Benjamin–Ono Equation in Weighted Sobolev
Spaces
Xueshang Feng
∗Abstract
Here we continue the study of the initial value problem for the third order Benjamin-Ono equation in the weighted Sobolev spacesHγs =HsT
L2γ, where
s >3, γ ≥0. The result is the proof of well-posedness of the afore mentioned problem in Hγs, s > 3, γ ∈[0,1]. The proof involves the use of parabolic regularization, the Riesz-Thorin interpolation theorem and the construction technique of auxiliary functions.
1
Introduction
In recent years, the initial value problem for Benjamin-Ono(BO) equation of deep water,
∂tu+∂x(2Hux+u2) = 0
has been investigated by many authors [1, 4, 5, 7, 8]. Iorio[5] established the well-posedness of the above BO equation inH2
γ, where γ ∈[0,1], by using Kato’s theory of linear evolution equations of “hyperbolic type”. Ponce [7] proved the global well-posedness of the BO equation in Hs, s ≥ 3/2. In [3], the authors obtained the
∗Supported by National Natural Science Foundation of China
Received by the editors May 1996. Communicated by J. Mawhin.
1991Mathematics Subject Classification : 35Q20, 35G25, 35D99.
Key words and phrases : The third order BO equation; Initial value problem; Weighted Sobolev spaces; Global well-posedness.
global existence and uniqueness both in the usual Sobolev space Hs and weighted Sobolev spaces Hs
γ for the third order BO equation
∂tu=−∂x(u3+ 3uHux+ 3H(uux)−4∂x2u), t >0, x∈R (1.1)
u(x,0) =φ(x) (1.2)
which can be considered as the high order approximation of the BO equation. Here
u = u(x, t) is real valued, ∂k x = ∂
k
∂xk and the subscripts appended to u stand for
partial differentiations, and H denotes the Hilbert transform
(Hf)(x) =P.V.
Z f(y)
π(y−x)dy In this paper, Hs
γ =Hs T
L2
γ with the following norm
||u||2γ,s=||f||2s+|ωγf|22, ωγ(x) = (1 +x2)γ
where|| • ||s and| • |2 are the usual norms of the Sobolev spacesHs(R) and Lebesgue
space L2(R), respectively. Throughout the paper, | • |
p will denote the usual norm of the Lebesgue spaces Lp(R), p∈[1,∞]. IfX and Y are Banach spaces we denote the set of all bounded operators fromX toY byB(X, Y). IfX =Y we writeB(X) instead of B(X, Y). As usual [A, B] will indicate the commutator of two linear operators(not necessarily continuous).
The BO equation arises in the study of unidirectional propagation of nonlinear dispersive waves or long nonlinear sausage-wave propagation in a magnetic slab in an incompressible plasma of the solar atmosphere, and presents the interesting fact that operrators modelling the dispersive effect in the BO equation or the effect of derivative nonlinearity in (1.1) are nonlocal. In [3, 5], the authors have shown that
u ∈ C([0, T];H4 3 2
) or u ∈ C([0, T];H4
2) solves equation (1.1) or the BO equation
if and only if u ≡ 0, and pointed out in [3] that this phenomena are caused by the nonlocal terms modelling the equations. The papers [1, 7] are devoted to the proof of the well-posedness of the global solutions to the initial value problem of the BO equation in Hs, for s ≥ 3
2. Many physicists such as Ruderman (see [8]
and references therein) have used this equation to study the wave propagation in a magnetically structured atmosphere and found its solutions in the form of periodic waves of permanent shape numerically. As the high order approximation of the BO equation, the third order BO equation (1.1) is interesting in its own rights and therefore it is of importance to study its well-posedness whether mathematically or physically considered. I would here like to apologize if I miss citing other papers on the BO equation.
Our aim here is to investigate the global well-posedness of problem (1.1), (1.2) in the weighted Sobolev spaces Hs
γ as done in [5] for the BO equation. Our main result is as follows:
Theorem 1.1. Let φ ∈ Hs
γ, s ≥ 4 and γ ∈ [0,1]. Then for all fixed µ ≥ 0 there exists a unique uµ ∈ C([0, T];Hs) for any given T > 0 such that ∂tuµ ∈
C([0, T];Hs−4) and ω′
γ∂xkuµ∈C([0, T];L2) fork ≤s/2 and the following (2.6), (2.7) are satisfied. Moreover,
Furthermore, the map φ 7→ uµ is continuous in the sense that if φn → φ in Hγs−2 and un
γ, uµ are the corresponding solutions then
lim
n→∞sup[0,T]||u n
µ(t)−uµ(t)||γ,s−2= 0
wheres ≥6.
In fact, we shall show that the map φ7→u is globally Lipschitz. In our cases, it is difficult to apply Kato’s theory of linear evolution equations of “hyperbolic type” because of the appearance of the derivative nonlinear nonsmooth terms. The proof in this paper is very technical and long. In the following two sections we shall finish the proof of Theorem 1.1 by presenting a series of statements.
2
Preliminary results
In this part, a series of lemmas are given for the purpose of proving Theorem 1.1. The lemmas below are established following the same lines as those of [5].
Lemma 2.1. Supposea(x) is defined onRand|a(x)−a(y)| ≤M|x−y|, x, y∈
R. LetTǫ(f) =R|x−y|≥ǫ
na(x)−a(y)
(x−y)2 o
f(y)dy. Then
|Tǫ(f)|p ≤Ap|f|p, for all f ∈Lp(R) with Ap independent of ǫ, 1< p <∞.
For details and further results of this kind see Calderon [2] and P.238 of Stein [9]. The following well-known Gagliardo-Nirenberg’s inequality will be unexplainedly used throughout the paper.
Lemma 2.2. For f ∈Hm(R) and p≥2, then we have
|∂xdf|p ≤2 (p−2)
2p |∂k
xu|λ2|u|1−λ (2.1)
whered, m, k ∈NS
{0}, λ= (d+12−1p)/k, and 0≤d < k ≤m. More precisely, forf ∈Hd+1(R) we have
|∂d
xf|∞ ≤ |f| 1 2(d+1)
2 |∂xd+1f| 2d+1 2(d+1)
2 (2.2)
Proof. (2.1) has been well-known. To prove (2.2), noting that
|f(x)|2 = 2 Z x
−∞ffxdx≤2 Z x
−∞|ffx|dx
|f(x)|2 =−2 Z ∞
x ffxdx≤2 Z ∞
x |ffx|dx We get by using the Cauchy-Schwartz inequality
|f|2 ∞ ≤
Z
|ffx|dx≤ |f|2|fx|2 (2.3)
which gives (2.2) for the case d=0. By using (2.1) and (2.3) we get
|∂xdu|2∞ ≤ |∂xdu|2|∂xd+1u|2 ≤ |∂xd+1u| 2d+1
d+1 2 |u|
1
which finishes the proof of (2.2).
It should be pointed out that inequality (2.2) can be used to improve the unique-ness result for the smallunique-ness case of the initial data in [3].
Lemma 2.3. Let λ∈ [0,∞), γ ∈[0,1], µ >0, and Qµ =−µ∂x4+ 4∂x3. Then
Sµ(t) =exp(tQµ)∈B(Hγs, Hγs+λ) for all t >0, s≥0 and satisfies the estimate
||Sµ(t)φ||γ,s≤G(t, µ, λ, γ)||φ||γ,s (2.4) for all φ ∈ Hs
γ, where G(t, µ, λ, γ) is locally integrable with respect to t. Moreover the map t ∈(0,∞)7→ Sµ(t)φ is continuous with respect to the topology of Hγs+λ.
Proof. Due to [3], where the Lemma is established in the case ofHs
0 =Hs, s∈R,
it suffices to examineSµ(t) as an operator fromHγsintoHγ0 =L2γ. Ifγ = 0 we obtain
H0
0 = L2 so that ||Sµ(t)φ||0,0 ≤ ||φ||0,0. Next, if γ = 1 and φ ∈ H1s = Hs
T
L2 1 we
have
(ω1Sµ(t)φ)∧(ξ) =Eµ(ξ, t)(1−∂ξ2) ˆφ−(∂ξ2Eµ(ξ, t)) ˆφ−2(∂ξEµ(ξ, t))∂ξφˆ (2.5) where Eµ(ξ, t) = exp[−(µξ4 + 4iξ3)t]. The result then follows by combining (2.5) with the formulas
∂ξEµ(ξ, t) =−(4µξ3+ 12iξ2)tEµ(ξ, t)
∂ξ2Eµ(ξ, t) =−12t(µξ2+ 2iξ)Eµ(ξ, t) + 16t2ξ4(µξ+ 3i)Eµ(ξ, t) and the estimate
0≤ξ2λexp(−µtξ4)≤λλexp(λ 2)(2µt)
−λ/2
The statement in the Lemma is now an easy consequence of the Riesz-Thorin interpolation theorem.
We are now in position to establish a local existence for the following parabolic regularized problem
∂tuµ=−µ∂x4uµ−∂x(u3µ+ 3uµH∂xuµ+ 3H(uµ∂xuµ)−4∂x2uµ) (2.6)
uµ(x,0) =φ(x) (2.7)
in Hs
γ, γ∈ [0,1] and µ >0. Let T be a positive number and consider the set
Xγ,s(T) = n
f ∈C([0, T];Hs
γ); ||f(t)− Sµ(t)φ||γ,s≤ ||φ||γ,s, t∈[0, T] o
which becomes a complete metric space when provided with the distance obtained from the sup norm. Note that if f ∈Xγ,s(T), then by Lemma 2.3 it is not difficult to show that one can choose T >0 such that the map
(Af)(t) =Sµ(t)φ− Z t
0 Sµ(t−τ)∂x
f3+ 3fHfx+ 3H(ffx)
dτ
defines a contraction in Xγ,s(T). In view of the uniqueness and regularity results established in [3] we have
Corollary 2.4. Let φ ∈ Hs
γ, s ≥ 4, γ ∈ [0,1], µ > 0. Then there exist a
Letµ >0 be fixed. From now on, unless otherwise specifed, we write u=uµfor simplicity. The next step is to establish global existence inHs
γ, s≥4 andγ ∈[0,1]. This has already been done in [3] in the casesγ = 0 and γ = 1 (the result for γ = 1 will be re-obtained below from a slightly different point of view). In order to obtain global estimates for the Hs
γ norm of the solution it suffices to study what happens in L2
γ since the Hs result was proved in [3]. Now, by Corollary 2.4 and integration by parts, at least formally we have
∂t||u(t)||2γ,0+ 2µ
Z t
0
Z
ωγ2u22xdx
=−4µ
Z
uu2xωγωγ′′dx−4µ Z
uu2x(ωγ′)2dx−8µ Z
uxu2xωγωγ′′dx
−8 Z
ωγ2u∂x(u3)dx−3 Z
ω2γu2H(∂x2u)dx+ 6 Z
ωγωγ′u2Huxdx
+2 Z
(ωγ2)′′′u2dx−24 Z
ωγω′γuu2xdx−3 Z
ω2γuH(∂x2(u2))dx (2.8)
In order to extend the local solution to the whole time interval [0, T], for any fixed
T >0 we must estimate the right-hand side of (2.8). For this purpose we need the following Lemmas.
Lemma 2.5. Let u∈H4
γ, γ ∈[0,1]. Then ωγ/2u2x∈L2 and satisfies
|ωγ
2u2x|2 ≤C||u|| 1 2 4|ωγu|
1 2
2 ≤C||u||γ,4 (2.9)
whereC is a generic constant.
Proof. By definition, for u∈S(R)(the Schwartz space) we have
|ωγ
2u2x| 2 2 =
Z
ωγu22xdx= Z
ωγ′′uu2x dx+ 2 Z
ω′γuu3xdx+ Z
ωγuu4xdx
≤ |ωγ′′u|2|u2x|2+ 2|ωγ′u|2|u3x|2+|ωγu|2|u4x|2 ≤C||u||4|ωγu|2 (2.10)
which combines a limiting argument to yield the result.
Lemma 2.6. Let f ∈Hs
γ, γ ∈[0,1]. Then x(1 +x2)γ∂xkf ∈L2 and
|x(1 +x2)(r−1)∂xkf|2 ≤C(r)||f||γ,s, 2k ≤s (2.11) whereC(γ) is a constant depending only on γ. The same results are true ifx(x2+
1)γ−1 is replaced by ω′ γ(x).
Proof. A simple limiting argument shows that it suffices to prove (2.11) for
f ∈ S(R). Assume therefore that this is the case and integrate by parts to obtain fork = 1
|x(1 +x2)γ−1fx|22 =− 3
X
j=1
Ij
where
I1 = 2
Z
(1 +x2)γf(x)[x(1 +x2)γ−2]fxdx
I2 = 2(2γ −2)
Z
I3 =
Z
(1 +x3)γf(x)[x2(1 +x2)γ−2]∂x2f(x)dx
It is easy to verify that the expressions inside the square brackets in all the three ingegrals are bounded of x. By this remark and the Cauchy-Schwartz inequality we obtain
|I1| ≤2|fx|2|ωγ|2, |I2| ≤ |4γ−4||fx|2|ωγf|2, |I3| ≤ |∂x2f|2|ωγf|2
Now, we have
|x(1 +x2)γ−1fx|2 ≤ {(2 +|4γ−4|)|fx|2+|f2x|2} |ωγf|2 ≤ C(γ)||f||2|ωγf|2
which finishes the proof. For the general case, a similar argument yields
|x(1 +x2)γ−1∂xkf|22 =
2k X
j=k Z
(1 +x2)γf(x)pj(x)∂xjf(x)dx
where pj(x) is bounded inx. So the result follows.
Corollary 2.7. Let f ∈ H4
γ, γ ∈ [0,1]. Then ωγ2fx ∈ L2, ωγ
2fx ∈ L ∞ and
satisfy the estimates
|ωγ
2fx|2 ≤C(γ)||f|| 1 2 2|ωγf|
1 2
2 (2.12)
|ωγ
2fx|∞≤C(γ)||f|| 1 4 4|ωγf|
1 2
2 (2.13)
where C(γ)>0 depends only on γ.
Proof. Similar to the proof of (2.9), it suffices to prove (2.12) for f ∈ S(R). If this is the case, then we have
|ωγ
2fx| 2 2=
Z
ωγfxfxdx=− Z
ωγff2xdx− Z
ωγ′ffxdx
≤|fx|2|ω′γf|2 +|f2x|2|ωγf|2
≤C(γ)||f||2|ωγf|2
which gives (2.12). By (2.2), (2.9), (2.11) and (2.12) we obtain
|ωγ
2fx| 2
∞≤ |ωγ2fx|2|ω′γ
2fx+ω
γ
2f2x|2 ≤C(γ)||f||4|ωγf|2 This finishes (2.13).
Lemma 2.8. Letf ∈Hγs, γ∈[0,1]. Then [ωγ
2, H∂x]f ∈ L 2 and
|[ωγ
2, H∂x]f|2 ≤C(γ)|f|2 (2.14) where C(γ) depends only on γ.
Proof. First observe that it suffices to prove (2.14) in case of f ∈ S(R). Now [ωγ
2, H∂x]f = [ω
γ
2, H]∂xf−H[∂x, ω
γ
2]f (2.15)
For the second commutator in (2.15) we have
H[∂x, ωγ
2]f =H(ω ′
γ
For the first commutator in (2.15) we integrate by parts once to obtain
[ωγ
2, H]∂xf =P.V. 1
π
Z ωγ
2(x)−ω
γ
2(y)
(x−y)2 f(y)dy−H(ω ′
γ
2f) (2.17)
From (2.15)-(2.17) we know that
[ωγ
2, H∂x]f =P.V. 1
π
Z ωγ
2(x)−ω
γ
2(y)
(x−y)2 f(y)dy−2H(ω ′
γ
2f) (2.18)
Sinceω′γ
2 is bounded andω
γ
2 is Lipschitz, by Lemma 2.1 and the fact thatH ∈B(L 2)
it follows from (2.18) that
|[ωγ, H∂x2]f|2 ≤C(γ)|f|2
which finishes the proof.
From the proof above we see that (2.14) holds for all f ∈L2.
Lemma 2.9. Let f ∈H2
γ, γ ∈[0,1]. Then [ωγ, H∂x2]f ∈L2 and
|[ωγ, H∂x2]f|2 ≤C(γ)||f||γ,2 (2.19)
whereC(γ)≥0 depends only on γ.
Proof. First note that it suffices to prove (2.19) for f ∈S(R). Evidently
[ωγ, H∂x2]f = [ωγ, H]∂x2f −H[∂x2, ωγ]f (2.20) So it suffices to establish (2.19) with [ωγ, H∂x2]freplaced byH[∂x2, ωγ]f and [ωγ, H]∂x2f. The result then follows for the first of these functions by combining the formula
H[∂x2, ωγ]f =H(ωγ′′f + 2ωγ′∂xf) (2.21)
with Lemma 2.6 and the facts that ω′′
γ ∈ L∞ and H ∈ B(L2). In order to handle [ωγ, H]∂x2f we integrate by parts once to obtain
[ωγ, H]∂x2f =H(ωγ′∂xf)−P.V. 1
π
Z ω
γ(x)−ωγ(y)
(x−y)2 f(y)dy (2.22)
In view of Lemma 2.6 the first term on the right-hand side of (2.22) satisfies
|H(ω′
γ∂xf)|2 ≤C(γ)||f||γ,2, so it remains to bound the second. Sinceω′γ is bounded if and only ifγ ∈[0,12], in this caseωγ is Lipschitz. Hence, by Lemma 2.1 we have
P.V.1 π
Z ω
γ(x)−ωγ(y)
(x−y)2 f(y)dy
≤A|f|2 (2.23)
where A > 0 is a generic constant, γ ∈[0,21]. In the case of γ ∈(12,1] we integrate by parts once in (2.22) to get
[ωγ, H]∂x2f =−P.V. 1
π
Z
K(x, y) = 2(ωγ(x)−ωγ(y))−2ω ′
γ(y)(x−y)−ω′′γ(y)(x−y)2
(x−y)3(1 +y2)γ (2.25)
where g(y) = (1 +y2)γf(y) ∈ L2. If γ ∈ (1
2,1], it is not difficult to verify that
K(x, y) is a Hilbert-Schmidt kernel. Thus
2,1]. Now the result of the lemma follows
from (2.20)-(2.26).
Now we are in position to estimate each term in the right-hand side of (2.8) as follows: and µvarying in any bounded interval. That is,
whereC(µ,||φ||s)>0 depends in an nondecreasing way onµand||φ||s, respectively. Ifµ remains in a bounded set [0, µ0], one can choose C(µ,||φ||s) depending only on
||φ||s. With this remark in mind, apply Gronwall’s inequality to (2.8)′ to obtain
||u(t)||γ,0≤C(µ, γ, T,||φ||γ,4), t∈[0, T] (2.28)
whereC(µ, γ, T,||φ||γ,4) has the same property as that of (2.27).
Now we return to the global existence result. Ifµ >0, the global existence result follows from (2.27) and (2.28).The existence result in the caseµ= 0 can be proved by the standard limiting argument.
Proof of uniqueness. Here the proof is reproduced from [3]. Let u, v be two solutions to problem (2.6), (2.7) with the initial dataφ(x), ψ(x)∈Hs, respectively. Puttingw=u−v, then w satisfies the following equation
wt=−µ∂x4w+ 4∂x3w−∂x
(u3−v3) + 3wHux+ 3vHwx+ 3
2H∂x(u
2−v2) (2.29)
w(x,0) =φ(x)−ψ(x) (2.30)
For µ > 0 fixed, the uniqueness result follows from the standard energy estimate. So in the following we only give the proof forµ= 0.
From (2.29), a direct calculation of (2.29) and use of Lemma 2.2 yield
d dt
Z
w2dx= 2
Z
−1
2w
2(u2+uv+v2)
x+ 3wwxHux+ 3vwxHwx
− 3
2(u+v)wxHwx− 3
2(ux+vx)wHwx
dx (2.31)
Now it is not difficult to find that
|w|22 ≤ |w(0)|2+C(||φ||2,||ψ||2)
Z t
0 ||w(t)|| 2
1dt (2.32)
Since s≥ 4, we know that|ut|L∞([0,T]×R), |vt|L∞([0,T]×R), |∂k
xu(x, t)|L∞([0,T]×R) and
|∂k
xv(x, t)|L∞
([0,T]×R) are bounded for 0 ≤ k ≤ s−1. Multiply equation (2.29) by
K4(u)−K4(v) to know
Z
wt(K4(u)−K4(v))dx= 0 (2.33)
whereK4(u) =u3+ 3uHux+ 3H(uux)−4u2x.
The bound for the term in the right-hand side of (2.33) is carried out as follows.
1) Z t
0
Z
wt(u3−v3)dxdt
= 1 2 Z
w2(u2+uv+v2)dx t
0
− 1
2 Z t
0
Z
w2(u2+uv+v2) tdxdt
≤C(|w(t)|22+|w(0)|22+ Z t
0 |w(τ)| 2
2)
Adjusting the value ofδ, from (2.32)-(2.36) we derive
||w(t)||21 ≤C(||φ||1,||ψ||1)||w(0)||21+C(||φ||4,||ψ||4)
Z t
0 ||w(τ)|| 2
1dτ (2.37)
Applying Gronwall’s inequality to (2.37) to yield
||w(t)||21≤C(T,||φ||4,||ψ||4)||w(0)||21, t∈[0, T] (2.38)
which implies the uniqueness.
Up to now we have proved the following
Corollary 2.10. Let φ ∈ Hs
γ, s ≥ 4 and γ ∈ [0,1]. Then for all fixed
µ ≥ 0 there exists a unique uµ ∈ C([0, T];Hγs) for any given T > 0 such that
∂tuµ ∈ C([0, T];Hs−4) and ω′γ∂xkuµ ∈ C([0, T];L2) for 2k ≤ s and (2.6), (2.7) are satisfied.
3
Proof of Theorem 1.1 Continued
In this section we continue the proof of Theorem 1.1.
Lemma 3.1. Let φ, ψ ∈ Hs
γ, s ≥ 4 and γ ∈ [0,1], µ ≥ 0. Let uµ, vµ be the solutions(obtained in Corollary 2.10) to (2.6) withφ, ψ as the initial data, respectively. Suppose w=uµ−vµ. Then we have
where C depends in an nondecreasing way on its arguments.
Proof. By the assumptions of the Lemma we know that w satisfies problem (2.29), (2.30). Multiply (2.29) byω2
γw and integrate by parts to obtain
−6
The estimate for each Bj is as follows. By using (2.9), (2.11), (2.12), (2.13) and (2.14), we obtain
Considering (3.1) and (3.2) we know that
d
Integrate (3.3) with respect tot to yield the result.
Lemma 3.2. Let the assumptions in Lemma 3.1 be satisfied and s ≥ 6. Then we have
whereC depends in an increasing way on its arguments and remains bounded when-ever its arguments stay in a bounded set.
Proof. The proof of (3.4) for µ > 0 is simple and is omitted. But the proof for
µ= 0 is technical. For s= 6, a complicated calculation yields
d
right hand side. Hence, (3.4) holds for s = 6. For general s we can prove (3.4) by induction on s. The details are omitted.
The Completion of Proof of Theorem 1.1. By Lemma 2.2 we have
2s−3 4
Z
(u+v)∂xs−3wH∂xs−2wdx
≤C(|φ|2,|ψ|2)|∂ s−3
x w|∞|∂xs−2w|2
≤C|w|2(s1−2)|∂s−2
x w| 2s−5
2(s−2)+1
2 ≤
1 2|∂
s−2
x w|22+C|w|22 (3.5)
From (2.38), (3.4) and (3.5) there appears that
|∂xs−2w|22 ≤C||w(0)||2s−2+C
Z t
0 ||w(t)|| 2
s−2dt (3.6)
Combining (2.38), (3.1), (3.6) with Gronwall’s inequality we can obtain
||w(t)||2γ,s−2 ≤C(T,||φ||γ,s,||ψ||γ,s)||w(0)||2γ,s−2, t∈[0, T]
which implies the last statement of Theorem 1.1.
References
[1] Abdelouhab L. et al., Nonlocal models for nonlinear, dispersive waves, Physica D, 40(1989), 360-392
[2] Calderon A. P., Commutators of singular integral operators, Proc. Nat. Acad. Sci., 53(1965), 1092-1099
[3] Feng X., Han X., On the Cauchy problem for the third order Benjamin-Ono equation, J. London Math. Soc., 53(1996), 512-528
[4] Iorio R. J., On the Cauchy problem for the Benjamin-Ono equations, Comm. PDEs, 11(1986), 1031-1086
[5] Iorio R. J., The Benjamin-Ono equation in weighted Sobolev spaces, J. Math. Anal. Appl., 157(1991), 577-590
[6] Kato T., Linear evolution equations of “hyperbolic” type, I, J. Math. Soc. Japan, 25(1973), 648-666
[7] Ponce G., On the global well-posedness of the Benjamin-Ono equation, Differ-ential and Integral Equations, 4(1991), 527-542
[8] Ruderman M. S., Nonlinear sausage-wave propagation in a magnetic slab in an incompressible fluid, J. Plasma Physics, 49(1993), 271-282
Laboratory of Numerical Study for Helioshperic Physics CAS, P.O.Box 8701, Beijing 100080
