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Peakons of the Camassa–Holm equation
Applied Mathematical Modelling 26 (2002) 473–480 www.elsevier.com/locate/apm
Peakons of the Camassa–Holm equation Zhengrong Liu a
a,1
, Tifei Qian
b,*
Department of Mathematics, Yunnan University and Institute of Applied Mathematics of Yunnan Province, Kunming, Yunnan 650091, China b Allstate Insurance Company, AEI4-M30, 1500 W Shure Dr, Arlington Heights, IL 60004, USA Received 13 October 2000; received in revised form 1 August 2001; accepted 8 October 2001
Abstract From the mathematical point of view, we use the bifurcation method of phase plane to study the CH equation given by Camassa and Holm [Phys. Rev. Lett. 71(11) (1993) 1661]. In case of k ¼ 0, they showed that it has solitary waves of the form cejxctj , which were called ‘‘peakons’’. When k 6¼ 0, we obtain the peakons of the form ð3k=2Þ expðx ðkt=2ÞÞ k in three different ways, which are also applicable to the case of k ¼ 0. Note that our approaches can also be used to study the traveling waves in various nonlinear integrable equations. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Peakon; Camassa–Holm equation; Hamiltonian system; Phase portrait bifurcation
1. Introduction Camassa and Holm [3] recently gave a dispersive shallow water equation ut þ 2kux uxxt þ 3uux ¼ 2ux uxx þ uuxxx ;
ð1Þ
which is called the CH equation. In the case of k ¼ 0, they showed that the solitary wave solution of Eq. (1) is uðx; tÞ ¼ cejxctj ;
ð2Þ
which were named ‘‘peakons’’ to single them out of general solitary wave solutions since they have a corner at the peak of height c. They [4] presented numerical solutions of the time-dependent *
Corresponding author. Address: Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA. E-mail addresses: [email protected] (Z. Liu), [email protected] (T. Qian). 1 Partially supported by the National Natural Science Foundation of China. 0307-904X/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 1 ) 0 0 0 8 6 - 5
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form and a discussion of the CH equation as a time-dependent Hamiltonian system. Cooper and Shepard [7] derived an approximate solitary wave solution to the CH equation using some variational functions. An exact solution was given for k ¼ 0, and the optimal variational solution was obtained for k 6¼ 0. In Ref. [1] the CH equation was studied as a complex integrable Hamiltonian system on a Riemann surface and new solutions were obtained. For a general k, Boyd [2] used a perturbation series, which converges even at the peakon limit, to give three analytical representations of the spatially periodic generalization of the peakons, the so-called ‘‘coshoidal waves’’. Constantin [5,6] gave a mathematical description of the existence of interacting solitary waves, which preserve their forms through collisions for k ¼ 0. Here, if we let u ¼ /ðnÞ with n ¼ x ct as in Ref. [1], then Eq. (1) becomes c/0 þ 2k/0 þ c/000 þ 3//0 ¼ 2/0 /00 þ //000 :
ð3Þ
Integrating it once we get 3 1 ð/ cÞ/00 ¼ /2 þ ð2k cÞ/ ð/0 Þ2 : 2 2 0 Let y ¼ / , we obtain a planar integrable system ( d/=dn ¼ y; 2 2 ; dy=dn ¼ 3=2/ þð2kcÞ/1=2y /c
ð4Þ
ð5Þ
with the first integral H ð/; yÞ ¼ ð/ cÞy 2 /3 þ ðc 2kÞ/2 ¼ h:
ð6Þ
When only bounded traveling waves are considered, it is well known that the periodic and homoclinic orbits of Eq. (5) respectively correspond to the periodic and single solitary waves of Eq. (1). We call the single solitary wave the general solitary wave. After the transformation dn ¼ ð/ cÞds, Eq. (5) becomes a Hamiltonian system d/=ds ¼ ð/ cÞy; ð7Þ dy=ds ¼ 3=2/2 þ ð2k cÞ/ 1=2y 2 : Since the first integral (6) of system (5) is the same as the Hamiltonian of Eq. (7), Eq. (5) has the same topological phase portrait as Eq. (7) except the singular line / ¼ c. Therefore we should be able to obtain the topological properties of Eq. (5) from the phase portrait of Eq. (7). From the properties of the phase portraits of Eq. (5), we conjecture that the CH equation (1) has a peakon when c ¼ ðk=2Þ, k 6¼ 0. Then we show the conjecture is true. In fact, we obtain the peakon in two different ways, i.e., from the limit of general single solitary wave solutions and from the limit of periodic cusp wave solutions respectively. Finally we give a simpler way to obtain the peakon. Of course, all our three approaches to obtain the peakon are applicable for the case of k ¼ 0.
2. Bifurcation of the phase portraits of Eq. (7) Using the bifurcation methods of dynamical systems [8], we analyze the equilibrium points and bifurcation curves of Eq. (7).
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475
It is obvious that Eq. (7) has two equilibrium points O(0,0) and Að2ðc2kÞ ; 0Þ on the /-axis when pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 c 6¼ 2k, and two equilibrium points B ðc; cðc þ 4kÞÞ on the line / ¼ c when cðc þ 4kÞ > 0. Let /0 denote either of the two zeroes of f ð/Þ ¼ 32/2 þ ð2k cÞ/; then the eigenvalue of the linearized system of Eq. (7) at ð/0 ; 0Þ (either O or A) are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1;2 ¼ ð/0 cÞf 0 ð/0 Þ;
ð8Þ ð9Þ
while at B
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k3;4 ¼ cðc þ 4kÞ:
ð10Þ
At O, A, and B the Hamiltonians Hð/; yÞ respectively are h0 ¼ H ðOÞ ¼ 0; h1 ¼ H ðAÞ ¼
4ðc 2kÞ3 ; 27
ð11Þ ð12Þ
and h2 ¼ H ðBþ Þ ¼ H ðB Þ ¼ 2kc2 :
ð13Þ
Therefore, Eq. (7) has only one equilibrium point O(0,0) on /-axis when c ¼ 2k, and no equilibrium point on / ¼ c when cðc þ 4kÞ < 0. When c ¼ 4k, A and B coincide on the line / ¼ c. When k ¼ 0 or c ¼ 0 we have h2 ¼ 0. And h1 ¼ h2 when c ¼ k=2. Hence on the ðk; cÞ-plane we have five bifurcation curves, namely c ¼ 0, c ¼ k=2, c ¼ 2k, k ¼ 0, and c ¼ 4k, which divide the parameter plane into 10 separate regions. Further analysis yields the bifurcation of phase portraits of Eq. (7), which is shown in Fig. 1 for c P 0. Note that when c < 0, the bifurcation of phase portraits is symmetric on the ðk; cÞ-plane with respect to the origin. It was shown in Ref. [3] that peakons exist for the CH equation in the case of k ¼ 0. Comparing the other phase portraits with the one when k ¼ 0 in Fig. 1, we see the heteroclinic connections of the points O with Bþ , Bþ with B and B with O for k ¼ 0, and A with Bþ , Bþ with B and B with A for c ¼ k=2. We conjecture that such a heteroclinic connection in the phase portrait of Eq. (7) represents a peakon for the CH equation (1). Thus Eq. (1) has a peakon when c ¼ k=2 for k 6¼ 0. This turns out to be the case as illustrated in the following discussion. 3. Peakons from the limit of general single solitary waves Without loss of generality, we consider the case of c > 0. When ðk; cÞ 2 ðBÞ in Fig. 1, system (7) has a homoclinic orbit at Að2ðc 2kÞ=3; 0Þ around the origin O(0,0). Since system (5) has the same topological phase portraits as system (7) except the line / ¼ c, system (5) has a homoclinic orbit at Að2ðc 2kÞ=3; 0Þ around the origin O(0,0). Since a homoclinic orbit of Eq. (5) corresponds to a general single solitary wave of Eq. (1), we suspect that when c ! k=2, the limit of the general single solitary wave solutions becomes the peakon.
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Fig. 1. Bifurcation of phase portraits of Eq. (7) for c P 0.
By Eqs. (6) and (12), the homoclinic orbit can be represented as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3/ þ 4k 2cÞ 3/ þ c 2k pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi : y¼ 3 3 /c
ð14Þ
Substituting Eq. (14) into the first equation of system (5) and integrating along the homoclinic orbit, we obtain pffiffiffiffiffiffiffiffiffiffi Z 2kc Z 0 3 1 sc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ pffiffiffi ds for n < 0; ð15Þ ð3s þ 4k 2cÞ 3s þ c 2k 3 3 n / and Z
/ 2kc 3
pffiffiffiffiffiffiffiffiffiffi Z n sc 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ pffiffiffi ds for n P 0: ð3s þ 4k 2cÞ 3s þ c 2k 3 3 0
ð16Þ
From Eqs. (15) and (16) we then have w2 / c ð6/ þ 8k 4cÞa ð2c kÞa ¼ ejnj ; a a 2a ð/ cÞ 2ðc þ 4kÞ 2c k X
ð17Þ
Z. Liu, T. Qian / Appl. Math. Modelling 26 (2002) 473–480
477
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c þ 4k ; a¼ 3ð2k cÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3/ þ c 2k pffiffiffi w¼ þ 3; /c
ð18Þ
ð19Þ
and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3/ þ c 2k 9ð2k cÞ X¼ þ : /c c þ 4k
ð20Þ
Obviously, lim a ¼ 1;
c!k=2
ð21Þ
lim
w2 ¼ 1; X2a
ð22Þ
lim
ð6/ þ 8k 4cÞa 2ð/ þ kÞ ¼ ; 3k 2ðc þ 4kÞa
ð23Þ
lim
ð2c kÞa ¼ expð lim lnð2c kÞÞ ¼ 1: c!k=2ða1Þ 2c k
ð24Þ
c!k=2
c!k=2
and c!k=2
Therefore letting c ! k=2 in Eq. (17) we get 3k jnj e k; 2 which implies that for k > 0 the CH equation (1) has a peakon /ðnÞ ¼
ð25Þ
3k jxktj ð26Þ e 2 k: 2 Hence we obtain the peakon from the limit of general single solitary wave solutions. Using the same approach, one proves that Eq. (26) also holds for k < 0, and one can obtain the peakon (2) for k ¼ 0. uðx; t; kÞ ¼
4. Peakons from the limit of periodic cusp waves From Eqs. (10) and (12), and Fig. 1 we know that when ðk; cÞ 2 ðAÞ or (D) and h 2 ð2kc2 ; 0Þ, system (5) has a family of periodic orbits. When h ! 2kc2 , the periodic orbits lose smooth property and become periodic cusp orbits [1].
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Let Ckc be the limiting curve of the periodic orbits when h ! 2kc2 . By Eqs. (6) and (13) Ckc consists of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 2 ¼ /2 þ 2k/ þ 2kc for kðk 2cÞ k 6 / < c; and / ¼ c;
for jyj 6
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðc þ 4kÞ:
ð27Þ ð28Þ
Substituting (27) into ðd/=dnÞ ¼ y and integrating along Ckc we have Z c Z 0 ds pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ds for n < 0; s2 þ 2ks þ 2kc / n
ð29Þ
and Z
/ c
ds pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 s þ 2ks þ 2kc
Z
n
ds
for n P 0:
ð30Þ
0
From Eqs. (29) and (30) we obtain the periodic cusp wave solutions uðn; k; cÞ ¼ u0 ðn 2nT ; c; kÞ;
ð31Þ
where n ¼ 0; 1; 2; . . . ; n ¼ x ct 2 ðð2n 1ÞT ; ð2n þ 1ÞT Þ, and a0 kðk 2cÞ jnj u0 ðn; k; cÞ ¼ ejnj þ e k; 2a0 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 ¼ cðc þ 4kÞ þ c þ k; and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 þ 4kc þ c þ k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ ln : kðk 2cÞ
ð32Þ ð33Þ
ð34Þ
Similarly, for either k > 0 and c < 4k, or k < 0 and k=2 < c < 0, Eqs. (31)–(34) also hold. Obviously T ! 1 when either k ! 0 or c ! k=2. Thus we have lim uðx; t; k; cÞ ¼ lim u0 ðn; k; cÞ ¼ cejxctj ; k!0
k!0
ð35Þ
and lim uðx; t; k; cÞ ¼ lim u0 ðn; k; cÞ ¼
c!k=2
c!k=2
3k expðjx ðkt=2ÞjÞ k: 2
ð36Þ
Hence we have obtained the peakons (2) and (26) again from the limit of the periodic cusp waves when k ! 0 and k ! k=2, respectively. 5. Peakons obtained in a simpler way As illustrated in the last two sections, one can obtain peakons from the limit of general single solitary waves and the limit of periodic cusp waves. In this section, we will give a simpler way to obtain the peakon.
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479
Let Ck be the limiting curve of the homoclinic orbits of Eq. (5) when c ! k=2 with k 6¼ 0. From Eqs. (6), (12) and (13) we know that Ck consists of three line segments k y ¼ ð/ þ kÞ for k 6 / < ; 2
ð37Þ
3k for jyj 6 : 2
ð38Þ
and k /¼ ; 2
Substituting Eq. (37) into the first equation of (5) and integrating along Ck , we have Z k2 Z 0 ds ¼ ds for n < 0; / sþk n
ð39Þ
and Z
/
k 2
ds ¼ s k
Z
n
ds for n P 0:
ð40Þ
0
From Eqs. (39) and (40) we immediately obtain the peakon (26). When k ! 0, let C0 be the limiting curve of the homoclinic orbits of Eq. (5). By Eqs. (6), (11) and (13), C0 consists of y ¼ / for 0 6 / < c;
ð41Þ
/ ¼ c;
ð42Þ
and for jyj 6 jcj:
Similarly we have Z 0 Z c ds ds; ¼ s / n
for n < 0;
ð43Þ
and Z c
/
ds ¼ s
Z
n
ds;
for n P 0;
ð44Þ
0
which immediately yield the peakon (2).
6. Conclusions From the above analysis we conclude that 1. The CH equation (1) has peakons if either the parameter k ¼ 0, or the wave speed c ¼ k=2 and k 6¼ 0. 2. From the mathematical point of view, the corner at the peak of a peakon was caused by the singular line / ¼ c in Eq. (5).
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3. The peakon comes from the limit of general single solitary waves or periodic cusp waves as shown in Sections 3 and 4. 4. Peakons of the CH equation (1) are represented by heteroclinic connections in the phase portrait of system (7) as illustrated in Section 5.
Acknowledgement Liu would like to thank Prof. Jeff Xia for his kind invitation to visit the Department of Mathematics of Northwestern University, where this paper was completed. Qian is grateful to Prof. Xia for his kind support.
References [1] M.S. Alber, R. Camassa, D.D. Holm, J.E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDEs, Lett. Math. Phys. 32 (2) (1994) 137–151. [2] J.P. Boyd, Peakons and coshoidal waves: travelling wave solutions of the Camassa–Holm equation, Appl. Math. Compt. 81 (2–3) (1997) 173–187. [3] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (11) (1993) 1661–1664. [4] R. Camassa, D.D. Holm, J.M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994) 1–33. [5] A. Constantin, The Hamiltonian structure of the Camassa–Holm equation, Exposition. Math. 15 (1) (1997) 53–85. [6] A. Constantin, Soliton interactions for the Camassa–Holm equation, Exposition. Math. 15 (3) (1997) 251–264. [7] F. Cooper, H. Shepard, Solitons in the Camassa–Holm shallow water equation, Phys. Lett. A 194 (4) (1994) 246–250. [8] J. Guckenheimer, P. Holmes, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.
Peakons of the Camassa–Holm equation Zhengrong Liu a
a,1
, Tifei Qian
b,*
Department of Mathematics, Yunnan University and Institute of Applied Mathematics of Yunnan Province, Kunming, Yunnan 650091, China b Allstate Insurance Company, AEI4-M30, 1500 W Shure Dr, Arlington Heights, IL 60004, USA Received 13 October 2000; received in revised form 1 August 2001; accepted 8 October 2001
Abstract From the mathematical point of view, we use the bifurcation method of phase plane to study the CH equation given by Camassa and Holm [Phys. Rev. Lett. 71(11) (1993) 1661]. In case of k ¼ 0, they showed that it has solitary waves of the form cejxctj , which were called ‘‘peakons’’. When k 6¼ 0, we obtain the peakons of the form ð3k=2Þ expðx ðkt=2ÞÞ k in three different ways, which are also applicable to the case of k ¼ 0. Note that our approaches can also be used to study the traveling waves in various nonlinear integrable equations. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Peakon; Camassa–Holm equation; Hamiltonian system; Phase portrait bifurcation
1. Introduction Camassa and Holm [3] recently gave a dispersive shallow water equation ut þ 2kux uxxt þ 3uux ¼ 2ux uxx þ uuxxx ;
ð1Þ
which is called the CH equation. In the case of k ¼ 0, they showed that the solitary wave solution of Eq. (1) is uðx; tÞ ¼ cejxctj ;
ð2Þ
which were named ‘‘peakons’’ to single them out of general solitary wave solutions since they have a corner at the peak of height c. They [4] presented numerical solutions of the time-dependent *
Corresponding author. Address: Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA. E-mail addresses: [email protected] (Z. Liu), [email protected] (T. Qian). 1 Partially supported by the National Natural Science Foundation of China. 0307-904X/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 1 ) 0 0 0 8 6 - 5
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Z. Liu, T. Qian / Appl. Math. Modelling 26 (2002) 473–480
form and a discussion of the CH equation as a time-dependent Hamiltonian system. Cooper and Shepard [7] derived an approximate solitary wave solution to the CH equation using some variational functions. An exact solution was given for k ¼ 0, and the optimal variational solution was obtained for k 6¼ 0. In Ref. [1] the CH equation was studied as a complex integrable Hamiltonian system on a Riemann surface and new solutions were obtained. For a general k, Boyd [2] used a perturbation series, which converges even at the peakon limit, to give three analytical representations of the spatially periodic generalization of the peakons, the so-called ‘‘coshoidal waves’’. Constantin [5,6] gave a mathematical description of the existence of interacting solitary waves, which preserve their forms through collisions for k ¼ 0. Here, if we let u ¼ /ðnÞ with n ¼ x ct as in Ref. [1], then Eq. (1) becomes c/0 þ 2k/0 þ c/000 þ 3//0 ¼ 2/0 /00 þ //000 :
ð3Þ
Integrating it once we get 3 1 ð/ cÞ/00 ¼ /2 þ ð2k cÞ/ ð/0 Þ2 : 2 2 0 Let y ¼ / , we obtain a planar integrable system ( d/=dn ¼ y; 2 2 ; dy=dn ¼ 3=2/ þð2kcÞ/1=2y /c
ð4Þ
ð5Þ
with the first integral H ð/; yÞ ¼ ð/ cÞy 2 /3 þ ðc 2kÞ/2 ¼ h:
ð6Þ
When only bounded traveling waves are considered, it is well known that the periodic and homoclinic orbits of Eq. (5) respectively correspond to the periodic and single solitary waves of Eq. (1). We call the single solitary wave the general solitary wave. After the transformation dn ¼ ð/ cÞds, Eq. (5) becomes a Hamiltonian system d/=ds ¼ ð/ cÞy; ð7Þ dy=ds ¼ 3=2/2 þ ð2k cÞ/ 1=2y 2 : Since the first integral (6) of system (5) is the same as the Hamiltonian of Eq. (7), Eq. (5) has the same topological phase portrait as Eq. (7) except the singular line / ¼ c. Therefore we should be able to obtain the topological properties of Eq. (5) from the phase portrait of Eq. (7). From the properties of the phase portraits of Eq. (5), we conjecture that the CH equation (1) has a peakon when c ¼ ðk=2Þ, k 6¼ 0. Then we show the conjecture is true. In fact, we obtain the peakon in two different ways, i.e., from the limit of general single solitary wave solutions and from the limit of periodic cusp wave solutions respectively. Finally we give a simpler way to obtain the peakon. Of course, all our three approaches to obtain the peakon are applicable for the case of k ¼ 0.
2. Bifurcation of the phase portraits of Eq. (7) Using the bifurcation methods of dynamical systems [8], we analyze the equilibrium points and bifurcation curves of Eq. (7).
Z. Liu, T. Qian / Appl. Math. Modelling 26 (2002) 473–480
475
It is obvious that Eq. (7) has two equilibrium points O(0,0) and Að2ðc2kÞ ; 0Þ on the /-axis when pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 c 6¼ 2k, and two equilibrium points B ðc; cðc þ 4kÞÞ on the line / ¼ c when cðc þ 4kÞ > 0. Let /0 denote either of the two zeroes of f ð/Þ ¼ 32/2 þ ð2k cÞ/; then the eigenvalue of the linearized system of Eq. (7) at ð/0 ; 0Þ (either O or A) are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1;2 ¼ ð/0 cÞf 0 ð/0 Þ;
ð8Þ ð9Þ
while at B
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k3;4 ¼ cðc þ 4kÞ:
ð10Þ
At O, A, and B the Hamiltonians Hð/; yÞ respectively are h0 ¼ H ðOÞ ¼ 0; h1 ¼ H ðAÞ ¼
4ðc 2kÞ3 ; 27
ð11Þ ð12Þ
and h2 ¼ H ðBþ Þ ¼ H ðB Þ ¼ 2kc2 :
ð13Þ
Therefore, Eq. (7) has only one equilibrium point O(0,0) on /-axis when c ¼ 2k, and no equilibrium point on / ¼ c when cðc þ 4kÞ < 0. When c ¼ 4k, A and B coincide on the line / ¼ c. When k ¼ 0 or c ¼ 0 we have h2 ¼ 0. And h1 ¼ h2 when c ¼ k=2. Hence on the ðk; cÞ-plane we have five bifurcation curves, namely c ¼ 0, c ¼ k=2, c ¼ 2k, k ¼ 0, and c ¼ 4k, which divide the parameter plane into 10 separate regions. Further analysis yields the bifurcation of phase portraits of Eq. (7), which is shown in Fig. 1 for c P 0. Note that when c < 0, the bifurcation of phase portraits is symmetric on the ðk; cÞ-plane with respect to the origin. It was shown in Ref. [3] that peakons exist for the CH equation in the case of k ¼ 0. Comparing the other phase portraits with the one when k ¼ 0 in Fig. 1, we see the heteroclinic connections of the points O with Bþ , Bþ with B and B with O for k ¼ 0, and A with Bþ , Bþ with B and B with A for c ¼ k=2. We conjecture that such a heteroclinic connection in the phase portrait of Eq. (7) represents a peakon for the CH equation (1). Thus Eq. (1) has a peakon when c ¼ k=2 for k 6¼ 0. This turns out to be the case as illustrated in the following discussion. 3. Peakons from the limit of general single solitary waves Without loss of generality, we consider the case of c > 0. When ðk; cÞ 2 ðBÞ in Fig. 1, system (7) has a homoclinic orbit at Að2ðc 2kÞ=3; 0Þ around the origin O(0,0). Since system (5) has the same topological phase portraits as system (7) except the line / ¼ c, system (5) has a homoclinic orbit at Að2ðc 2kÞ=3; 0Þ around the origin O(0,0). Since a homoclinic orbit of Eq. (5) corresponds to a general single solitary wave of Eq. (1), we suspect that when c ! k=2, the limit of the general single solitary wave solutions becomes the peakon.
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Z. Liu, T. Qian / Appl. Math. Modelling 26 (2002) 473–480
Fig. 1. Bifurcation of phase portraits of Eq. (7) for c P 0.
By Eqs. (6) and (12), the homoclinic orbit can be represented as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3/ þ 4k 2cÞ 3/ þ c 2k pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi : y¼ 3 3 /c
ð14Þ
Substituting Eq. (14) into the first equation of system (5) and integrating along the homoclinic orbit, we obtain pffiffiffiffiffiffiffiffiffiffi Z 2kc Z 0 3 1 sc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ pffiffiffi ds for n < 0; ð15Þ ð3s þ 4k 2cÞ 3s þ c 2k 3 3 n / and Z
/ 2kc 3
pffiffiffiffiffiffiffiffiffiffi Z n sc 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ pffiffiffi ds for n P 0: ð3s þ 4k 2cÞ 3s þ c 2k 3 3 0
ð16Þ
From Eqs. (15) and (16) we then have w2 / c ð6/ þ 8k 4cÞa ð2c kÞa ¼ ejnj ; a a 2a ð/ cÞ 2ðc þ 4kÞ 2c k X
ð17Þ
Z. Liu, T. Qian / Appl. Math. Modelling 26 (2002) 473–480
477
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c þ 4k ; a¼ 3ð2k cÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3/ þ c 2k pffiffiffi w¼ þ 3; /c
ð18Þ
ð19Þ
and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3/ þ c 2k 9ð2k cÞ X¼ þ : /c c þ 4k
ð20Þ
Obviously, lim a ¼ 1;
c!k=2
ð21Þ
lim
w2 ¼ 1; X2a
ð22Þ
lim
ð6/ þ 8k 4cÞa 2ð/ þ kÞ ¼ ; 3k 2ðc þ 4kÞa
ð23Þ
lim
ð2c kÞa ¼ expð lim lnð2c kÞÞ ¼ 1: c!k=2ða1Þ 2c k
ð24Þ
c!k=2
c!k=2
and c!k=2
Therefore letting c ! k=2 in Eq. (17) we get 3k jnj e k; 2 which implies that for k > 0 the CH equation (1) has a peakon /ðnÞ ¼
ð25Þ
3k jxktj ð26Þ e 2 k: 2 Hence we obtain the peakon from the limit of general single solitary wave solutions. Using the same approach, one proves that Eq. (26) also holds for k < 0, and one can obtain the peakon (2) for k ¼ 0. uðx; t; kÞ ¼
4. Peakons from the limit of periodic cusp waves From Eqs. (10) and (12), and Fig. 1 we know that when ðk; cÞ 2 ðAÞ or (D) and h 2 ð2kc2 ; 0Þ, system (5) has a family of periodic orbits. When h ! 2kc2 , the periodic orbits lose smooth property and become periodic cusp orbits [1].
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Z. Liu, T. Qian / Appl. Math. Modelling 26 (2002) 473–480
Let Ckc be the limiting curve of the periodic orbits when h ! 2kc2 . By Eqs. (6) and (13) Ckc consists of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 2 ¼ /2 þ 2k/ þ 2kc for kðk 2cÞ k 6 / < c; and / ¼ c;
for jyj 6
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðc þ 4kÞ:
ð27Þ ð28Þ
Substituting (27) into ðd/=dnÞ ¼ y and integrating along Ckc we have Z c Z 0 ds pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ds for n < 0; s2 þ 2ks þ 2kc / n
ð29Þ
and Z
/ c
ds pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 s þ 2ks þ 2kc
Z
n
ds
for n P 0:
ð30Þ
0
From Eqs. (29) and (30) we obtain the periodic cusp wave solutions uðn; k; cÞ ¼ u0 ðn 2nT ; c; kÞ;
ð31Þ
where n ¼ 0; 1; 2; . . . ; n ¼ x ct 2 ðð2n 1ÞT ; ð2n þ 1ÞT Þ, and a0 kðk 2cÞ jnj u0 ðn; k; cÞ ¼ ejnj þ e k; 2a0 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 ¼ cðc þ 4kÞ þ c þ k; and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 þ 4kc þ c þ k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ ln : kðk 2cÞ
ð32Þ ð33Þ
ð34Þ
Similarly, for either k > 0 and c < 4k, or k < 0 and k=2 < c < 0, Eqs. (31)–(34) also hold. Obviously T ! 1 when either k ! 0 or c ! k=2. Thus we have lim uðx; t; k; cÞ ¼ lim u0 ðn; k; cÞ ¼ cejxctj ; k!0
k!0
ð35Þ
and lim uðx; t; k; cÞ ¼ lim u0 ðn; k; cÞ ¼
c!k=2
c!k=2
3k expðjx ðkt=2ÞjÞ k: 2
ð36Þ
Hence we have obtained the peakons (2) and (26) again from the limit of the periodic cusp waves when k ! 0 and k ! k=2, respectively. 5. Peakons obtained in a simpler way As illustrated in the last two sections, one can obtain peakons from the limit of general single solitary waves and the limit of periodic cusp waves. In this section, we will give a simpler way to obtain the peakon.
Z. Liu, T. Qian / Appl. Math. Modelling 26 (2002) 473–480
479
Let Ck be the limiting curve of the homoclinic orbits of Eq. (5) when c ! k=2 with k 6¼ 0. From Eqs. (6), (12) and (13) we know that Ck consists of three line segments k y ¼ ð/ þ kÞ for k 6 / < ; 2
ð37Þ
3k for jyj 6 : 2
ð38Þ
and k /¼ ; 2
Substituting Eq. (37) into the first equation of (5) and integrating along Ck , we have Z k2 Z 0 ds ¼ ds for n < 0; / sþk n
ð39Þ
and Z
/
k 2
ds ¼ s k
Z
n
ds for n P 0:
ð40Þ
0
From Eqs. (39) and (40) we immediately obtain the peakon (26). When k ! 0, let C0 be the limiting curve of the homoclinic orbits of Eq. (5). By Eqs. (6), (11) and (13), C0 consists of y ¼ / for 0 6 / < c;
ð41Þ
/ ¼ c;
ð42Þ
and for jyj 6 jcj:
Similarly we have Z 0 Z c ds ds; ¼ s / n
for n < 0;
ð43Þ
and Z c
/
ds ¼ s
Z
n
ds;
for n P 0;
ð44Þ
0
which immediately yield the peakon (2).
6. Conclusions From the above analysis we conclude that 1. The CH equation (1) has peakons if either the parameter k ¼ 0, or the wave speed c ¼ k=2 and k 6¼ 0. 2. From the mathematical point of view, the corner at the peak of a peakon was caused by the singular line / ¼ c in Eq. (5).
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Z. Liu, T. Qian / Appl. Math. Modelling 26 (2002) 473–480
3. The peakon comes from the limit of general single solitary waves or periodic cusp waves as shown in Sections 3 and 4. 4. Peakons of the CH equation (1) are represented by heteroclinic connections in the phase portrait of system (7) as illustrated in Section 5.
Acknowledgement Liu would like to thank Prof. Jeff Xia for his kind invitation to visit the Department of Mathematics of Northwestern University, where this paper was completed. Qian is grateful to Prof. Xia for his kind support.
References [1] M.S. Alber, R. Camassa, D.D. Holm, J.E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDEs, Lett. Math. Phys. 32 (2) (1994) 137–151. [2] J.P. Boyd, Peakons and coshoidal waves: travelling wave solutions of the Camassa–Holm equation, Appl. Math. Compt. 81 (2–3) (1997) 173–187. [3] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (11) (1993) 1661–1664. [4] R. Camassa, D.D. Holm, J.M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech. 31 (1994) 1–33. [5] A. Constantin, The Hamiltonian structure of the Camassa–Holm equation, Exposition. Math. 15 (1) (1997) 53–85. [6] A. Constantin, Soliton interactions for the Camassa–Holm equation, Exposition. Math. 15 (3) (1997) 251–264. [7] F. Cooper, H. Shepard, Solitons in the Camassa–Holm shallow water equation, Phys. Lett. A 194 (4) (1994) 246–250. [8] J. Guckenheimer, P. Holmes, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.