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Orbital stability of peakons for a generalized CH equation
Applied Mathematics and Computation 232 (2014) 183–190
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Orbital stability of peakons for a generalized CH equation Zheng-yong Ouyang a,⇑, Shan Zheng b a b
Department of Mathematics, Foshan University, Foshan City, Guangdong Province 528000, PR China Guangzhou Maritime College, Guangzhou City, Guangdong Province 510725, PR China
a r t i c l e
i n f o
Keywords: Peakons Orbital stability Generalized Camassa–Holm equation
a b s t r a c t In this Letter, two different kinds of peaked traveling wave solutions for a generalized Camassa–Holm equation are considered, the orbital stability of these peaked traveling wave solutions are proved. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Camassa and Holm [1] derived a shallow water wave equation
ut þ 2kux uxxt þ 3uux ¼ 2ux uxx þ uuxxx ;
ð1Þ
which is called Camassa–Holm equation (CH equation). For k ¼ 0, they showed that Eq. (1) has peaked solitary wave solutions (peakons)
u1 ðx; tÞ ¼ cejxctj ;
ð2Þ
which have discontinuous first derivative at the wave peak in contrast to the smoothness of most previously known specious of solitary waves and thus are called peakons. Eq. (1) is integrable in the sense of an infinite-dimensional Hamiltonian system and arises as model for shallow water waves [1,2]. A more in-depth discussion of this essential aspect of the solution is performed in[3,4]. The peakons capture a characteristic of the traveling waves of greatest height – exact traveling solutions of the governing equations for water waves with a peak at their crest [5–7]. Simper approximate shallow water models (like the classical Korteweg–de Vries equation) do not present traveling wave solutions with this feature. In the sense of papers [8,9], the peakons are to be understood as weak solutions. As pointed out above, the governing equations for water waves admit peaked and smooth solutions. The contrast between these two classes is very big since the smooth solutions are known to be real-analytic cf. the discussion in [10]. The discovery of this special kind of peakons from CH equation has attracted much attention of mathematicians and physicists, many authors have come to study the CH equation and different generalized forms of CH equation. There are two important aspects on investigation of CH equation and its modified forms, one is under what conditions, the CH equation and its modified forms have solutions of peakons, another is that if the peakons obtained are orbital stable. For instance, Constantin investigated peakons (2) and proved that the peakons are orbital stable [11]. When k – 0 for CH equation, Zhang [12] obtained a more general expression of peakons as follows
uðx; tÞ ¼ ðk þ cÞejxctj k;
⇑ Corresponding author. E-mail address: [email protected] (Z.-y. Ouyang). 0096-3003/$ - see front matter Ó 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2014.01.062
ð3Þ
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Z.-y. Ouyang, S. Zheng / Applied Mathematics and Computation 232 (2014) 183–190
which means that peakons exist for CH equation for k takes any values, obviously (2) are special case of (3). The author of this letter considered the orbital stability of such peakons (3) with non-vanishing boundary, and proved peakons (3) are orbital stable in the norm of H1 [13]. Based on the above study, Liu and Qian [14] studied a generalized CH equation
ut þ 2kux uxxt þ auux ¼ 2ux uxx þ uuxxx ;
ð4Þ
when a ¼ 3 in (4), Eq. (4) is the CH equation namely. In case of a – 3, they considered if Eq. (4) has solutions of peakons, they found that when the coefficient a changes, there still exist peakons for Eq. (4), and the results are as follows: (i) In case of a – 3 and k – 0, Eq. (4) has peakons
x Bt ; u1 ðx; tÞ ¼ B exp A
ð5Þ
qffiffi 6k where B ¼ 3a ; A ¼ 3a.
(ii) In case of k – 0, Eq. (4) has peakons
x Ct 2 ; u2 ðx; tÞ ¼ C 3 exp A
ð6Þ
qffiffi 2k where C ¼ 1þa ; A ¼ 3a.
Although solutions (5) and (6) are all peaked solitary waves, there exist some differ- ences between peakons (5) and (6), it is obviously that peakons (5) vanish and peakons (6) tend to an non-zero constant as jxj ! 1. The orbital stability problems of the peakons (5) and (6) have not been solved yet, so it is interesting to consider the stability of the two different kinds of peakons. As already observed by Benjamin and coworkers [15,16], a solitary wave cannot be stable in the strictest sense of the word. To understand this, consider two solitary waves of different height, centered initially at the same point. Science the two waves have different amplitudes they have different velocities according to the expressions (5) and (6). As time passes the two waves will apart, no matter how small the initial difference was. However, in the situation just described, it is evident that two solitary waves with slightly differing height will stay similar in shape during the time evolution. Measuring the difference in shape therefore will give an acceptable notion of stability. This sense of orbital stability was introduced by Benjamin [15]. We say a solitary wave is orbital stable if a solution u of Eq. (4) that is initially sufficiently close to a solitary wave will always stay close to a translation of the solitary wave during the time of evolution. A more mathematically precise definition is as follows. For any > 0, consider the tube
n o U ¼ u 2 H1 : inf jju ss uc jjH1 < ; s
where
ss f ðxÞ ¼ f ðx þ sÞ is a translation of f. The set U is an neighborhood of the collection of all translates of uc .
Definition 1.1. The solitary wave is stable if for all > 0, there exists a d > 0, such that if u0 ¼ uð; 0Þ 2 U d , for all t 2 R. The solitary uc is unstable if uc is not stable. The following two quantities E and F are critically important to the proof of peakons,
EðuÞ ¼
Z
R
ðu2 þ A2 u2x Þdx and FðuÞ ¼
Z
R
ðu3 þ A2 uu2x Þdx;
these are constants of the motion,that is, along solutions these expressions are independent of time. The role of these integrals of motion in stability considerations was pioneered in [11]. 1 R Now we denote jjujj2H1 ¼ R ðu2 þ A2 u2x Þdx. It is obvious that jjujj2H1 6 max 1; A2 jjujj2H1 and jjujj2H1 6 maxf1; A2 gjjujj2H1 . A
A
A
From the relationship between the norm jjujjH1 and jjujjH1 , we only need to prove peakons (5) and (6) are stable on the norm jjujjH1 . A
A
2. Stability of peakons vanishing as jxj ! ‘ In this section we discuss the orbital stability of peakons (5) with vanishing boundary for Eq. (4). We have the following theorem. Theorem 2.1. If u 2 Cð½0; TÞ; H1 ðRÞÞ is a solution to (4) with
jjuð; 0Þ u1 jjH1 < A
4 81B3 A3=2
;
0<<
pffiffiffi AB;
then
jjuð; tÞ u1 ð nðtÞÞjjH1 < for t 2 ð0; TÞ; A
where nðtÞ 2 R is any point where the function uð; tÞ attains its maximum.
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Lemma 2.1. For every u 2 H1 and n 2 R,
jjuð; tÞ u1 ð nÞjj2H1 ¼ EðuÞ Eðu1 Þ 4ABðuðn; tÞ BÞ: A
Proof. By direct calculation we get Eðu1 Þ ¼ 2AB2 and
jjuð; tÞ u1 ð nÞjj2H1 ¼ EðuÞ þ Eðu1 Þ 2 A
¼ EðuÞ þ Eðu1 Þ 2
Z
uðx;tÞu1 ðx nÞdx 2A2
R
Z
Z
ux ðx; tÞu1x ðx nÞdx
R
uðx;tÞu1 ðx nÞdx 2A2
R
Z
n
ux ðx; tÞu1x ðx nÞdx 2A2
1
¼ EðuÞ þ Eðu1 Þ 4ABuðn;tÞ ¼ EðuÞ Eðu1 Þ 4ABðuðn; tÞ BÞ:
Z
1
ux ðx;tÞu1x ðx nÞdx
n
pffiffiffi 1 Lemma 2.2. For u 2 H1 , let M ¼ maxx2R fuðx; tÞg. If jju0 u1 jjH1 6 d for some d ¼ d1 AB where d1 < 20 , then A
pffiffiffiffiffiffi ffiffiffi pffiffiffiffiffiffi p 4 jEðuÞ Eðu1 Þj 6 dð2 2AB þ dÞ and jM Bj 6 2 Bd= A: Proof. (1) Since
Eðu1 Þ ¼ 2AB2 ;
Eðuðx; 0ÞÞ ¼ Eðuðx; tÞÞ;
we obtain immediately
jEðuÞ Eðu1 Þj ¼ jEðu0 Þ Eðu1 Þj ¼ ðjju0 jjH1 jju1 jjH1 Þðjju0 jjH1 þ jju1 jjH1 Þ 6 jju0 u1 jjH1 ðjju0 u1 jjH1 þ 2jju1 jjH1 Þ A A A A A A A pffiffiffiffiffiffi 6 dðd þ 2jju1 jjH1 Þ ¼ dð2 2AB þ dÞ: A
ð7Þ
(2) For any given t 2 ð0; TÞ, let M be taken at x ¼ n, we define
gðx; tÞ ¼
uðx; tÞ Aux ðx; tÞ;
x < n;
uðx; tÞ þ Aux ðx; tÞ;
x > n:
we have
Z
g 2 ðx; tÞdx ¼
R
Z
n
ðuðx; tÞ Aux ðx; tÞÞ2 dx þ
1
¼
Z
R
Z
1
ðuðx; tÞ þ Aux ðx; tÞÞ2 dx
n
2 2 ðu2 ðx; tÞ þ A2 u2x ðx; tÞÞdx Au2 ð; tÞjn1 þ A2 u2 ð; tÞj1 n ¼ EðuÞ 2Au ðn; tÞ ¼ EðuÞ 2AM
ð8Þ
and
Z
uðx; tÞg 2 ðx; tÞdx ¼
R
Z
n
uðx; tÞðuðx; tÞ Aux ðx; tÞÞ2 dx þ
1
Z
Z
1
uðx; tÞðuðx; tÞ þ Aux ðx; tÞÞ2 dx
n
2 2 4 3 uðx; tÞðu2 ðx; tÞ þ A2 u2x ðx; tÞÞdx Au3 ð; tÞjn1 þ Au3 ð; tÞj1 n ¼ FðuÞ Au ðn; tÞ 3 3 3 4 3 ¼ FðuÞ AM : 3
¼
R
ð9Þ
So, from (8) and (9) we get
M3
3 3 EðuÞM þ FðuÞ 6 0: 2A 2A
Identity (8) shows that for all uð; tÞ 2 H1 ðRÞ
1 pffiffiffiffiffiffiffiffiffiffi 1 supx2R juðx; tÞj 6 pffiffiffiffiffiffi EðuÞ ¼ pffiffiffiffiffiffi jjujjH1 : A 2A 2A Since
Fðu1 Þ ¼
4 3 AB 3
and Fðuðx; 0ÞÞ ¼ Fðuðx; tÞÞ;
ð10Þ
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by direct calculation we have
Z Z FðuÞ 4 AB3 ¼ jFðu0 Þ Fðu1 Þj ¼ ðu0 u1 Þðu2 þ A2 u2 Þdx þ u1 ðu2 þ A2 u2 u2 A2 u2 Þdx 0 0x 0 0x 1 1x 3 R R Z Z 6 ðu0 u1 Þðu20 þ A2 u20x Þdx þ u1 ððu0 u1 Þ2 þ A2 ðu0x u1x Þ2 Þdx R R Z 2 þ u1 ð2ðu0 u1 Þu1 þ 2A ðu0x u1x Þu1x Þdx R
6 jju0 u1 jjL1 Eðu0 Þ þ jju1 jjL1 jju0 u1 jj2H1 þ 2jju1 jjL1 jju0 u1 jjH1 jju1 jjH1 A
A
A
! pffiffiffiffiffiffi pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 2 1 d2 2 2 2 6 pffiffiffiffiffiffi dð2AB þ 2 2ABd þ d Þ þ Bd þ 2 2AB d ¼ d 3 2AB þ 3Bd þ pffiffiffiffiffiffi : 2A 2A
From (7) and (11) we know that EðuÞ is near 2AB2 and FðuÞ is near 43 AB3 , with FðuÞ ¼
MP
R R
ð11Þ
uðu2 þ u2x Þdx 6 MEðuÞ we have
FðuÞ 1 > B: EðuÞ 2
ð12Þ
According to (10) consider the cubic polynomial
PðyÞ ¼ y3
3 3 EðuÞy þ FðuÞ: 2A 2A
In case EðuÞ ¼ Eðu1 Þ ¼ 2AB2 and FðuÞ ¼ Fðu1 Þ ¼ 43 AB3 , PðyÞ takes the form
P0 ðyÞ ¼ y3 3B2 y þ 2B3 ¼ ðy BÞ2 ðy þ 2BÞ: Because of the estimates on EðuÞ and FðuÞ, there is a root of P near y ¼ 2B, and there may be two roots of P near y ¼ B. But (10) and (12) show that there must be two roots near y ¼ B and M must lie between the two roots. The two roots are closer to y ¼ B than the roots of the cubic
P1 ðyÞ ¼ y3
pffiffiffiffiffiffi pffiffiffiffiffiffi 3 3 4 3 1 yð2AB2 þ 2 2ABd þ d2 Þ þ AB 3 2AB2 d 3Bd2 pffiffiffiffiffiffi d3 ; 2A 2A 3 2A
pffiffiffi 1 whose graph on Rþ lies below the graph of PðyÞ. Notice that, for d ¼ d1 AB; d1 < 20 , we have
P1 ðBÞ ¼ B3 P2 ð1Þ < 0 and
pffiffiffiffiffi ffiffiffi pffiffiffiffiffiffi p 4 P1 ðB 2 Bd= AÞ ¼ B3 P2 ð1 2 d1 Þ > 0; where
pffiffiffi pffiffiffi 3 3 4 1 P2 ðyÞ ¼ y3 yð2 þ 2 2d1 þ d21 Þ þ 3 2d1 3d21 pffiffiffi d31 : 2 2 3 2
pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi So M lies in the interval ðB 2 Bd= 4 A; B þ 2 Bd= 4 AÞ. This completes the proof. 4
Proof of Theorem 2.1. Let d ¼ 81B3 A3=2 , by assumption
jEðuÞ Eðu1 Þj 6
pffiffiffi 2 2 2 2 þ 2 81 81
juðnðtÞ; tÞ Bj 6
22 : 9AB
<
h
pffiffiffi pffiffiffi 1 AB so that d < 20 B A, then from Lemma 2.2, we obtain
and
Combining the above inequalities and Lemma 2.1, we find
jjuðx; tÞ u1 ðx nðtÞÞjj2H1 ¼ EðuÞ Eðu1 Þ 4ABðuðnðtÞ; tÞ BÞ 6 jEðuÞ Eðu1 Þj þ 4jABðuðnðtÞ; tÞ BÞj A pffiffiffi 2 2 2 2 8 2 2 6 2þ þ < : 9 81 81 That is
jjuðx; tÞ u1 ðx nðtÞÞjjH1 < : A
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3. Stability of peakons non-vanishing as jxj ! ‘ In this section we discuss the stability of peakons (6) with non-vanishing boundary for Eq. (4).The solutions u2 ðx; tÞ approach the constant 2C as jxj ! 1. From the structure of peakons (6) we know that u2 ðx; tÞ may be regarded as the addition
j and a constant 2C, so it can be rewritten as the following form of functions 3C exp j xCt A
u2 ðx; tÞ ¼ v ðx; tÞ 2C;
ð13Þ
where v ðx; tÞ ¼ 3C expðj xCt jÞ, and Eðv Þ and Fðv Þ for v ðx; tÞ are conserved. So let A e ðx; tÞ 2C with u e ðx; tÞ 2 Cð½0; TÞ; H1 ðRÞÞ and Eð u e Þ and F u e Þ are conserved, 2C is the constant in (6)}. X 1 ¼ fuðx; tÞ : uðx; tÞ ¼ u e e Note u0 ¼ uðx; 0Þ and u 0 ¼ u ðx; 0Þ, we have the following results from the definition of orbital stability. e ðx; tÞ 2C is a solution of CH equation (4). For any Theorem 3.1. If uðx; tÞ ¼ u such that if
pffiffiffi AC, there exists a d ¼
with 0 < < 3
4
2187C 3 A3=2
jju0 u2 jjH1 6 d; A
then
jjuð; tÞ u2 ð nðtÞÞjjH1 < for t 2 ð0; TÞ; A
where nðtÞ 2 R is any point where the function uð; tÞ attains its maximum. Remark 3.1. In Theorem 3.1 uðx; tÞ is called a solution to (4) if uðx; tÞ 2 X 1 and uðx; tÞ is a solution of (4) in the sense of distribution. As (13) are solutions of (4), substituting (13) into (4) we get that v ðx; tÞ ¼ 3C expðj xCt jÞ are solutions of equation A
v t þ 2ðk aCÞv x v xxt þ avv x þ 2C v xxx ¼ 2v x v xx þ vv xxx :
ð14Þ
e ðx; tÞ is a solution of Eq. (14) on the assumption that uðx; tÞ ¼ u e ðx; tÞ 2C is a solution to Eq. (4). Since Similarly, u
e 0 v jjH1 ; jju0 u2 jjH1 ¼ jj u A
A
e ð; tÞ v ð nðtÞÞjjH1 jjuð; tÞ u2 ð nðtÞÞjjH1 ¼ jj u A
A
e ð; tÞ attaining its maximum at nðtÞ, the question of stability of u2 ðx; tÞ for Eq. and uð; tÞ attaining its maximum at nðtÞ means u (4) can be reduced to the question of stability of v ðx; tÞ ¼ 3C expðj xCt jÞ for Eq. (14) from the analysis above. A Lemma 3.1. For every u 2 X 1 and n 2 R
e ð; tÞ v ð nÞjj2H1 ¼ Eð u e Þ Eðv Þ 12ACð u e ðn; tÞ 3CÞ: jjuð; tÞ u2 ð nÞjj2H1 ¼ jj u A
A
Proof. By direct calculation we get Eðv Þ ¼ 18AC 2 and
e ð; tÞ v ð nÞjj2H1 ¼ Eð u e Þ þ Eðv Þ 2 jj u
Z
A
e Þ þ Eðv Þ 2 ¼ Eð u
Z
R
R
e ðx; tÞv ðx nÞdx 2A2 u e ðx; tÞv ðx nÞdx 2A2 u
Z R
Z
e x ðx; tÞv x ðx nÞdx u n
1
e x ðx; tÞv x ðx nÞdx 2A2 u
e Þ þ Eðv Þ 12AC u e ðn; tÞ ¼ Eð u e Þ Eðv Þ 12ACð u e ðn; tÞ 3CÞ: ¼ Eð u
Z n
1
e x ðx; tÞv x ðx nÞdx u
pffiffiffi 1 e 2C in X 1 , let M ¼ maxx2R f u e ðx; tÞg. If jju0 u2 jjH1 6 d for some d ¼ d1 AC where d1 < 10 Lemma 3.2. For u ¼ u , then A
pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffi e Þ Eðv Þj 6 dðd þ 6 2ACÞ and jM 3Cj 6 2 3Cd= 4 A: jEð u Proof. (1) Since
Eðv Þ ¼ 18AC 2 ;
e ðx; 0ÞÞ ¼ Eð u e ðx; tÞÞ Eð u
and
e 0 v jjH1 ; jju0 u2 jjH1 ¼ jj u A
A
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Z.-y. Ouyang, S. Zheng / Applied Mathematics and Computation 232 (2014) 183–190
we obtain immediately
e Þ Eðv Þj ¼ jEð u e 0 Þ Eðv Þj ¼ ðjj u e 0 jjH1 jjv jjH1 Þðjj u e 0 jjH1 þ jjv jjH1 Þ 6 jj u e 0 v jjH1 ðjj u e 0 v jjH1 þ 2jjv jjH1 Þ jEð u A A A A A A A pffiffiffiffiffiffi ¼ jju0 u2 jjH1 ðjju0 u2 jjH1 þ 2jjv jjH1 Þ 6 dðd þ 2jjv jjH1 Þ ¼ dðd þ 6 2ACÞ: A
A
A
A
ð15Þ
(2) For any given t 2 ð0; TÞ, let M be taken at x ¼ n, we define
gðx; tÞ ¼
e ðx; tÞ A u e x ðx; tÞ; u e e x ðx; tÞ; u ðx; tÞ þ A u
x < n; x > n:
we have
Z
g 2 ðx; tÞdx ¼
R
¼
Z Z
n 1
R
2
e ðx; tÞ A u e x ðx; tÞÞ dx þ ðu
Z n
1
2
e ðx; tÞ þ A u e x ðx; tÞÞ dx ðu
2 e 2 ðx; tÞ þ A2 u e 2x ðx; tÞÞdx A u e 2 ð; tÞjn1 þ A u e 2 ð; tÞj1 e e2 e ðu n ¼ Eð u Þ 2A u ðn; tÞ ¼ Eð u Þ 2AM
ð16Þ
and
Z R
e ðx; tÞg 2 ðx; tÞdx ¼ u
Z
n
1
e ðx; tÞð u e ðx; tÞ A u e x ðx; tÞÞ2 dx þ u
Z
Z n
1
e ðx; tÞð u e ðx; tÞ þ A u e x ðx; tÞÞ2 dx u
2 3 2 3 4 3 e ðx; tÞð u e 2 ðx; tÞ þ A2 u e 2x ðx; tÞÞdx A u e ð; tÞjn1 þ A u e ð; tÞj1 e e u n ¼ Fð u Þ A u ðn; tÞ 3 3 3 R 4 e Þ AM 3 : ¼ Fð u 3
¼
ð17Þ
From (16) and (17) we get
M3
3 3 e ÞM þ e Þ 6 0: Eð u Fð u 2A 2A
ð18Þ
e ð; tÞ 2 H1 ðRÞ Identity (16) shows that for all u
1 e ðx; tÞj 6 pffiffiffiffiffiffi supx2R j u 2A
qffiffiffiffiffiffiffiffiffiffi 1 e Þ ¼ pffiffiffiffiffiffi jj u e jjH1 : Eð u A 2A
Since
Fðv Þ ¼ 36AC 3
e ðx; 0ÞÞ ¼ Fð u e ðx; tÞÞ; and Fð u
by direct calculation we have
Z Z e 3 e 0 Þ Fðv Þj ¼ ð u e 0 v Þð u e 20 þ A2 u e 20x Þdx þ v ð u e 20 þ A2 u e 20x v 2 A2 v 2x Þdx Fð u Þ 36AC ¼ jFð u R R Z Z 2 2 2 2 e 0 v Þð u e0 þ A u e 0x Þdx þ v ðð u e 0 v Þ þ A2 ð u e 0x v x Þ2 Þdx 6 ðu R R Z 2 e 0 v Þv þ 2A ð u e 0x v x Þv x Þdx þ v ð2ð u R
e 0 v jjL1 Eð u e 0 Þ þ jjv jjL1 jj u e 0 v jj2H1 þ 2jjv jjL1 jj u e 0 v jjH1 jjv jjH1 6 jj u A
A
A
e 0 Þ þ jjv jjL1 jju0 u2 jj2H1 þ 2jjv jjL1 jju0 u2 jjH1 jjv jjH1 ¼ jju0 u2 jjL1 Eð u A
A
A
! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 1 d2 6 pffiffiffiffiffiffi dð18AC 2 þ 6 2ACd þ d2 Þ þ 3Cd2 þ 18 2AC 2 d ¼ d 27 2AC 2 þ 9Cd þ pffiffiffiffiffiffi : 2A 2A e Þ is near 18AC 2 and Fð u e Þ is near 36AC 3 , with Fð u eÞ ¼ From (15) and (19) we know that Eð u have
MP
eÞ 3 Fð u > C: eÞ 2 Eð u
According to (18) consider the cubic polynomial
QðyÞ ¼ y3
3 3 e Þy þ e Þ: Eð u Fð u 2A 2A
e Þ ¼ Eðv Þ ¼ 18AC 2 and Fð u e Þ ¼ Fðv Þ ¼ 36AC 3 , it takes the form In case Eð u
R R
ð19Þ
eðu e 2 þ A2 u e 2x Þdx 6 MEð u e Þ we u
ð20Þ
Z.-y. Ouyang, S. Zheng / Applied Mathematics and Computation 232 (2014) 183–190
189
Q 0 ðyÞ ¼ y3 27C 2 y þ 54C 3 ¼ ðy 3CÞ2 ðy þ 6CÞ: e Þ and Fð u e Þ, there is a root of Q near y ¼ 6C, and there may be two roots of Q near y ¼ 3C. But Because of the estimates on Eð u (18) and (20) show that there must be two roots near y ¼ 3C and M must lie between the two roots. The two roots are closer to y ¼ 3C than the roots of the cubic
Q 1 ðyÞ ¼ y3
pffiffiffiffiffiffi pffiffiffiffiffiffi 3 3 1 yð18AC 2 þ 6 2ACd þ d2 Þ þ 36AC 3 27 2AC 2 d 9Cd2 pffiffiffiffiffiffi d3 ; 2A 2A 2A
pffiffiffi 1 whose graph on Rþ lies below the graph of Q ðyÞ. Notice that, for d ¼ d1 AC; d1 < 10 , we have
Q 1 ð3CÞ ¼ 27C 3 Q 2 ð1Þ < 0 and
pffiffiffiffiffiffiffiffiffi p ffiffiffi pffiffiffiffiffiffiffiffiffiffi 4 Q 1 ð3C 2 3Cd= AÞ ¼ 27C 3 Q 2 ð1 2 d1 =3Þ > 0; where
! pffiffiffi pffiffiffi ! 3 2 2 1 3 4 pffiffiffi 1 2 3 Q 2 ðyÞ ¼ y3 y 2 þ d1 þ d21 þ 2d1 d21 d : 2 9 2 3 3 3 54 1 pffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffi So M lies in the interval ð3C 2 3Cd= 4 A; 3C þ 2 3Cd= 4 AÞ. This completes the proof. h 4
Proof of Theorem 3.1. Let d ¼ 2187C 3 A3=2 , by assumption
e Þ Eðv Þj 6 jEð u
pffiffiffi pffiffiffi 1 AC so that d < 10 AC, then from Lemma 3.2, we obtain
<3
pffiffiffi 2 2 2 2 þ 2 81 81
and
e ðnðtÞ; tÞ 3Cj 6 ju
22 : 27AC
Combining the above inequalities and Lemma 3.1, we find
e Þ Eðv Þ 12ACð u e ðnðtÞ; tÞ 3CÞ 6 jEð u e Þ Eðv Þj þ 12ACjð u e ðnðtÞ; tÞ 3CÞj jjuðx; tÞ u2 ðx nðtÞÞjj2H1 ¼ Eð u A pffiffiffi 2 2 2 2 8 2 2 6 2þ þ < : 9 81 81 That is
jjuðx; tÞ u2 ðx nðtÞÞjjH1 < :
A
4. Conclusion This paper deal with the stability of such peakons as (5) and (6), one of which tend to 0 as jxj ! 1 and the other have nonvanishing boundry. By some transformation and the method in [11] we prove the orbital stability of these peakons. The methods used in this paper could be extended to multipeakons [17,18]. Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 11226303) and Guangdong Province (No. 2013KJCX0189). The authors thank the editors for their hard working and also gratefully acknowledge helpful comments and suggestions by reviewers. References [1] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (11) (1993) 1661–1664. [2] R.S. Johnson, Camassa Holm, Korteweg de Vries and related models for water waves, J. Fluid Mech. 455 (2002) 63–82. [3] A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm and Degasperis–Processi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186. [4] S. Israwi, Variable depth KdV equations and generalizations to more nonlinear regimes, M2AN Math. Model. Numer. Anal. 44 (2010) 347–370. [5] J.F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1) (1996) 1–48. [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006) 523–535. [7] A. Constantin, J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. 44 (2007) 423–431. [8] A. Constantin, L. Molinet, Global weak solutions for a shallow water equation, Commun. Math. Phys. 211 (2000) 45–61. [9] A. Bressan, A. Constantin, Global conservative solutions of the Camassa Holm equation, Arch. Ration. Mech. Anal. 183 (2007) 215–239.
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A. Constantin, J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math. 173 (2011) 559–568. A. Constantin, W. Strauss, Stability of peakons, Commun. Pure Appl. Math. 53 (2000) 603–610. W.L. Zhang, General expressions of peaked travelling wave solutions of CH- and CH equations, Sci. China Ser. A Math. 47 (6) (2004) 862–873. Z.Y. Ouyang, S. Zhen, Z.R. Liu, Orbital stability of peakons with nonvanishing boundary for CH and CH- equations, Phys. Lett. A 372 (2008) 7046–7050. Z.R. Liu, T.F. Qian, Peakons of the Camassa–Holm equation, Appl. Math. Modelling 26 (2002) 473–480. T.B. Benjamin, The stability of solitary waves, Proc. R. Soc. Lond. A 328 (1972) 153–183. T.B. Benjamin, J.B. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. A 272 (1972) 47–78. K. El Dika, L. Molinet, Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys Eng. Sci. 365 (2007) 2313–2331. [18] K. El Dika, L. Molinet, Stability of multipeakons, Ann. I.H. Poincar AN 26 (2009) 1517–1532.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Orbital stability of peakons for a generalized CH equation Zheng-yong Ouyang a,⇑, Shan Zheng b a b
Department of Mathematics, Foshan University, Foshan City, Guangdong Province 528000, PR China Guangzhou Maritime College, Guangzhou City, Guangdong Province 510725, PR China
a r t i c l e
i n f o
Keywords: Peakons Orbital stability Generalized Camassa–Holm equation
a b s t r a c t In this Letter, two different kinds of peaked traveling wave solutions for a generalized Camassa–Holm equation are considered, the orbital stability of these peaked traveling wave solutions are proved. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Camassa and Holm [1] derived a shallow water wave equation
ut þ 2kux uxxt þ 3uux ¼ 2ux uxx þ uuxxx ;
ð1Þ
which is called Camassa–Holm equation (CH equation). For k ¼ 0, they showed that Eq. (1) has peaked solitary wave solutions (peakons)
u1 ðx; tÞ ¼ cejxctj ;
ð2Þ
which have discontinuous first derivative at the wave peak in contrast to the smoothness of most previously known specious of solitary waves and thus are called peakons. Eq. (1) is integrable in the sense of an infinite-dimensional Hamiltonian system and arises as model for shallow water waves [1,2]. A more in-depth discussion of this essential aspect of the solution is performed in[3,4]. The peakons capture a characteristic of the traveling waves of greatest height – exact traveling solutions of the governing equations for water waves with a peak at their crest [5–7]. Simper approximate shallow water models (like the classical Korteweg–de Vries equation) do not present traveling wave solutions with this feature. In the sense of papers [8,9], the peakons are to be understood as weak solutions. As pointed out above, the governing equations for water waves admit peaked and smooth solutions. The contrast between these two classes is very big since the smooth solutions are known to be real-analytic cf. the discussion in [10]. The discovery of this special kind of peakons from CH equation has attracted much attention of mathematicians and physicists, many authors have come to study the CH equation and different generalized forms of CH equation. There are two important aspects on investigation of CH equation and its modified forms, one is under what conditions, the CH equation and its modified forms have solutions of peakons, another is that if the peakons obtained are orbital stable. For instance, Constantin investigated peakons (2) and proved that the peakons are orbital stable [11]. When k – 0 for CH equation, Zhang [12] obtained a more general expression of peakons as follows
uðx; tÞ ¼ ðk þ cÞejxctj k;
⇑ Corresponding author. E-mail address: [email protected] (Z.-y. Ouyang). 0096-3003/$ - see front matter Ó 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2014.01.062
ð3Þ
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which means that peakons exist for CH equation for k takes any values, obviously (2) are special case of (3). The author of this letter considered the orbital stability of such peakons (3) with non-vanishing boundary, and proved peakons (3) are orbital stable in the norm of H1 [13]. Based on the above study, Liu and Qian [14] studied a generalized CH equation
ut þ 2kux uxxt þ auux ¼ 2ux uxx þ uuxxx ;
ð4Þ
when a ¼ 3 in (4), Eq. (4) is the CH equation namely. In case of a – 3, they considered if Eq. (4) has solutions of peakons, they found that when the coefficient a changes, there still exist peakons for Eq. (4), and the results are as follows: (i) In case of a – 3 and k – 0, Eq. (4) has peakons
x Bt ; u1 ðx; tÞ ¼ B exp A
ð5Þ
qffiffi 6k where B ¼ 3a ; A ¼ 3a.
(ii) In case of k – 0, Eq. (4) has peakons
x Ct 2 ; u2 ðx; tÞ ¼ C 3 exp A
ð6Þ
qffiffi 2k where C ¼ 1þa ; A ¼ 3a.
Although solutions (5) and (6) are all peaked solitary waves, there exist some differ- ences between peakons (5) and (6), it is obviously that peakons (5) vanish and peakons (6) tend to an non-zero constant as jxj ! 1. The orbital stability problems of the peakons (5) and (6) have not been solved yet, so it is interesting to consider the stability of the two different kinds of peakons. As already observed by Benjamin and coworkers [15,16], a solitary wave cannot be stable in the strictest sense of the word. To understand this, consider two solitary waves of different height, centered initially at the same point. Science the two waves have different amplitudes they have different velocities according to the expressions (5) and (6). As time passes the two waves will apart, no matter how small the initial difference was. However, in the situation just described, it is evident that two solitary waves with slightly differing height will stay similar in shape during the time evolution. Measuring the difference in shape therefore will give an acceptable notion of stability. This sense of orbital stability was introduced by Benjamin [15]. We say a solitary wave is orbital stable if a solution u of Eq. (4) that is initially sufficiently close to a solitary wave will always stay close to a translation of the solitary wave during the time of evolution. A more mathematically precise definition is as follows. For any > 0, consider the tube
n o U ¼ u 2 H1 : inf jju ss uc jjH1 < ; s
where
ss f ðxÞ ¼ f ðx þ sÞ is a translation of f. The set U is an neighborhood of the collection of all translates of uc .
Definition 1.1. The solitary wave is stable if for all > 0, there exists a d > 0, such that if u0 ¼ uð; 0Þ 2 U d , for all t 2 R. The solitary uc is unstable if uc is not stable. The following two quantities E and F are critically important to the proof of peakons,
EðuÞ ¼
Z
R
ðu2 þ A2 u2x Þdx and FðuÞ ¼
Z
R
ðu3 þ A2 uu2x Þdx;
these are constants of the motion,that is, along solutions these expressions are independent of time. The role of these integrals of motion in stability considerations was pioneered in [11]. 1 R Now we denote jjujj2H1 ¼ R ðu2 þ A2 u2x Þdx. It is obvious that jjujj2H1 6 max 1; A2 jjujj2H1 and jjujj2H1 6 maxf1; A2 gjjujj2H1 . A
A
A
From the relationship between the norm jjujjH1 and jjujjH1 , we only need to prove peakons (5) and (6) are stable on the norm jjujjH1 . A
A
2. Stability of peakons vanishing as jxj ! ‘ In this section we discuss the orbital stability of peakons (5) with vanishing boundary for Eq. (4). We have the following theorem. Theorem 2.1. If u 2 Cð½0; TÞ; H1 ðRÞÞ is a solution to (4) with
jjuð; 0Þ u1 jjH1 < A
4 81B3 A3=2
;
0<<
pffiffiffi AB;
then
jjuð; tÞ u1 ð nðtÞÞjjH1 < for t 2 ð0; TÞ; A
where nðtÞ 2 R is any point where the function uð; tÞ attains its maximum.
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Lemma 2.1. For every u 2 H1 and n 2 R,
jjuð; tÞ u1 ð nÞjj2H1 ¼ EðuÞ Eðu1 Þ 4ABðuðn; tÞ BÞ: A
Proof. By direct calculation we get Eðu1 Þ ¼ 2AB2 and
jjuð; tÞ u1 ð nÞjj2H1 ¼ EðuÞ þ Eðu1 Þ 2 A
¼ EðuÞ þ Eðu1 Þ 2
Z
uðx;tÞu1 ðx nÞdx 2A2
R
Z
Z
ux ðx; tÞu1x ðx nÞdx
R
uðx;tÞu1 ðx nÞdx 2A2
R
Z
n
ux ðx; tÞu1x ðx nÞdx 2A2
1
¼ EðuÞ þ Eðu1 Þ 4ABuðn;tÞ ¼ EðuÞ Eðu1 Þ 4ABðuðn; tÞ BÞ:
Z
1
ux ðx;tÞu1x ðx nÞdx
n
pffiffiffi 1 Lemma 2.2. For u 2 H1 , let M ¼ maxx2R fuðx; tÞg. If jju0 u1 jjH1 6 d for some d ¼ d1 AB where d1 < 20 , then A
pffiffiffiffiffiffi ffiffiffi pffiffiffiffiffiffi p 4 jEðuÞ Eðu1 Þj 6 dð2 2AB þ dÞ and jM Bj 6 2 Bd= A: Proof. (1) Since
Eðu1 Þ ¼ 2AB2 ;
Eðuðx; 0ÞÞ ¼ Eðuðx; tÞÞ;
we obtain immediately
jEðuÞ Eðu1 Þj ¼ jEðu0 Þ Eðu1 Þj ¼ ðjju0 jjH1 jju1 jjH1 Þðjju0 jjH1 þ jju1 jjH1 Þ 6 jju0 u1 jjH1 ðjju0 u1 jjH1 þ 2jju1 jjH1 Þ A A A A A A A pffiffiffiffiffiffi 6 dðd þ 2jju1 jjH1 Þ ¼ dð2 2AB þ dÞ: A
ð7Þ
(2) For any given t 2 ð0; TÞ, let M be taken at x ¼ n, we define
gðx; tÞ ¼
uðx; tÞ Aux ðx; tÞ;
x < n;
uðx; tÞ þ Aux ðx; tÞ;
x > n:
we have
Z
g 2 ðx; tÞdx ¼
R
Z
n
ðuðx; tÞ Aux ðx; tÞÞ2 dx þ
1
¼
Z
R
Z
1
ðuðx; tÞ þ Aux ðx; tÞÞ2 dx
n
2 2 ðu2 ðx; tÞ þ A2 u2x ðx; tÞÞdx Au2 ð; tÞjn1 þ A2 u2 ð; tÞj1 n ¼ EðuÞ 2Au ðn; tÞ ¼ EðuÞ 2AM
ð8Þ
and
Z
uðx; tÞg 2 ðx; tÞdx ¼
R
Z
n
uðx; tÞðuðx; tÞ Aux ðx; tÞÞ2 dx þ
1
Z
Z
1
uðx; tÞðuðx; tÞ þ Aux ðx; tÞÞ2 dx
n
2 2 4 3 uðx; tÞðu2 ðx; tÞ þ A2 u2x ðx; tÞÞdx Au3 ð; tÞjn1 þ Au3 ð; tÞj1 n ¼ FðuÞ Au ðn; tÞ 3 3 3 4 3 ¼ FðuÞ AM : 3
¼
R
ð9Þ
So, from (8) and (9) we get
M3
3 3 EðuÞM þ FðuÞ 6 0: 2A 2A
Identity (8) shows that for all uð; tÞ 2 H1 ðRÞ
1 pffiffiffiffiffiffiffiffiffiffi 1 supx2R juðx; tÞj 6 pffiffiffiffiffiffi EðuÞ ¼ pffiffiffiffiffiffi jjujjH1 : A 2A 2A Since
Fðu1 Þ ¼
4 3 AB 3
and Fðuðx; 0ÞÞ ¼ Fðuðx; tÞÞ;
ð10Þ
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by direct calculation we have
Z Z FðuÞ 4 AB3 ¼ jFðu0 Þ Fðu1 Þj ¼ ðu0 u1 Þðu2 þ A2 u2 Þdx þ u1 ðu2 þ A2 u2 u2 A2 u2 Þdx 0 0x 0 0x 1 1x 3 R R Z Z 6 ðu0 u1 Þðu20 þ A2 u20x Þdx þ u1 ððu0 u1 Þ2 þ A2 ðu0x u1x Þ2 Þdx R R Z 2 þ u1 ð2ðu0 u1 Þu1 þ 2A ðu0x u1x Þu1x Þdx R
6 jju0 u1 jjL1 Eðu0 Þ þ jju1 jjL1 jju0 u1 jj2H1 þ 2jju1 jjL1 jju0 u1 jjH1 jju1 jjH1 A
A
A
! pffiffiffiffiffiffi pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 2 1 d2 2 2 2 6 pffiffiffiffiffiffi dð2AB þ 2 2ABd þ d Þ þ Bd þ 2 2AB d ¼ d 3 2AB þ 3Bd þ pffiffiffiffiffiffi : 2A 2A
From (7) and (11) we know that EðuÞ is near 2AB2 and FðuÞ is near 43 AB3 , with FðuÞ ¼
MP
R R
ð11Þ
uðu2 þ u2x Þdx 6 MEðuÞ we have
FðuÞ 1 > B: EðuÞ 2
ð12Þ
According to (10) consider the cubic polynomial
PðyÞ ¼ y3
3 3 EðuÞy þ FðuÞ: 2A 2A
In case EðuÞ ¼ Eðu1 Þ ¼ 2AB2 and FðuÞ ¼ Fðu1 Þ ¼ 43 AB3 , PðyÞ takes the form
P0 ðyÞ ¼ y3 3B2 y þ 2B3 ¼ ðy BÞ2 ðy þ 2BÞ: Because of the estimates on EðuÞ and FðuÞ, there is a root of P near y ¼ 2B, and there may be two roots of P near y ¼ B. But (10) and (12) show that there must be two roots near y ¼ B and M must lie between the two roots. The two roots are closer to y ¼ B than the roots of the cubic
P1 ðyÞ ¼ y3
pffiffiffiffiffiffi pffiffiffiffiffiffi 3 3 4 3 1 yð2AB2 þ 2 2ABd þ d2 Þ þ AB 3 2AB2 d 3Bd2 pffiffiffiffiffiffi d3 ; 2A 2A 3 2A
pffiffiffi 1 whose graph on Rþ lies below the graph of PðyÞ. Notice that, for d ¼ d1 AB; d1 < 20 , we have
P1 ðBÞ ¼ B3 P2 ð1Þ < 0 and
pffiffiffiffiffi ffiffiffi pffiffiffiffiffiffi p 4 P1 ðB 2 Bd= AÞ ¼ B3 P2 ð1 2 d1 Þ > 0; where
pffiffiffi pffiffiffi 3 3 4 1 P2 ðyÞ ¼ y3 yð2 þ 2 2d1 þ d21 Þ þ 3 2d1 3d21 pffiffiffi d31 : 2 2 3 2
pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi So M lies in the interval ðB 2 Bd= 4 A; B þ 2 Bd= 4 AÞ. This completes the proof. 4
Proof of Theorem 2.1. Let d ¼ 81B3 A3=2 , by assumption
jEðuÞ Eðu1 Þj 6
pffiffiffi 2 2 2 2 þ 2 81 81
juðnðtÞ; tÞ Bj 6
22 : 9AB
<
h
pffiffiffi pffiffiffi 1 AB so that d < 20 B A, then from Lemma 2.2, we obtain
and
Combining the above inequalities and Lemma 2.1, we find
jjuðx; tÞ u1 ðx nðtÞÞjj2H1 ¼ EðuÞ Eðu1 Þ 4ABðuðnðtÞ; tÞ BÞ 6 jEðuÞ Eðu1 Þj þ 4jABðuðnðtÞ; tÞ BÞj A pffiffiffi 2 2 2 2 8 2 2 6 2þ þ < : 9 81 81 That is
jjuðx; tÞ u1 ðx nðtÞÞjjH1 < : A
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3. Stability of peakons non-vanishing as jxj ! ‘ In this section we discuss the stability of peakons (6) with non-vanishing boundary for Eq. (4).The solutions u2 ðx; tÞ approach the constant 2C as jxj ! 1. From the structure of peakons (6) we know that u2 ðx; tÞ may be regarded as the addition
j and a constant 2C, so it can be rewritten as the following form of functions 3C exp j xCt A
u2 ðx; tÞ ¼ v ðx; tÞ 2C;
ð13Þ
where v ðx; tÞ ¼ 3C expðj xCt jÞ, and Eðv Þ and Fðv Þ for v ðx; tÞ are conserved. So let A e ðx; tÞ 2C with u e ðx; tÞ 2 Cð½0; TÞ; H1 ðRÞÞ and Eð u e Þ and F u e Þ are conserved, 2C is the constant in (6)}. X 1 ¼ fuðx; tÞ : uðx; tÞ ¼ u e e Note u0 ¼ uðx; 0Þ and u 0 ¼ u ðx; 0Þ, we have the following results from the definition of orbital stability. e ðx; tÞ 2C is a solution of CH equation (4). For any Theorem 3.1. If uðx; tÞ ¼ u such that if
pffiffiffi AC, there exists a d ¼
with 0 < < 3
4
2187C 3 A3=2
jju0 u2 jjH1 6 d; A
then
jjuð; tÞ u2 ð nðtÞÞjjH1 < for t 2 ð0; TÞ; A
where nðtÞ 2 R is any point where the function uð; tÞ attains its maximum. Remark 3.1. In Theorem 3.1 uðx; tÞ is called a solution to (4) if uðx; tÞ 2 X 1 and uðx; tÞ is a solution of (4) in the sense of distribution. As (13) are solutions of (4), substituting (13) into (4) we get that v ðx; tÞ ¼ 3C expðj xCt jÞ are solutions of equation A
v t þ 2ðk aCÞv x v xxt þ avv x þ 2C v xxx ¼ 2v x v xx þ vv xxx :
ð14Þ
e ðx; tÞ is a solution of Eq. (14) on the assumption that uðx; tÞ ¼ u e ðx; tÞ 2C is a solution to Eq. (4). Since Similarly, u
e 0 v jjH1 ; jju0 u2 jjH1 ¼ jj u A
A
e ð; tÞ v ð nðtÞÞjjH1 jjuð; tÞ u2 ð nðtÞÞjjH1 ¼ jj u A
A
e ð; tÞ attaining its maximum at nðtÞ, the question of stability of u2 ðx; tÞ for Eq. and uð; tÞ attaining its maximum at nðtÞ means u (4) can be reduced to the question of stability of v ðx; tÞ ¼ 3C expðj xCt jÞ for Eq. (14) from the analysis above. A Lemma 3.1. For every u 2 X 1 and n 2 R
e ð; tÞ v ð nÞjj2H1 ¼ Eð u e Þ Eðv Þ 12ACð u e ðn; tÞ 3CÞ: jjuð; tÞ u2 ð nÞjj2H1 ¼ jj u A
A
Proof. By direct calculation we get Eðv Þ ¼ 18AC 2 and
e ð; tÞ v ð nÞjj2H1 ¼ Eð u e Þ þ Eðv Þ 2 jj u
Z
A
e Þ þ Eðv Þ 2 ¼ Eð u
Z
R
R
e ðx; tÞv ðx nÞdx 2A2 u e ðx; tÞv ðx nÞdx 2A2 u
Z R
Z
e x ðx; tÞv x ðx nÞdx u n
1
e x ðx; tÞv x ðx nÞdx 2A2 u
e Þ þ Eðv Þ 12AC u e ðn; tÞ ¼ Eð u e Þ Eðv Þ 12ACð u e ðn; tÞ 3CÞ: ¼ Eð u
Z n
1
e x ðx; tÞv x ðx nÞdx u
pffiffiffi 1 e 2C in X 1 , let M ¼ maxx2R f u e ðx; tÞg. If jju0 u2 jjH1 6 d for some d ¼ d1 AC where d1 < 10 Lemma 3.2. For u ¼ u , then A
pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffi e Þ Eðv Þj 6 dðd þ 6 2ACÞ and jM 3Cj 6 2 3Cd= 4 A: jEð u Proof. (1) Since
Eðv Þ ¼ 18AC 2 ;
e ðx; 0ÞÞ ¼ Eð u e ðx; tÞÞ Eð u
and
e 0 v jjH1 ; jju0 u2 jjH1 ¼ jj u A
A
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Z.-y. Ouyang, S. Zheng / Applied Mathematics and Computation 232 (2014) 183–190
we obtain immediately
e Þ Eðv Þj ¼ jEð u e 0 Þ Eðv Þj ¼ ðjj u e 0 jjH1 jjv jjH1 Þðjj u e 0 jjH1 þ jjv jjH1 Þ 6 jj u e 0 v jjH1 ðjj u e 0 v jjH1 þ 2jjv jjH1 Þ jEð u A A A A A A A pffiffiffiffiffiffi ¼ jju0 u2 jjH1 ðjju0 u2 jjH1 þ 2jjv jjH1 Þ 6 dðd þ 2jjv jjH1 Þ ¼ dðd þ 6 2ACÞ: A
A
A
A
ð15Þ
(2) For any given t 2 ð0; TÞ, let M be taken at x ¼ n, we define
gðx; tÞ ¼
e ðx; tÞ A u e x ðx; tÞ; u e e x ðx; tÞ; u ðx; tÞ þ A u
x < n; x > n:
we have
Z
g 2 ðx; tÞdx ¼
R
¼
Z Z
n 1
R
2
e ðx; tÞ A u e x ðx; tÞÞ dx þ ðu
Z n
1
2
e ðx; tÞ þ A u e x ðx; tÞÞ dx ðu
2 e 2 ðx; tÞ þ A2 u e 2x ðx; tÞÞdx A u e 2 ð; tÞjn1 þ A u e 2 ð; tÞj1 e e2 e ðu n ¼ Eð u Þ 2A u ðn; tÞ ¼ Eð u Þ 2AM
ð16Þ
and
Z R
e ðx; tÞg 2 ðx; tÞdx ¼ u
Z
n
1
e ðx; tÞð u e ðx; tÞ A u e x ðx; tÞÞ2 dx þ u
Z
Z n
1
e ðx; tÞð u e ðx; tÞ þ A u e x ðx; tÞÞ2 dx u
2 3 2 3 4 3 e ðx; tÞð u e 2 ðx; tÞ þ A2 u e 2x ðx; tÞÞdx A u e ð; tÞjn1 þ A u e ð; tÞj1 e e u n ¼ Fð u Þ A u ðn; tÞ 3 3 3 R 4 e Þ AM 3 : ¼ Fð u 3
¼
ð17Þ
From (16) and (17) we get
M3
3 3 e ÞM þ e Þ 6 0: Eð u Fð u 2A 2A
ð18Þ
e ð; tÞ 2 H1 ðRÞ Identity (16) shows that for all u
1 e ðx; tÞj 6 pffiffiffiffiffiffi supx2R j u 2A
qffiffiffiffiffiffiffiffiffiffi 1 e Þ ¼ pffiffiffiffiffiffi jj u e jjH1 : Eð u A 2A
Since
Fðv Þ ¼ 36AC 3
e ðx; 0ÞÞ ¼ Fð u e ðx; tÞÞ; and Fð u
by direct calculation we have
Z Z e 3 e 0 Þ Fðv Þj ¼ ð u e 0 v Þð u e 20 þ A2 u e 20x Þdx þ v ð u e 20 þ A2 u e 20x v 2 A2 v 2x Þdx Fð u Þ 36AC ¼ jFð u R R Z Z 2 2 2 2 e 0 v Þð u e0 þ A u e 0x Þdx þ v ðð u e 0 v Þ þ A2 ð u e 0x v x Þ2 Þdx 6 ðu R R Z 2 e 0 v Þv þ 2A ð u e 0x v x Þv x Þdx þ v ð2ð u R
e 0 v jjL1 Eð u e 0 Þ þ jjv jjL1 jj u e 0 v jj2H1 þ 2jjv jjL1 jj u e 0 v jjH1 jjv jjH1 6 jj u A
A
A
e 0 Þ þ jjv jjL1 jju0 u2 jj2H1 þ 2jjv jjL1 jju0 u2 jjH1 jjv jjH1 ¼ jju0 u2 jjL1 Eð u A
A
A
! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 1 d2 6 pffiffiffiffiffiffi dð18AC 2 þ 6 2ACd þ d2 Þ þ 3Cd2 þ 18 2AC 2 d ¼ d 27 2AC 2 þ 9Cd þ pffiffiffiffiffiffi : 2A 2A e Þ is near 18AC 2 and Fð u e Þ is near 36AC 3 , with Fð u eÞ ¼ From (15) and (19) we know that Eð u have
MP
eÞ 3 Fð u > C: eÞ 2 Eð u
According to (18) consider the cubic polynomial
QðyÞ ¼ y3
3 3 e Þy þ e Þ: Eð u Fð u 2A 2A
e Þ ¼ Eðv Þ ¼ 18AC 2 and Fð u e Þ ¼ Fðv Þ ¼ 36AC 3 , it takes the form In case Eð u
R R
ð19Þ
eðu e 2 þ A2 u e 2x Þdx 6 MEð u e Þ we u
ð20Þ
Z.-y. Ouyang, S. Zheng / Applied Mathematics and Computation 232 (2014) 183–190
189
Q 0 ðyÞ ¼ y3 27C 2 y þ 54C 3 ¼ ðy 3CÞ2 ðy þ 6CÞ: e Þ and Fð u e Þ, there is a root of Q near y ¼ 6C, and there may be two roots of Q near y ¼ 3C. But Because of the estimates on Eð u (18) and (20) show that there must be two roots near y ¼ 3C and M must lie between the two roots. The two roots are closer to y ¼ 3C than the roots of the cubic
Q 1 ðyÞ ¼ y3
pffiffiffiffiffiffi pffiffiffiffiffiffi 3 3 1 yð18AC 2 þ 6 2ACd þ d2 Þ þ 36AC 3 27 2AC 2 d 9Cd2 pffiffiffiffiffiffi d3 ; 2A 2A 2A
pffiffiffi 1 whose graph on Rþ lies below the graph of Q ðyÞ. Notice that, for d ¼ d1 AC; d1 < 10 , we have
Q 1 ð3CÞ ¼ 27C 3 Q 2 ð1Þ < 0 and
pffiffiffiffiffiffiffiffiffi p ffiffiffi pffiffiffiffiffiffiffiffiffiffi 4 Q 1 ð3C 2 3Cd= AÞ ¼ 27C 3 Q 2 ð1 2 d1 =3Þ > 0; where
! pffiffiffi pffiffiffi ! 3 2 2 1 3 4 pffiffiffi 1 2 3 Q 2 ðyÞ ¼ y3 y 2 þ d1 þ d21 þ 2d1 d21 d : 2 9 2 3 3 3 54 1 pffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffi So M lies in the interval ð3C 2 3Cd= 4 A; 3C þ 2 3Cd= 4 AÞ. This completes the proof. h 4
Proof of Theorem 3.1. Let d ¼ 2187C 3 A3=2 , by assumption
e Þ Eðv Þj 6 jEð u
pffiffiffi pffiffiffi 1 AC so that d < 10 AC, then from Lemma 3.2, we obtain
<3
pffiffiffi 2 2 2 2 þ 2 81 81
and
e ðnðtÞ; tÞ 3Cj 6 ju
22 : 27AC
Combining the above inequalities and Lemma 3.1, we find
e Þ Eðv Þ 12ACð u e ðnðtÞ; tÞ 3CÞ 6 jEð u e Þ Eðv Þj þ 12ACjð u e ðnðtÞ; tÞ 3CÞj jjuðx; tÞ u2 ðx nðtÞÞjj2H1 ¼ Eð u A pffiffiffi 2 2 2 2 8 2 2 6 2þ þ < : 9 81 81 That is
jjuðx; tÞ u2 ðx nðtÞÞjjH1 < :
A
4. Conclusion This paper deal with the stability of such peakons as (5) and (6), one of which tend to 0 as jxj ! 1 and the other have nonvanishing boundry. By some transformation and the method in [11] we prove the orbital stability of these peakons. The methods used in this paper could be extended to multipeakons [17,18]. Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 11226303) and Guangdong Province (No. 2013KJCX0189). The authors thank the editors for their hard working and also gratefully acknowledge helpful comments and suggestions by reviewers. References [1] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (11) (1993) 1661–1664. [2] R.S. Johnson, Camassa Holm, Korteweg de Vries and related models for water waves, J. Fluid Mech. 455 (2002) 63–82. [3] A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm and Degasperis–Processi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186. [4] S. Israwi, Variable depth KdV equations and generalizations to more nonlinear regimes, M2AN Math. Model. Numer. Anal. 44 (2010) 347–370. [5] J.F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1) (1996) 1–48. [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006) 523–535. [7] A. Constantin, J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. 44 (2007) 423–431. [8] A. Constantin, L. Molinet, Global weak solutions for a shallow water equation, Commun. Math. Phys. 211 (2000) 45–61. [9] A. Bressan, A. Constantin, Global conservative solutions of the Camassa Holm equation, Arch. Ration. Mech. Anal. 183 (2007) 215–239.
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