (4.4), which was shown in [20,21] by using Painlevé analysis.The b-equation with c0 = γ = 0 admits peakon solutions for any b ∈ R, cf. [20,32,33].

Let α �= 0 be given. We can also rewrite (4.4) in the following nonlocal form [25]:{

∂t u + u∂x u − γ

α2 ∂x u + (1 − α2∂2x )−1∂x (

b2 u2 + (3−b)α2

2 (∂x u)2 + (c0 + γ

α2 )u) = 0,

u(0, x) = u0(x),

(4.5)

where t ∈ R, x ∈ R(or T) and u0 ∈ Cω.It is not difficult to obtain the unique analytic solution of (4.4) when α �= 0 by the above

similar way.

Remark 4.1 Some classical water wave equations as the special cases of (4.4) when α �= 0are as follows:

(i) If b = 2, then Eq. (4.4) becomes the Dullin-Gottwald-Holm equation;(ii) If b = 2 and γ = 0, then Eq. (4.4) becomes the Camassa–Holm equation; its analyticity

of periodic solutions is essentially obtained in [29] mentioned in the introduction.(iii) If b = 2 and c0 = γ = 0, then Eq. (4.4) becomes the Degasperis–Procesi equation.

Remark 4.2 If α = 0 and b = 2, then Eq. (4.4) becomes the well-known KdV equation. Ourapproach in this paper does not apply to Eq. (4.4) when α = 0. Note that the solution to theCauchy problem of the KdV equation with an analytic initial profile is analytic in the spacevariable for a fixed time [48]. However, analyticity in the time variable fails [3,38]. The proofof the preservation of analyticity in space for KdV equation relies on the associated inversespectral problem (the width of the time-invariant spectral gaps encodes the smoothness inspace) and this property holds also for CH (see the considerations in the paper [9]). In thiscontext, the approach presented in this paper is not dependent upon integrability properties.

Acknowledgments This work was partially supported by NNSFC (No. 10971235), RFDP (No.200805580014), NCET-08-0579 and the key project of Sun Yat-sen University. The authors thank the refereesfor their valuable comments and suggestions.

References

1. Baouendi, S., Goulaouic, C.: Remarks on the abstract form of nonlinear Cauchy–Kowalevski theo-rems. Comm. Partial Differ. Equ. 2, 1151–1162 (1977)

2. Baouendi, S., Goulaouic, C.: Sharp estimates for analytic pseudodifferential operators and applicationsto Cauchy problems. J. Differ. Equ. 48, 241–268 (1983)

3. Byers, P., Himonas, A.: Non-analytic solutions of the KdV equation. Abstr. Appl. Anal. 453–460 (2004)4. Beals, R., Scattinger, D., Szmigielski, J.: Acoustic scatting and the extended Korteweg–de Vries hierar-

chy. Adv. Math. 140, 190–206 (1998)5. Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev.

Lett. 71, 1661–1664 (1993)

123

1126 K. Yan, Z. Yin

6. Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)

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

8. Constantin, A.: The Hamiltonian structure of the Camassa–Holm equation. Expo. Math. 15, 53–85 (1997)9. Constantin, A.: On the inverse spectral problem for the Camassa–Holm equation. J. Funct. Anal. 155, 352–

363 (1998)10. Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond.

A 457, 953–970 (2001)11. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)12. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi

equations. Arch. Rat. Mech. Anal. 1992, 165–186 (2009)13. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta

Math. 181, 229–243 (1998)14. Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–

431 (2007)15. Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann.

Math. 172 (2010) (to appear)16. Constantin, A., Ivanov, R.: On an integrable two-component Camass-Holm shallow water system. Phys.

Lett. A 372, 7129–7132 (2008)17. Constantin, A., Strauss, W.: Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 (2000)18. Constantin, A., Strauss, W.: Stability of a class of solitary waves in compressible elastic rods. Phys. Lett.

A 270, 140–148 (2000)19. Dai, H.: Model equations for nolinear dispersive waves in compressible Mooney-Rivlin rod. Acta

Mech. 127, 193–207 (1998)20. Degasperis, A., Holm, D.D., Hone, A.N.W.: Integral and non-integrable equations with peakons. Non-

linear Phys. Theory Exp. II, 37–43 (2003)21. Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and

Perturbation Theory, pp. 23–37. World Scientific, NJ (1999)22. Dullin, H.R., Gottwald, G.A., Holm, D.D.: Camassa–Holm, Korteweg-de Vries-5 and other asymptoti-

cally equivalent equations for shallow water waves. Fluid Dyn. Res. 33, 73–79 (2003)23. Dullin, H.R., Gottwald, G.A., Holm, D.D.: On asymptotically equivalent shallow water wave equa-

tions. Phys. D 190, 1–14 (2004)24. 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)25. Escher, J., Yin, Z.: Well-posedness, blow-up phenomena and global solutions for the b-equation. J. Reine

Angew. Math. 624, 51–80 (2008)26. Falqui, G.: On a Camassa–Holm type equation with two dependent variables. J. Phys. A Math.

Gen. 39, 327–342 (2006)27. Fokas, A., Fuchssteiner, B.: symplectic, their Backlund transformation and hereditary symmetries. Phys.

D 4, 47–66 (1981)28. Guan, C., Yin, Z.: Global existence and blow-up phenomena for an integrable two-component Camassa–

Holm shallow water system. J. Differ. Equ. 248, 2003–2014 (2010)29. Himonas, A.A., Misiolek, G.: Analyticity of the Cauchy problem for an integrable evolution equa-

tion. Math. Ann. 327, 575–584 (2003)30. Holm, D., Marsden, J., Ratiu, T.: The Euler-Poincaré equations and semidirect products with applications

to continuum theories. Adv. Math. 137, 1–81 (1998)31. Holm, D., Naraigh, L., Tronci, C.: Singular solution of a modified two-component Camassa–Holm equa-

tion. Phy. Rev. E 79, 1–13 (2009)32. Holm, D.D., Staley, M.F.: Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM

J. Appl. Dyn. Sys. 2, 323–380 (2003)33. Holm, D.D., Staley, M.F.: Nonlinear balances and exchange of stability in dynamics of solitons, peakons,

ramps/ cliffs and leftons in a 1-1 nonlinear evolutionary PDE. Phys. Lett. A 308, 437–444 (2003)34. Hu, Q., Yin, Z.: Well-posedness and blowup phenomena for a 2-component periodic Camassa–Holm

equation. Proc. R. Soc. Edinb. (2010) (to appear)35. Ionescu-Krus, D.: Variational derivation of the Camassa–Holm shallow water equation. J. Nonlinear

Math. Phys. 14, 303–312 (2007)36. Ivanov, R.: Water waves and integrability. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng.

Sci. 365, 2267–2280 (2007)

123


37. Johnson, R.: Camassa–Holm, Korteweg-de Vries and related models for water waves. J. Fluid.Mech. 457, 63–82 (2002)

38. Kato, T., Masuda, K.: Nonlinear evolution equations and analyticity I. Ann. Inst. H. Poincaré Anal. NonLinéaire 3, 455–467 (1986)

39. Marsden, J., Ratiu, T.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechan-ics and Systems, 2nd edn. Texts in Applied Mathematics, vol. 17. Springer, New York (1999)

40. Mikhailov, A.V., Novikov, V.S.: Perturbative symmetry approach. J. Phys. A 35, 4775–4790 (2002)41. Nirenberg, L.: An abstract form of the nonlinear Cauchy-Kowalevski theorem. J. Differ. Geom. 6, 561–

576 (1972)42. Nishida, T.: A note on a theorem of Nirenberg. J. Differ. Geom. 12, 629–633 (1977)43. Ovsiannikov, L.V.: Non-local Cauchy problems in fluid dynamics. Actes Congr. Int. Math. Nice 3, 137–

142 (1970)44. Ovsiannikov L.V.: A nonlinear Cauchy problems in a scale of Banach spaces. Dokl. Akad. Nauk. SSSR

200 (1971)45. Popowicz, Z.: A 2-component or N = 2 supersymmetric Camassa–Holm equation. Phys. Lett. A 354,

110–114 (2006)46. Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)47. Treves F.: Ovsiannikov theorem and hyperdifferential operators. Notas Matematica 46 (1968)48. Trubowitz, E.: The inverse problem for periodic potentials. Comm. Pure Appl. Math. 30, 321–337 (1977)49. Whitham, G.B.: Linear and Nonlinear Waves. Wiley-Interscience, New York (1974)

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