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A note on exact travelling wave solutions for the modified Camassa–Holm and Degasperis–Procesi equations
Applied Mathematics and Computation 218 (2011) 2269–2276
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A note on exact travelling wave solutions for the modified Camassa–Holm and Degasperis–Procesi equations Xijun Deng School of Information and Mathematics, Yangtze University, Jingzhou, Hubei 434023, PR China
a r t i c l e
i n f o
a b s t r a c t Exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations by Liu et al. (2010) [Y.F. Liu, X.Y. Zhu, J.X. He, Factorization technique and new exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations, Appl. Math. Comput. 217 (2010) 1658–1665] are investigated. Liu et al. has used the factorization technique to reduce the modified Camassa–Holm and Degasperis–Procesi equations to first-order ordinary differential equations, and then derived some exact travelling wave solutions by direct integral method. In this note, we will explain that the implementation of the so-called factorization technique is completely unnecessary. Moreover, based on the method of complete discrimination system for polynomial, we shall demonstrate that the general explicit exact solution and its classification for the above two types of equations can be obtained directly and many exact solutions by Liu et al. are our special cases. Besides, some known results in previously relevant literatures are extended and some simple remarks are also made. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Modified Camassa–Holm and Degasperis– Procesi equations Factorization technique Method of complete discrimination system for polynomial Travelling wave solutions
The modified Camassa–Holm (CH) and Degasperis–Procesi (DP) equations are of the following forms [1,2]
ut uxxt ¼ uuxxx þ 2ux uxx 3u2 ux ; 2
ut uxxt ¼ uuxxx þ 3ux uxx 4u ux :
ð1Þ ð2Þ
Recently, in [3] Liu et al. studied and looked for the travelling wave solutions of Eqs. (1) and (2). They first rewrite Eqs. (1) and (2) as a general form
ut uxxt ¼ uuxxx þ bux uxx ðb þ 1Þu2 ux ;
ð3Þ
then using the travelling wave transformation u(x, t) = U(n), n = x + ct in Eq. (3), which leads to the following third-order ordinary differential equation (ODE): 0
UU 000 þ bU U 00 ðb þ 1ÞU 2 U 0 cU 0 þ cU 000 ¼ 0:
ð4Þ
Next, they applied the factorization technique [4] and reduced Eq. (4) to the two types of first-order ODE as follows
i 1h 3 C0 2U 2cU 2 þ ð4c þ 2c2 ÞU þ þ C 1 ¼ 0; 4 Uþc 2U 2 C0 4U 3cU þ 5c þ 2c2 þ þ C 1 ¼ 0; U 02 15 ðU þ cÞ2
U 02
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.07.044
ð5Þ ð6Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
where C0, C1 are arbitrary integral constants. However, we find that the implementation of factorization technique is completely unnecessary, since Eq. (4) can be reduced to first-order ODE by direct integral method. That means, of course, that the factorization technique may be very effective for other nonlinear wave equations which cannot be transformed into the firstorder ODE. Let us demonstrate this fact. For example, when b = 2, Eq. (4) becomes
UU 000 þ 2U 0 U 00 3U 2 U 0 cU 0 þ cU 000 ¼ 0:
ð7Þ
Integrating (7) once, we have
ðU þ cÞU 00 þ
ðU 0 Þ2 ¼ U 3 þ cU þ g; 2
ð8Þ
where g is a integration constant. Multiplying Eq. (8) on 2U0 and integrating it once leads to
ðU þ cÞU 02 ¼
1 4 U þ cU 2 þ gU þ h; 2
ð9Þ
namely 3
U 02 ¼
h þ cðc2 þ c2 gÞ 2g 2c2 c3 1 3 ½U cU 2 þ ð2c þ c2 ÞU þ ; þ 2 4 Uþc
ð10Þ c4
c3 þ2c2 2g
where h is an arbitrary integration constant. Denote that C 0 ¼ h c3 2 þ cg; C 1 ¼ , it is easy to see that Eq. (10) 4 coincides with Eq. (5). Similarly, Eq. (6) can be derived by a completely similar analysis. The authors [3] solved the first-order ODE (5) and (6) directly, and obtained some exact solitary wave solutions for the modified CH and DP Eqs. (1) and (2). We were in doubt yet by what method it is that so many complicated exact solutions for Eqs. (5) and (6) described in [3] can be solved. Consequently we can expect that maybe some exact solutions are wrong. Just to be on the safe side we have checked all solutions obtained by Liu et al. [3], and find that obviously some solutions [3] pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8þ3cþ cð8þ3cÞ do not satisfy Eqs. (1) and (2). For instance, setting c ¼ 52 leads to < 0, then we can verify that the solution (22) cþ2 described in [3] is wrong. Similarly, setting c = 3 yields that the solution (35) in [3] is also wrong. Therefore, we were surprised that why the authors [3] claims that ‘‘some new exact solitary wave solutions for the modified CH and DP equations are obtained by factorization technique and direct integral’’. It is well-known that solving the exact travelling wave solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. Many powerful and elegant techniques have been presented, such as inverse scattering method [5], Hirota bilinear method [6], Backlund transformation method [7], homogenous balance method [8], F-expansion method [9], dynamical system approach [10,11], the method of complete discrimination system for polynomial [12,13], sub-equation method [14,15], tanh function method [16], similarity transform method [17,18] and so on. The method of complete discrimination system for polynomial method [12,13] will be the primary tools employed in the present note. In fact, the general explicit solutions and its classification for Eqs. (1) and (2) can be obtained by using the method of complete discrimination system for polynomial [12,13]. Let us demonstrate it below. Noticing that in the case C0 – 0, it seems to us that the exact explicit solutions for Eqs. (5) and (6) can not be obtained. Thus we assume that C0 = 0 throughout the whole paper. For convenience we first rewrite Eqs. (5) and (6) as the general form
U 02 ¼
2ðb þ 1Þ 3 3c 3ð2c2 þ 2c þ bcÞ U U2 þ U þ C2 : 3ðb þ 2Þ bþ1 bðb þ 1Þ
ð11Þ
Let
d2 ¼
3c ; bþ1
d1 ¼
3ð2c2 þ 2c þ bcÞ ; bðb þ 1Þ
d0 ¼ C 2 ;
ð12Þ
then Eq. (12) becomes
Z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dU 2ðb þ 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðn n0 Þ: 3ðb þ 2Þ U 3 þ d2 U 2 þ d1 U þ d0
ð13Þ
Denote that F(U) = U3 + d2U2 + d1U + d0, whose complete discrimination system is given by [12,13] 3
2d2 d1 d2 M ¼ 27 þ d0 27 3
!2
2
d 4 d1 2 3
!3
2
;
D ¼ d1
d2 ; 3
ð14Þ
According to the method of complete discrimination system for polynomial [12,13], we can obtain the corresponding exact solutions for Eqs. (1) and (2) in the following four cases.
X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
2271
Case 1. D = 0, D < 0. In this case, we have F(U) = (U a)2(U b), a – b. If U > b, Eq. (11) admits three types of exact explicit solutions as follows
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ðb þ 1Þða bÞ ðn n0 Þ ; a > b; 6ðb þ 2Þ "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ðb þ 1Þða bÞ 2 UðnÞ ¼ b þ ða bÞcoth ðn n0 Þ ; a > b; 6ðb þ 2Þ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi " # ðb þ 1Þðb aÞ ðn n0 Þ ; a < b: UðnÞ ¼ b þ ðb aÞ tan2 6ðb þ 2Þ 2
UðnÞ ¼ b þ ða bÞtanh
ð15Þ
ð16Þ
ð17Þ
When b = 2, from (14) and D = 0, D < 0 wepcan ffiffiffi derive pthat ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c 2 ð3; 0Þ;
d0 ¼
7c3 18c2 4 2cðc þ 3Þ cðc þ 3Þ : 27
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thus, we have a ¼ cr ; b ¼ c2r , where r ¼ 2cðc þ 3Þ. If a ¼ cþr > b ¼ c2r , then from (15) and (16) we can obtain that Eq. (1) 3 3 3 3 admits the following travelling wave solutions
rffiffiffi c 2r r 2 þ rtanh ðx þ ct n0 Þ ; uðx; tÞ ¼ 3 8 rffiffiffi c 2r r 2 þ rcoth ðx þ ct n0 Þ : uðx; tÞ ¼ 3 8
ð19Þ ð20Þ
If a ¼ cr < b ¼ cþ2r , then from (17) we can derive that Eq. (1) has the following trigonometric periodic singular wave solution 3 3
uðx; tÞ ¼
c þ 2r þ r tan2 3
rffiffiffi r ðx þ ct n0 Þ : 8
ð21Þ
When b = 3, from (14) and D = 0, D < 0 we can derive that
c 2 ð4; 0Þ;
d0 ¼
rffiffiffiffiffiffiffiffiffiffiffi 3c3 10c2 4D D : 27 32
Similarly, we can obtain that Eq. (2) has the solitary wave and singular wave solutions as follows
"rffiffiffiffiffiffi # 3c 2r r r 2 uðx; tÞ ¼ þ tanh ðx þ ct n0 Þ ; 12 4 30 "rffiffiffiffiffiffi # 3c 2r r r 2 uðx; tÞ ¼ þ coth ðx þ ct n0 Þ ; 12 4 30 "rffiffiffiffiffiffi # 3c þ 2r r r 2 uðx; tÞ ¼ þ tan ðx þ ct n0 Þ ; 12 4 30
ð22Þ ð23Þ ð24Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ 15cðc þ 4Þ. 2 2 2 Remark 1. Since tan2 ðx þ p2 Þ ¼rcot ffiffiffi ðxÞ, and 1 + tan x = csc x, from (21), we can know that the following functions
c þ 2r r þ rcot2 ðx þ ct n0 Þ ; 3 8 rffiffiffi cr r þ rcsc2 ðx þ ct n0 Þ ; uðx; tÞ ¼ 3 8 rffiffiffi cr r þ r sec2 ðx þ ct n0 Þ ; uðx; tÞ ¼ 3 8
uðx; tÞ ¼
ð25Þ ð26Þ ð27Þ
are also the exact solutions of Eq. (1). Likewise, from (24) we can obtain that
"rffiffiffiffiffiffi # 3c þ 2r r r þ cot2 ðx þ ct n0 Þ ; 12 4 30 "rffiffiffiffiffiffi # 3c r r r uðx; tÞ ¼ þ csc2 ðx þ ct n0 Þ ; 12 4 30 "rffiffiffiffiffiffi # 3c r r r uðx; tÞ ¼ þ sec2 ðx þ ct n0 Þ ; 12 4 30
uðx; tÞ ¼
are also the exact solutions of Eq. (2).
ð28Þ ð29Þ ð30Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
Remark 2. Note that tanh2x + sech2x = 1, it is easy to see that the above solutions (19) and (22) agree well with the solutions (26) and (38) obtained in [3], respectively. If taking c = 2, n0 = 0, (19) and (20) become
2 1 ðx 2tÞ ; uðx; tÞ ¼ 2sech 2 2 1 ðx 2tÞ ; uðx; tÞ ¼ 2csch 2
ð31Þ ð32Þ
respectively. Setting c ¼ 52 ; n0 ¼ 0, (22) and (23) become
15 5 2 1 sech x t ; 8 2 2 15 1 5 2 csch x t ; uðx; tÞ ¼ 8 2 2 uðx; tÞ ¼
ð33Þ ð34Þ
respectively. The above solutions (31)–(34) have been obtained in [10], and this implies that these solutions are only particular cases of our results. Setting c ¼ 13 ; n0 ¼ 0, (19) and (20) become
1 1 1 2 1 4sech pffiffiffi x t ; 3 3 6 1 1 1 2 3 þ 4coth pffiffiffi x t ; uðx; tÞ ¼ 3 3 6
uðx; tÞ ¼
ð35Þ ð36Þ
respectively. Setting c ¼ 14 ; n0 ¼ 0, (22) and (23) become
1 1 1 2 4 15sech pffiffiffi x t ; 16 4 8 1 1 1 2 11 þ 15coth pffiffiffi x t ; uðx; tÞ ¼ 16 4 8 uðx; tÞ ¼
ð37Þ ð38Þ
respectively. It is easily seen that the above solutions (35), (36), (37) and (38) are in agreement with the solutions (1.5), (1.7), (1.9) and (1.11) obtained in [11], respectively. Remark 3. Note that tanh2x = tanh2jxj, coth2x = coth2jxj, we find that the following functions
rffiffiffi c 2r r 2 þ rtanh jx þ ctj þ g0 ; 3 8 rffiffiffi c 2r r 2 uðx; tÞ ¼ þ rcoth jx þ ctj þ g0 ; 3 8
uðx; tÞ ¼
ð39Þ ð40Þ
are also the solutions of Eq. (1), and
"rffiffiffiffiffiffi # 3c 2r r r 2 uðx; tÞ ¼ þ tanh jx þ ctj þ g0 ; 12 4 30 "rffiffiffiffiffiffi # 3c 2r r r 2 uðx; tÞ ¼ þ coth jx þ ctj þ g0 ; 12 4 30
ð41Þ ð42Þ
are also the solutions of Eq. (2), where g0 is arbitrary constant. Let us show that the peakon wavepsolutions (9) and (10) described in [10] are particular cases of (40) and (42), respecffiffiffi 1 tively. Denoting c ¼ 2; g0 ¼ coth ð 2Þ from (40) one can obtain that 2
uðx; tÞ ¼ 2csch
1 jx 2tj þ g0 ¼ 2
cosh 12 jx
2 2 : pffiffiffi 2tj þ 2 sinh 12 jx 2tj
ð43Þ
1
If denoting c ¼ 52 ; g0 ¼ coth ð73Þ from (42) one can get
15 5 75 2 1 csch jx tj þ g0 ¼ pffiffiffiffiffiffi uðx; tÞ ¼ 2 : pffiffiffiffiffiffi 8 2 2 1 5 2 15 cosh 2 jx 2 tj þ 35 sinh 12 jx 52 tj
In [3], the authors obtained the peakon wave solution of Eq. (1) as follows
ð44Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
u¼
2 1 x 2 2
2a21 2a22 þ ð2a22 2a21 Þtanh
t
½a2 þ a1 tanh j 12 x tj
ð45Þ
;
and the peakon wave solution of Eq. (2) as follows
i
2
h 2 1 2 5 15 a2 a1 tanh 2 x 4 t 1 u¼ ; 2 8 a2 þ a1 tanh j 12 x 54 tj
ð46Þ
where a1, a2 are constants. By means of the following two identities
2 a22 a21 ð1 tanh HÞ ða2 þ a1 tanh jHjÞ
2
2
¼ sech ðjHj þ H0 Þ;
H0 ¼ tanh1
a1 ; a2
a22 > a21 ;
ð47Þ
and
2 a22 a21 ð1 tanh HÞ ða2 þ a1 tanh jHjÞ2
2
¼ csch ðjHj þ H0 Þ;
H0 ¼ coth1
a1 ; a2
a22 < a21 ;
ð48Þ
we have that (45) is equal to Eq. (39) or Eq. (40) at c = 2, g0 = H0, while (46) is equal to Eq. (41) or Eq. (42) at c ¼ 52 ; g0 ¼ H0 . Case 2. D = 0,D = 0. In this case, we have F(U) = (U a)3. We can get the rational form of exact solutions for Eq. (11)
UðnÞ ¼ a þ
6ðbþ2Þ bþ1
ðn n0 Þ2
ð49Þ
:
When b = 2, from (14) and D = 0, D = 0, we can derive that c = 3 or c = 0, and F(U) = U3 + 3U2 + 3U + 1 or F(U) = U3. This implies that a = 1 or a = 0. Thus, Eq. (1) admits the exact rational form of solution
uðx; tÞ ¼ 1 þ
8 ðx 3t n0 Þ2
ð50Þ
and
uðx; tÞ ¼
8 ðx n0 Þ2
ð51Þ
:
It is easy to show that the peakon solution (1.6) and singular wave solution (1.8) obtained in [11] are particular cases of Eq. (50). Similarly, when b = 3, we can derive that Eq. (2) possess the rational form of solution
uðx; tÞ ¼ 1 þ
15 2
ðx 4t n0 Þ2
ð52Þ
and
uðx; tÞ ¼
15 2
ðx n0 Þ2
ð53Þ
:
Consequently the peakon solution (1.10) and singular solution (1.12) described in [11] are particular cases of Eq. (52). Case 3. D > 0,D < 0. In this case, we have F(U) = (U a)(U b)(U c), a < b < c. When a < U < b, Eq. (11) has exact solution
UðnÞ ¼ a þ ðb aÞsn
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðb þ 1Þðc aÞ ðn n0 Þ; m ; 6ðb þ 2Þ
ð54Þ
when U > c, Eq. (11) has exact solution
UðnÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n0 Þ; m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðbþ1ÞðcaÞ ðn n0 Þ; m cn2 6ðbþ2Þ
c bsn2
ðbþ1ÞðcaÞ ðn 6ðbþ2Þ
a where m2 ¼ b ca. Thus, when u 2 (a,b), Eq. (1) admits the following smooth periodic wave solution
ð55Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
! rffiffiffiffiffiffiffiffiffiffiffiffi ca ðx þ ct n0 Þ; m ; 8
uðx; tÞ ¼ a þ ðb aÞsn2
ð56Þ
When u 2 (c, + 1), Eq. (1) has periodic blow-up solution as follows
uðx; tÞ ¼
qffiffiffiffiffiffi
ðx þ ct n0 Þ; m qffiffiffiffiffiffi ; ca ðx þ ct n0 Þ; m cn2 8
c bsn2
ca 8
ð57Þ
a where m2 ¼ b ca. Similarly, we have that when u 2 (a,b), Eq. (2) admits the following smooth periodic wave solution
uðx; tÞ ¼ a þ ðb aÞsn
! rffiffiffiffiffiffiffiffiffiffiffiffi ca ðx þ ct n0 Þ; m : 30
2
ð58Þ
When u 2 (c + 1), Eq. (2) has periodic blow-up solution as follows
uðx; tÞ ¼
qffiffiffiffiffiffi
ðx þ ct n Þ; m 0 30 qffiffiffiffiffiffi ; ca cn2 ðx þ ct n0 Þ; m 30
c bsn2
ca
ð59Þ
a where m2 ¼ b ca. Using the identity
sn2 n þ cn2 n ¼ 1;
ð60Þ
we can obtain that the solution (23) in [3] is equal to (56) at a ¼ in [3] is equal to (57) at a ¼
3cqqm2 12
3cqþ2qm2 12
;b ¼ ;c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cð5þ2cÞ q2 3cqþ4c2 þ10c a ¼ 3cq ; b ¼ 0; c ¼ ; q ¼ cð80 þ 23cÞ. 3cq
3cþ11qqm2 12
2cðcþ2Þ ;b cþg
;q ¼
2cðcþ2Þ ; cg
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
15cðcþ4Þ ; m4 m2 þ1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 0; g ¼ cð8 þ 3cÞ; the solution (33)
the solution (34) in [3] is equal to (59) at
Remark 4. It should be noted that a, b, c are determined by the selection of constants c and d0. In fact, as long as the relevant parameters meet the condition D > 0, D < 0, we would always find the specific values of a,b,c. For example, as for Eq. (1) let d0 ¼ 0; c ¼ 32, clearly it satisfies the condition D > 0, D < 0. In this case, we have FðUÞ ¼ U 3 þ 32 U 2 34 U. This implies that pffiffiffiffi pffiffiffiffi pffiffiffiffi a ¼ 34 21 ; b ¼ 0; c ¼ 3þ4 21. So from (56) and (57), we can obtain that when u 2 34 21 ; 0 , Eq. (1) has elliptic smooth periodic wave solution
ffiffiffiffiffiffiffiffiffiffi 0sp 1 pffiffiffiffiffiffi ffiffiffiffiffiffi 3 21 2 @ 3 21 x t n0 ; mA; cn uðx; tÞ ¼ 4 2 16
ð61Þ
pffiffiffiffi 21
when u > 3þ4
, Eq. (1) has elliptic periodic singular wave solution
ffiffiffiffiffiffiffiffiffiffi 0sp 1 ffiffiffiffiffiffi pffiffiffiffiffiffi 21 3 þ 21 2 @ 3 x t n0 ; mA; uðx; tÞ ¼ cn 2 16 4
ð62Þ
pffiffiffiffi 21
where m2 ¼ 21þ3 42
. Similarly, we can obtain some special exact travelling wave solutions for Eq. (2).
Case 4. D < 0. In this case, we have F(U) = (U a)(U2 + pU + q), p2 4q < 0. We can obtain that Eq. (11) has the exact solution as follows
UðnÞ ¼ a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ pa þ q þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ pa þ q qffiffiffiffiffiffiffiffiffiffiffi ; 1 2ðbþ1Þ 2 ða þ pa þ qÞ4 ðn n0 Þ; m 1 cn 3ðbþ2Þ
ð63Þ
aþ2p where m2 ¼ 12 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi . Thus, Eq. (1) admits periodic blow-up solutions 2 a þpaþq
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ a a2 þ pa þ q þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ pa þ q
qffiffi ; 1 1 2 1 cn ða þ pa þ qÞ4 ðx þ ct n0 Þ; m 2
ð64Þ
and Eq. (2) has periodic blow-up solutions
uðx; tÞ ¼ a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ pa þ q þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ pa þ q qffiffiffiffi ; 1 8 ða2 þ pa þ qÞ4 ðx þ ct n0 Þ; m 1 cn 15
ð65Þ
X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
2275
aþ2p where m2 ¼ 12 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 a þpaþq
Remark 5. Similar to Remark 4, from the condition D < 0, we can also determine the concrete values of a, p, q and obtain numerous special exact travelling wave solutions for Eqs. (1) and (2). For example, as for Eq. (1) setting
d0 ¼ 0; c 2 1; 83 [ ð0; þ1Þ leads to a = 0, p = c, q = c2 + 2c, then from (64) we can derive that Eq. (1) has exact explicit periodic blow-up solutions
0 uðx; tÞ ¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB C 2 1C c2 þ 2cB 1 @ A; 2 þ2cÞ4 ffiffi ðx þ ct n0 Þ; m 1 cn ðc p 2
ð66Þ
2 c 80 where m2 ¼ 12 1 þ pffiffiffiffiffiffiffiffiffi . Similarly for Eq. (2) setting d0 ¼ 0; c 2 1; 23 [ ð0; þ1Þ leads to a ¼ 0; p ¼ 34 c; q ¼ 2c 4þ5c, 2 2
c þ2c
then from (65) we can derive that Eq. (2) has exact explicit periodic blow-up solutions
0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2c þ 5c B B uðx; tÞ ¼ @ 2
1 C 2 1C 1 A; 2 þ5cÞ4 ðx þ ct n Þ; m 1 cn 2ð2cpffiffiffiffi 0 15
ð67Þ
3c where m2 ¼ 12 1 þ pffiffiffiffiffiffiffiffiffiffiffi . 2 4
2c þ5c
Remark 6. In [19], the authors investigated the coexistence of multifarious explicit travelling wave solutions for Eqs. (1) and (2), by using the bifurcation method of dynamical systems. By careful comparing, we can find that the main results presented in [19] could be deduced from our results. Moreover, as the authors pointed out that in their paper and previous results, they only obtained explicit nonlinear wave solutions with the wave speed c satisfying c 2 [3, 0) for Eq. (1) and c 2 [4, 0) for Eq. (2). And they do not know whether Eqs. (1) and (2) admit explicit travelling wave solutions for c < 3 and c < 4, respectively. Obviously, we answer the doubts raised by them. For example, the solution (66) exists for c < 3 and the solution (67) exists for c < 4. As a matter of fact, for arbitrary c 2 R, the solutions (64) and (65) exist for Eqs. (1) and (2), respectively. Remark 7. In [19], the authors claims in Propositions 2 and 2⁄ that when c ¼ 13 and c ¼ 14, ‘‘three types of nonlinear wave solutions’’, namely smooth solitary wave solution, hyperbolic singular wave solution and trigonometric periodic singular wave solution coexist for Eqs. (1) and (2) respectively. However, we would like to point out here that it is incomplete. And in fact other three types of exact explicit solutions including elliptic smooth periodic wave solution, elliptic periodic singular wave solution and periodic blow-up solution also coexist for Eqs. (1) and (2). Letpffiffiffiffi us show it below. For example, as for pffiffiffiffi Eq. (1), let d0 ¼ 0; c ¼ 13,we have FðUÞ ¼ U U 2 þ 13 U 59 . This implies that a ¼ 1þ6 21 ; b ¼ 0; c ¼ 1þ6 21. So from (56), we pffiffiffiffi can know that when u 2 1þ6 21 ; 0 , Eq. (1) admits smooth periodic wave solutions
ffiffiffiffiffiffiffiffiffiffi 0sp 1 pffiffiffiffiffiffi ffiffiffiffiffiffi 1 21 2 @ 1 21 x t n0 ; mA; cn uðx; tÞ ¼ 6 3 24
ð68Þ
pffiffiffiffi pffiffiffiffi where m2 ¼ 21þ42 21. When u 2 1þ6 21 ; þ1 , Eq. (1) admits periodic singular wave solutions
ffiffiffiffiffiffiffiffiffiffi 0sp 1 pffiffiffiffiffiffi ffiffiffiffiffiffi 1 þ 21 2 @ 1 21 x t n0 ; mA; cn uðx; tÞ ¼ 6 3 24
ð69Þ
pffiffiffiffi 2 where m2 ¼ 21þ42 21. Let d0 ¼ 79 ; c ¼ 13, we have FðUÞ ¼ U 3 þ U3 5U 79. Then from (64), we can get that Eq. (1) has periodic 9 blow-up solution
uðx; tÞ ¼ 1
pffiffi pffiffiffi 4 7 2 7 þ qffiffi 1 3
; 3 1 28 4 1 cn x 13 t n0 ; m 2 9
ð70Þ
pffiffi ffiffi . where m2 ¼ 2 4p75 pffiffiffiffi 7 33 Similarly, let d0 ¼ 0; c ¼ 14, we can know that when u 2 33 ; 0 , Eq. (2) admits smooth periodic wave solutions 32
ffiffiffiffiffiffiffiffiffiffi 0sp 1 ffiffiffiffiffiffi pffiffiffiffiffiffi 33 3 3 33 2 @ 1 uðx; tÞ ¼ x t n0 ; mA; cn 4 160 32
pffiffiffiffi pffiffiffiffi 33 where m2 ¼ 33þ66 33. When u 2 3þ3 ; þ1 , Eq. (2) admits periodic singular wave solutions 32
ð71Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
ffiffiffiffiffiffiffiffiffiffi 0sp 1 pffiffiffiffiffiffi ffiffiffiffiffiffi 3 þ 3 33 2 @ 1 33 x t n0 ; mA; cn uðx; tÞ ¼ 32 4 160
ð72Þ
pffiffiffiffi where m2 ¼ 33þ66 33. Let d0 ¼ 29 ; c ¼ 14, then from (65), we can get that Eq. (2) has periodic blow-up solution 32
qffiffiffiffi rffiffiffiffiffiffi 99 99 8 þ uðx; tÞ ¼ 1 qffiffiffiffi 1
; 32 1 cn 8 99 4 x 14 t n0 ; m 15 32
ð73Þ
pffiffiffiffi 51 pffiffiffiffi where m2 ¼ 122422 . 22
Remark 8. More recently, some authors have investigated and obtained many exact solutions of Eqs. (1) and (2), for instance [20–22]. Through careful comparative observations, we can find that most of exact solutions described in [20–22] are particular cases of our results. Moreover, compared with the method used in [20–22], which must have the aid of Computer symbolic computation, the advantages of our method are very simple and direct. It also reminds us that before searching for exact solutions of various nonlinear wave equations by using symbolic computation we should remember it will be better to use the direct method. In summary, it is shown in this note that the method of complete discrimination system for polynomial is very effective and direct, and in some sense that it can give systematic classification of explicit exact travelling wave solutions for some types of nonlinear wave equations. Acknowledgement This work was partially supported by the Research Foundation of Education Bureau of Hubei Province, China (Grant Nos. Q20101309, and D20101303). References [1] A.M. Wazwaz, Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations, Phys. Lett. A 352 (2006) 500–504. [2] A.M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations, Appl. Math. Comput. 186 (2007) 130–141. [3] Y.F. Liu, X.Y. Zhu, J.X. He, Factorization technique and new exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations, Appl. Math. Comput. 217 (2010) 1658–1665. [4] D.S. Wang, H. Li, Single and multi-solitary wave solutions to a class of nonlinear evolution equations, J. Math. Anal. Appl. 343 (2008) 273–298. [5] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, 1991. [6] K.W. Chow, A class of doubly periodic waves for nonlinear evolution equations, Wave Motion 35 (2002) 71–90. [7] M.R. Miurs, Backlund Transformation, Springer, Berlin, 1978. [8] M.L. Wang, Exact solutions for a compound KdV–Burgers equation, Phys. Lett. A 213 (1996) 279–287. [9] D.S. Wang, H.Q. Zhang, Further improved F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equation, Chaos Soliton. Fract. 25 (2005) 601–610. [10] Z. Liu, Z. Ouyang, A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations, Phys. Lett. A 366 (2007) 377–381. [11] Q. Wang, M. Tang, New exact solutions for two nonlinear equations, Phys. Lett. A 372 (2008) 2995–3000. [12] C.S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta Phys. Sinica 54 (2005) 2505–2509 (in Chinese). [13] C.S. Liu, Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. Commun. 181 (2010) 317–324. [14] D.S. Wang, H.B. Li, Elliptic equation’s new solutions and their applications to two nonlinear partial differential equations, Appl. Math. Comput. 188 (2007) 762–771. [15] D.S. Wang, H.B. Li, Symbolic computation and non-travelling wave solutions of (2 + 1)-dimensional nonlinear evolution equations, Chaos Soliton. Fract. 38 (2008) 383–390. [16] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. [17] D.S. Wang, X.H. Hu, W.M. Liu, Localized nonlinear matter waves in two-component Bose–Einstein condensates with time- and space-modulated nonlinearities, Phys. Rev. A 82 (2010) 023612–023619. [18] D.S. Wang, X.H. Hu, J.P. Hu, W.M. Liu, Quantized quasi-two-dimensional Bose–Einstein condensates with spatially modulated nonlinearity, Phys. Rev. A 81 (2010) 025604–025607. [19] Z. Liu, J. Pan, Coexistence of multifarious explicit nonlinear wave solutions for modified forms of Camassa–Holm and Degaperis–Procesi equations, Int. J. Bifur. Chaos 19 (2009) 2267–2282. [20] H.C. Ma, Y.D. Yu, D.J. Ge, New exact traveling wave solutions for the modified form of Degasperis–Procesi equation, Appl. Math. Comput. 203 (2008) 792–798. [21] S. Liang, D. Jeffrey, New travelling wave solutions to modified CH and DP equations, Comput. Phys. Commun. 180 (2009) 1429–1433. [22] B. G Zhang, Z.R. Liu, J.F. Mao, New exact solutions for mCH and mDP equations by auxiliary equation method, Appl. Math. Comput. 217 (2010) 1306– 1314.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on exact travelling wave solutions for the modified Camassa–Holm and Degasperis–Procesi equations Xijun Deng School of Information and Mathematics, Yangtze University, Jingzhou, Hubei 434023, PR China
a r t i c l e
i n f o
a b s t r a c t Exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations by Liu et al. (2010) [Y.F. Liu, X.Y. Zhu, J.X. He, Factorization technique and new exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations, Appl. Math. Comput. 217 (2010) 1658–1665] are investigated. Liu et al. has used the factorization technique to reduce the modified Camassa–Holm and Degasperis–Procesi equations to first-order ordinary differential equations, and then derived some exact travelling wave solutions by direct integral method. In this note, we will explain that the implementation of the so-called factorization technique is completely unnecessary. Moreover, based on the method of complete discrimination system for polynomial, we shall demonstrate that the general explicit exact solution and its classification for the above two types of equations can be obtained directly and many exact solutions by Liu et al. are our special cases. Besides, some known results in previously relevant literatures are extended and some simple remarks are also made. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Modified Camassa–Holm and Degasperis– Procesi equations Factorization technique Method of complete discrimination system for polynomial Travelling wave solutions
The modified Camassa–Holm (CH) and Degasperis–Procesi (DP) equations are of the following forms [1,2]
ut uxxt ¼ uuxxx þ 2ux uxx 3u2 ux ; 2
ut uxxt ¼ uuxxx þ 3ux uxx 4u ux :
ð1Þ ð2Þ
Recently, in [3] Liu et al. studied and looked for the travelling wave solutions of Eqs. (1) and (2). They first rewrite Eqs. (1) and (2) as a general form
ut uxxt ¼ uuxxx þ bux uxx ðb þ 1Þu2 ux ;
ð3Þ
then using the travelling wave transformation u(x, t) = U(n), n = x + ct in Eq. (3), which leads to the following third-order ordinary differential equation (ODE): 0
UU 000 þ bU U 00 ðb þ 1ÞU 2 U 0 cU 0 þ cU 000 ¼ 0:
ð4Þ
Next, they applied the factorization technique [4] and reduced Eq. (4) to the two types of first-order ODE as follows
i 1h 3 C0 2U 2cU 2 þ ð4c þ 2c2 ÞU þ þ C 1 ¼ 0; 4 Uþc 2U 2 C0 4U 3cU þ 5c þ 2c2 þ þ C 1 ¼ 0; U 02 15 ðU þ cÞ2
U 02
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.07.044
ð5Þ ð6Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
where C0, C1 are arbitrary integral constants. However, we find that the implementation of factorization technique is completely unnecessary, since Eq. (4) can be reduced to first-order ODE by direct integral method. That means, of course, that the factorization technique may be very effective for other nonlinear wave equations which cannot be transformed into the firstorder ODE. Let us demonstrate this fact. For example, when b = 2, Eq. (4) becomes
UU 000 þ 2U 0 U 00 3U 2 U 0 cU 0 þ cU 000 ¼ 0:
ð7Þ
Integrating (7) once, we have
ðU þ cÞU 00 þ
ðU 0 Þ2 ¼ U 3 þ cU þ g; 2
ð8Þ
where g is a integration constant. Multiplying Eq. (8) on 2U0 and integrating it once leads to
ðU þ cÞU 02 ¼
1 4 U þ cU 2 þ gU þ h; 2
ð9Þ
namely 3
U 02 ¼
h þ cðc2 þ c2 gÞ 2g 2c2 c3 1 3 ½U cU 2 þ ð2c þ c2 ÞU þ ; þ 2 4 Uþc
ð10Þ c4
c3 þ2c2 2g
where h is an arbitrary integration constant. Denote that C 0 ¼ h c3 2 þ cg; C 1 ¼ , it is easy to see that Eq. (10) 4 coincides with Eq. (5). Similarly, Eq. (6) can be derived by a completely similar analysis. The authors [3] solved the first-order ODE (5) and (6) directly, and obtained some exact solitary wave solutions for the modified CH and DP Eqs. (1) and (2). We were in doubt yet by what method it is that so many complicated exact solutions for Eqs. (5) and (6) described in [3] can be solved. Consequently we can expect that maybe some exact solutions are wrong. Just to be on the safe side we have checked all solutions obtained by Liu et al. [3], and find that obviously some solutions [3] pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8þ3cþ cð8þ3cÞ do not satisfy Eqs. (1) and (2). For instance, setting c ¼ 52 leads to < 0, then we can verify that the solution (22) cþ2 described in [3] is wrong. Similarly, setting c = 3 yields that the solution (35) in [3] is also wrong. Therefore, we were surprised that why the authors [3] claims that ‘‘some new exact solitary wave solutions for the modified CH and DP equations are obtained by factorization technique and direct integral’’. It is well-known that solving the exact travelling wave solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. Many powerful and elegant techniques have been presented, such as inverse scattering method [5], Hirota bilinear method [6], Backlund transformation method [7], homogenous balance method [8], F-expansion method [9], dynamical system approach [10,11], the method of complete discrimination system for polynomial [12,13], sub-equation method [14,15], tanh function method [16], similarity transform method [17,18] and so on. The method of complete discrimination system for polynomial method [12,13] will be the primary tools employed in the present note. In fact, the general explicit solutions and its classification for Eqs. (1) and (2) can be obtained by using the method of complete discrimination system for polynomial [12,13]. Let us demonstrate it below. Noticing that in the case C0 – 0, it seems to us that the exact explicit solutions for Eqs. (5) and (6) can not be obtained. Thus we assume that C0 = 0 throughout the whole paper. For convenience we first rewrite Eqs. (5) and (6) as the general form
U 02 ¼
2ðb þ 1Þ 3 3c 3ð2c2 þ 2c þ bcÞ U U2 þ U þ C2 : 3ðb þ 2Þ bþ1 bðb þ 1Þ
ð11Þ
Let
d2 ¼
3c ; bþ1
d1 ¼
3ð2c2 þ 2c þ bcÞ ; bðb þ 1Þ
d0 ¼ C 2 ;
ð12Þ
then Eq. (12) becomes
Z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dU 2ðb þ 1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðn n0 Þ: 3ðb þ 2Þ U 3 þ d2 U 2 þ d1 U þ d0
ð13Þ
Denote that F(U) = U3 + d2U2 + d1U + d0, whose complete discrimination system is given by [12,13] 3
2d2 d1 d2 M ¼ 27 þ d0 27 3
!2
2
d 4 d1 2 3
!3
2
;
D ¼ d1
d2 ; 3
ð14Þ
According to the method of complete discrimination system for polynomial [12,13], we can obtain the corresponding exact solutions for Eqs. (1) and (2) in the following four cases.
X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
2271
Case 1. D = 0, D < 0. In this case, we have F(U) = (U a)2(U b), a – b. If U > b, Eq. (11) admits three types of exact explicit solutions as follows
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ðb þ 1Þða bÞ ðn n0 Þ ; a > b; 6ðb þ 2Þ "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ðb þ 1Þða bÞ 2 UðnÞ ¼ b þ ða bÞcoth ðn n0 Þ ; a > b; 6ðb þ 2Þ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi " # ðb þ 1Þðb aÞ ðn n0 Þ ; a < b: UðnÞ ¼ b þ ðb aÞ tan2 6ðb þ 2Þ 2
UðnÞ ¼ b þ ða bÞtanh
ð15Þ
ð16Þ
ð17Þ
When b = 2, from (14) and D = 0, D < 0 wepcan ffiffiffi derive pthat ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c 2 ð3; 0Þ;
d0 ¼
7c3 18c2 4 2cðc þ 3Þ cðc þ 3Þ : 27
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Thus, we have a ¼ cr ; b ¼ c2r , where r ¼ 2cðc þ 3Þ. If a ¼ cþr > b ¼ c2r , then from (15) and (16) we can obtain that Eq. (1) 3 3 3 3 admits the following travelling wave solutions
rffiffiffi c 2r r 2 þ rtanh ðx þ ct n0 Þ ; uðx; tÞ ¼ 3 8 rffiffiffi c 2r r 2 þ rcoth ðx þ ct n0 Þ : uðx; tÞ ¼ 3 8
ð19Þ ð20Þ
If a ¼ cr < b ¼ cþ2r , then from (17) we can derive that Eq. (1) has the following trigonometric periodic singular wave solution 3 3
uðx; tÞ ¼
c þ 2r þ r tan2 3
rffiffiffi r ðx þ ct n0 Þ : 8
ð21Þ
When b = 3, from (14) and D = 0, D < 0 we can derive that
c 2 ð4; 0Þ;
d0 ¼
rffiffiffiffiffiffiffiffiffiffiffi 3c3 10c2 4D D : 27 32
Similarly, we can obtain that Eq. (2) has the solitary wave and singular wave solutions as follows
"rffiffiffiffiffiffi # 3c 2r r r 2 uðx; tÞ ¼ þ tanh ðx þ ct n0 Þ ; 12 4 30 "rffiffiffiffiffiffi # 3c 2r r r 2 uðx; tÞ ¼ þ coth ðx þ ct n0 Þ ; 12 4 30 "rffiffiffiffiffiffi # 3c þ 2r r r 2 uðx; tÞ ¼ þ tan ðx þ ct n0 Þ ; 12 4 30
ð22Þ ð23Þ ð24Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ 15cðc þ 4Þ. 2 2 2 Remark 1. Since tan2 ðx þ p2 Þ ¼rcot ffiffiffi ðxÞ, and 1 + tan x = csc x, from (21), we can know that the following functions
c þ 2r r þ rcot2 ðx þ ct n0 Þ ; 3 8 rffiffiffi cr r þ rcsc2 ðx þ ct n0 Þ ; uðx; tÞ ¼ 3 8 rffiffiffi cr r þ r sec2 ðx þ ct n0 Þ ; uðx; tÞ ¼ 3 8
uðx; tÞ ¼
ð25Þ ð26Þ ð27Þ
are also the exact solutions of Eq. (1). Likewise, from (24) we can obtain that
"rffiffiffiffiffiffi # 3c þ 2r r r þ cot2 ðx þ ct n0 Þ ; 12 4 30 "rffiffiffiffiffiffi # 3c r r r uðx; tÞ ¼ þ csc2 ðx þ ct n0 Þ ; 12 4 30 "rffiffiffiffiffiffi # 3c r r r uðx; tÞ ¼ þ sec2 ðx þ ct n0 Þ ; 12 4 30
uðx; tÞ ¼
are also the exact solutions of Eq. (2).
ð28Þ ð29Þ ð30Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
Remark 2. Note that tanh2x + sech2x = 1, it is easy to see that the above solutions (19) and (22) agree well with the solutions (26) and (38) obtained in [3], respectively. If taking c = 2, n0 = 0, (19) and (20) become
2 1 ðx 2tÞ ; uðx; tÞ ¼ 2sech 2 2 1 ðx 2tÞ ; uðx; tÞ ¼ 2csch 2
ð31Þ ð32Þ
respectively. Setting c ¼ 52 ; n0 ¼ 0, (22) and (23) become
15 5 2 1 sech x t ; 8 2 2 15 1 5 2 csch x t ; uðx; tÞ ¼ 8 2 2 uðx; tÞ ¼
ð33Þ ð34Þ
respectively. The above solutions (31)–(34) have been obtained in [10], and this implies that these solutions are only particular cases of our results. Setting c ¼ 13 ; n0 ¼ 0, (19) and (20) become
1 1 1 2 1 4sech pffiffiffi x t ; 3 3 6 1 1 1 2 3 þ 4coth pffiffiffi x t ; uðx; tÞ ¼ 3 3 6
uðx; tÞ ¼
ð35Þ ð36Þ
respectively. Setting c ¼ 14 ; n0 ¼ 0, (22) and (23) become
1 1 1 2 4 15sech pffiffiffi x t ; 16 4 8 1 1 1 2 11 þ 15coth pffiffiffi x t ; uðx; tÞ ¼ 16 4 8 uðx; tÞ ¼
ð37Þ ð38Þ
respectively. It is easily seen that the above solutions (35), (36), (37) and (38) are in agreement with the solutions (1.5), (1.7), (1.9) and (1.11) obtained in [11], respectively. Remark 3. Note that tanh2x = tanh2jxj, coth2x = coth2jxj, we find that the following functions
rffiffiffi c 2r r 2 þ rtanh jx þ ctj þ g0 ; 3 8 rffiffiffi c 2r r 2 uðx; tÞ ¼ þ rcoth jx þ ctj þ g0 ; 3 8
uðx; tÞ ¼
ð39Þ ð40Þ
are also the solutions of Eq. (1), and
"rffiffiffiffiffiffi # 3c 2r r r 2 uðx; tÞ ¼ þ tanh jx þ ctj þ g0 ; 12 4 30 "rffiffiffiffiffiffi # 3c 2r r r 2 uðx; tÞ ¼ þ coth jx þ ctj þ g0 ; 12 4 30
ð41Þ ð42Þ
are also the solutions of Eq. (2), where g0 is arbitrary constant. Let us show that the peakon wavepsolutions (9) and (10) described in [10] are particular cases of (40) and (42), respecffiffiffi 1 tively. Denoting c ¼ 2; g0 ¼ coth ð 2Þ from (40) one can obtain that 2
uðx; tÞ ¼ 2csch
1 jx 2tj þ g0 ¼ 2
cosh 12 jx
2 2 : pffiffiffi 2tj þ 2 sinh 12 jx 2tj
ð43Þ
1
If denoting c ¼ 52 ; g0 ¼ coth ð73Þ from (42) one can get
15 5 75 2 1 csch jx tj þ g0 ¼ pffiffiffiffiffiffi uðx; tÞ ¼ 2 : pffiffiffiffiffiffi 8 2 2 1 5 2 15 cosh 2 jx 2 tj þ 35 sinh 12 jx 52 tj
In [3], the authors obtained the peakon wave solution of Eq. (1) as follows
ð44Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
u¼
2 1 x 2 2
2a21 2a22 þ ð2a22 2a21 Þtanh
t
½a2 þ a1 tanh j 12 x tj
ð45Þ
;
and the peakon wave solution of Eq. (2) as follows
i
2
h 2 1 2 5 15 a2 a1 tanh 2 x 4 t 1 u¼ ; 2 8 a2 þ a1 tanh j 12 x 54 tj
ð46Þ
where a1, a2 are constants. By means of the following two identities
2 a22 a21 ð1 tanh HÞ ða2 þ a1 tanh jHjÞ
2
2
¼ sech ðjHj þ H0 Þ;
H0 ¼ tanh1
a1 ; a2
a22 > a21 ;
ð47Þ
and
2 a22 a21 ð1 tanh HÞ ða2 þ a1 tanh jHjÞ2
2
¼ csch ðjHj þ H0 Þ;
H0 ¼ coth1
a1 ; a2
a22 < a21 ;
ð48Þ
we have that (45) is equal to Eq. (39) or Eq. (40) at c = 2, g0 = H0, while (46) is equal to Eq. (41) or Eq. (42) at c ¼ 52 ; g0 ¼ H0 . Case 2. D = 0,D = 0. In this case, we have F(U) = (U a)3. We can get the rational form of exact solutions for Eq. (11)
UðnÞ ¼ a þ
6ðbþ2Þ bþ1
ðn n0 Þ2
ð49Þ
:
When b = 2, from (14) and D = 0, D = 0, we can derive that c = 3 or c = 0, and F(U) = U3 + 3U2 + 3U + 1 or F(U) = U3. This implies that a = 1 or a = 0. Thus, Eq. (1) admits the exact rational form of solution
uðx; tÞ ¼ 1 þ
8 ðx 3t n0 Þ2
ð50Þ
and
uðx; tÞ ¼
8 ðx n0 Þ2
ð51Þ
:
It is easy to show that the peakon solution (1.6) and singular wave solution (1.8) obtained in [11] are particular cases of Eq. (50). Similarly, when b = 3, we can derive that Eq. (2) possess the rational form of solution
uðx; tÞ ¼ 1 þ
15 2
ðx 4t n0 Þ2
ð52Þ
and
uðx; tÞ ¼
15 2
ðx n0 Þ2
ð53Þ
:
Consequently the peakon solution (1.10) and singular solution (1.12) described in [11] are particular cases of Eq. (52). Case 3. D > 0,D < 0. In this case, we have F(U) = (U a)(U b)(U c), a < b < c. When a < U < b, Eq. (11) has exact solution
UðnÞ ¼ a þ ðb aÞsn
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðb þ 1Þðc aÞ ðn n0 Þ; m ; 6ðb þ 2Þ
ð54Þ
when U > c, Eq. (11) has exact solution
UðnÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n0 Þ; m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðbþ1ÞðcaÞ ðn n0 Þ; m cn2 6ðbþ2Þ
c bsn2
ðbþ1ÞðcaÞ ðn 6ðbþ2Þ
a where m2 ¼ b ca. Thus, when u 2 (a,b), Eq. (1) admits the following smooth periodic wave solution
ð55Þ
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X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
! rffiffiffiffiffiffiffiffiffiffiffiffi ca ðx þ ct n0 Þ; m ; 8
uðx; tÞ ¼ a þ ðb aÞsn2
ð56Þ
When u 2 (c, + 1), Eq. (1) has periodic blow-up solution as follows
uðx; tÞ ¼
qffiffiffiffiffiffi
ðx þ ct n0 Þ; m qffiffiffiffiffiffi ; ca ðx þ ct n0 Þ; m cn2 8
c bsn2
ca 8
ð57Þ
a where m2 ¼ b ca. Similarly, we have that when u 2 (a,b), Eq. (2) admits the following smooth periodic wave solution
uðx; tÞ ¼ a þ ðb aÞsn
! rffiffiffiffiffiffiffiffiffiffiffiffi ca ðx þ ct n0 Þ; m : 30
2
ð58Þ
When u 2 (c + 1), Eq. (2) has periodic blow-up solution as follows
uðx; tÞ ¼
qffiffiffiffiffiffi
ðx þ ct n Þ; m 0 30 qffiffiffiffiffiffi ; ca cn2 ðx þ ct n0 Þ; m 30
c bsn2
ca
ð59Þ
a where m2 ¼ b ca. Using the identity
sn2 n þ cn2 n ¼ 1;
ð60Þ
we can obtain that the solution (23) in [3] is equal to (56) at a ¼ in [3] is equal to (57) at a ¼
3cqqm2 12
3cqþ2qm2 12
;b ¼ ;c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cð5þ2cÞ q2 3cqþ4c2 þ10c a ¼ 3cq ; b ¼ 0; c ¼ ; q ¼ cð80 þ 23cÞ. 3cq
3cþ11qqm2 12
2cðcþ2Þ ;b cþg
;q ¼
2cðcþ2Þ ; cg
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
15cðcþ4Þ ; m4 m2 þ1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 0; g ¼ cð8 þ 3cÞ; the solution (33)
the solution (34) in [3] is equal to (59) at
Remark 4. It should be noted that a, b, c are determined by the selection of constants c and d0. In fact, as long as the relevant parameters meet the condition D > 0, D < 0, we would always find the specific values of a,b,c. For example, as for Eq. (1) let d0 ¼ 0; c ¼ 32, clearly it satisfies the condition D > 0, D < 0. In this case, we have FðUÞ ¼ U 3 þ 32 U 2 34 U. This implies that pffiffiffiffi pffiffiffiffi pffiffiffiffi a ¼ 34 21 ; b ¼ 0; c ¼ 3þ4 21. So from (56) and (57), we can obtain that when u 2 34 21 ; 0 , Eq. (1) has elliptic smooth periodic wave solution
ffiffiffiffiffiffiffiffiffiffi 0sp 1 pffiffiffiffiffiffi ffiffiffiffiffiffi 3 21 2 @ 3 21 x t n0 ; mA; cn uðx; tÞ ¼ 4 2 16
ð61Þ
pffiffiffiffi 21
when u > 3þ4
, Eq. (1) has elliptic periodic singular wave solution
ffiffiffiffiffiffiffiffiffiffi 0sp 1 ffiffiffiffiffiffi pffiffiffiffiffiffi 21 3 þ 21 2 @ 3 x t n0 ; mA; uðx; tÞ ¼ cn 2 16 4
ð62Þ
pffiffiffiffi 21
where m2 ¼ 21þ3 42
. Similarly, we can obtain some special exact travelling wave solutions for Eq. (2).
Case 4. D < 0. In this case, we have F(U) = (U a)(U2 + pU + q), p2 4q < 0. We can obtain that Eq. (11) has the exact solution as follows
UðnÞ ¼ a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ pa þ q þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ pa þ q qffiffiffiffiffiffiffiffiffiffiffi ; 1 2ðbþ1Þ 2 ða þ pa þ qÞ4 ðn n0 Þ; m 1 cn 3ðbþ2Þ
ð63Þ
aþ2p where m2 ¼ 12 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi . Thus, Eq. (1) admits periodic blow-up solutions 2 a þpaþq
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ a a2 þ pa þ q þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ pa þ q
qffiffi ; 1 1 2 1 cn ða þ pa þ qÞ4 ðx þ ct n0 Þ; m 2
ð64Þ
and Eq. (2) has periodic blow-up solutions
uðx; tÞ ¼ a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ pa þ q þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ pa þ q qffiffiffiffi ; 1 8 ða2 þ pa þ qÞ4 ðx þ ct n0 Þ; m 1 cn 15
ð65Þ
X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
2275
aþ2p where m2 ¼ 12 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 a þpaþq
Remark 5. Similar to Remark 4, from the condition D < 0, we can also determine the concrete values of a, p, q and obtain numerous special exact travelling wave solutions for Eqs. (1) and (2). For example, as for Eq. (1) setting
d0 ¼ 0; c 2 1; 83 [ ð0; þ1Þ leads to a = 0, p = c, q = c2 + 2c, then from (64) we can derive that Eq. (1) has exact explicit periodic blow-up solutions
0 uðx; tÞ ¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB C 2 1C c2 þ 2cB 1 @ A; 2 þ2cÞ4 ffiffi ðx þ ct n0 Þ; m 1 cn ðc p 2
ð66Þ
2 c 80 where m2 ¼ 12 1 þ pffiffiffiffiffiffiffiffiffi . Similarly for Eq. (2) setting d0 ¼ 0; c 2 1; 23 [ ð0; þ1Þ leads to a ¼ 0; p ¼ 34 c; q ¼ 2c 4þ5c, 2 2
c þ2c
then from (65) we can derive that Eq. (2) has exact explicit periodic blow-up solutions
0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2c þ 5c B B uðx; tÞ ¼ @ 2
1 C 2 1C 1 A; 2 þ5cÞ4 ðx þ ct n Þ; m 1 cn 2ð2cpffiffiffiffi 0 15
ð67Þ
3c where m2 ¼ 12 1 þ pffiffiffiffiffiffiffiffiffiffiffi . 2 4
2c þ5c
Remark 6. In [19], the authors investigated the coexistence of multifarious explicit travelling wave solutions for Eqs. (1) and (2), by using the bifurcation method of dynamical systems. By careful comparing, we can find that the main results presented in [19] could be deduced from our results. Moreover, as the authors pointed out that in their paper and previous results, they only obtained explicit nonlinear wave solutions with the wave speed c satisfying c 2 [3, 0) for Eq. (1) and c 2 [4, 0) for Eq. (2). And they do not know whether Eqs. (1) and (2) admit explicit travelling wave solutions for c < 3 and c < 4, respectively. Obviously, we answer the doubts raised by them. For example, the solution (66) exists for c < 3 and the solution (67) exists for c < 4. As a matter of fact, for arbitrary c 2 R, the solutions (64) and (65) exist for Eqs. (1) and (2), respectively. Remark 7. In [19], the authors claims in Propositions 2 and 2⁄ that when c ¼ 13 and c ¼ 14, ‘‘three types of nonlinear wave solutions’’, namely smooth solitary wave solution, hyperbolic singular wave solution and trigonometric periodic singular wave solution coexist for Eqs. (1) and (2) respectively. However, we would like to point out here that it is incomplete. And in fact other three types of exact explicit solutions including elliptic smooth periodic wave solution, elliptic periodic singular wave solution and periodic blow-up solution also coexist for Eqs. (1) and (2). Letpffiffiffiffi us show it below. For example, as for pffiffiffiffi Eq. (1), let d0 ¼ 0; c ¼ 13,we have FðUÞ ¼ U U 2 þ 13 U 59 . This implies that a ¼ 1þ6 21 ; b ¼ 0; c ¼ 1þ6 21. So from (56), we pffiffiffiffi can know that when u 2 1þ6 21 ; 0 , Eq. (1) admits smooth periodic wave solutions
ffiffiffiffiffiffiffiffiffiffi 0sp 1 pffiffiffiffiffiffi ffiffiffiffiffiffi 1 21 2 @ 1 21 x t n0 ; mA; cn uðx; tÞ ¼ 6 3 24
ð68Þ
pffiffiffiffi pffiffiffiffi where m2 ¼ 21þ42 21. When u 2 1þ6 21 ; þ1 , Eq. (1) admits periodic singular wave solutions
ffiffiffiffiffiffiffiffiffiffi 0sp 1 pffiffiffiffiffiffi ffiffiffiffiffiffi 1 þ 21 2 @ 1 21 x t n0 ; mA; cn uðx; tÞ ¼ 6 3 24
ð69Þ
pffiffiffiffi 2 where m2 ¼ 21þ42 21. Let d0 ¼ 79 ; c ¼ 13, we have FðUÞ ¼ U 3 þ U3 5U 79. Then from (64), we can get that Eq. (1) has periodic 9 blow-up solution
uðx; tÞ ¼ 1
pffiffi pffiffiffi 4 7 2 7 þ qffiffi 1 3
; 3 1 28 4 1 cn x 13 t n0 ; m 2 9
ð70Þ
pffiffi ffiffi . where m2 ¼ 2 4p75 pffiffiffiffi 7 33 Similarly, let d0 ¼ 0; c ¼ 14, we can know that when u 2 33 ; 0 , Eq. (2) admits smooth periodic wave solutions 32
ffiffiffiffiffiffiffiffiffiffi 0sp 1 ffiffiffiffiffiffi pffiffiffiffiffiffi 33 3 3 33 2 @ 1 uðx; tÞ ¼ x t n0 ; mA; cn 4 160 32
pffiffiffiffi pffiffiffiffi 33 where m2 ¼ 33þ66 33. When u 2 3þ3 ; þ1 , Eq. (2) admits periodic singular wave solutions 32
ð71Þ
2276
X. Deng / Applied Mathematics and Computation 218 (2011) 2269–2276
ffiffiffiffiffiffiffiffiffiffi 0sp 1 pffiffiffiffiffiffi ffiffiffiffiffiffi 3 þ 3 33 2 @ 1 33 x t n0 ; mA; cn uðx; tÞ ¼ 32 4 160
ð72Þ
pffiffiffiffi where m2 ¼ 33þ66 33. Let d0 ¼ 29 ; c ¼ 14, then from (65), we can get that Eq. (2) has periodic blow-up solution 32
qffiffiffiffi rffiffiffiffiffiffi 99 99 8 þ uðx; tÞ ¼ 1 qffiffiffiffi 1
; 32 1 cn 8 99 4 x 14 t n0 ; m 15 32
ð73Þ
pffiffiffiffi 51 pffiffiffiffi where m2 ¼ 122422 . 22
Remark 8. More recently, some authors have investigated and obtained many exact solutions of Eqs. (1) and (2), for instance [20–22]. Through careful comparative observations, we can find that most of exact solutions described in [20–22] are particular cases of our results. Moreover, compared with the method used in [20–22], which must have the aid of Computer symbolic computation, the advantages of our method are very simple and direct. It also reminds us that before searching for exact solutions of various nonlinear wave equations by using symbolic computation we should remember it will be better to use the direct method. In summary, it is shown in this note that the method of complete discrimination system for polynomial is very effective and direct, and in some sense that it can give systematic classification of explicit exact travelling wave solutions for some types of nonlinear wave equations. Acknowledgement This work was partially supported by the Research Foundation of Education Bureau of Hubei Province, China (Grant Nos. Q20101309, and D20101303). References [1] A.M. Wazwaz, Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations, Phys. Lett. A 352 (2006) 500–504. [2] A.M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations, Appl. Math. Comput. 186 (2007) 130–141. [3] Y.F. Liu, X.Y. Zhu, J.X. He, Factorization technique and new exact solutions for the modified Camassa–Holm and Degasperis–Procesi equations, Appl. Math. Comput. 217 (2010) 1658–1665. [4] D.S. Wang, H. Li, Single and multi-solitary wave solutions to a class of nonlinear evolution equations, J. Math. Anal. Appl. 343 (2008) 273–298. [5] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, 1991. [6] K.W. Chow, A class of doubly periodic waves for nonlinear evolution equations, Wave Motion 35 (2002) 71–90. [7] M.R. Miurs, Backlund Transformation, Springer, Berlin, 1978. [8] M.L. Wang, Exact solutions for a compound KdV–Burgers equation, Phys. Lett. A 213 (1996) 279–287. [9] D.S. Wang, H.Q. Zhang, Further improved F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equation, Chaos Soliton. Fract. 25 (2005) 601–610. [10] Z. Liu, Z. Ouyang, A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations, Phys. Lett. A 366 (2007) 377–381. [11] Q. Wang, M. Tang, New exact solutions for two nonlinear equations, Phys. Lett. A 372 (2008) 2995–3000. [12] C.S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta Phys. Sinica 54 (2005) 2505–2509 (in Chinese). [13] C.S. Liu, Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. Commun. 181 (2010) 317–324. [14] D.S. Wang, H.B. Li, Elliptic equation’s new solutions and their applications to two nonlinear partial differential equations, Appl. Math. Comput. 188 (2007) 762–771. [15] D.S. Wang, H.B. Li, Symbolic computation and non-travelling wave solutions of (2 + 1)-dimensional nonlinear evolution equations, Chaos Soliton. Fract. 38 (2008) 383–390. [16] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. [17] D.S. Wang, X.H. Hu, W.M. Liu, Localized nonlinear matter waves in two-component Bose–Einstein condensates with time- and space-modulated nonlinearities, Phys. Rev. A 82 (2010) 023612–023619. [18] D.S. Wang, X.H. Hu, J.P. Hu, W.M. Liu, Quantized quasi-two-dimensional Bose–Einstein condensates with spatially modulated nonlinearity, Phys. Rev. A 81 (2010) 025604–025607. [19] Z. Liu, J. Pan, Coexistence of multifarious explicit nonlinear wave solutions for modified forms of Camassa–Holm and Degaperis–Procesi equations, Int. J. Bifur. Chaos 19 (2009) 2267–2282. [20] H.C. Ma, Y.D. Yu, D.J. Ge, New exact traveling wave solutions for the modified form of Degasperis–Procesi equation, Appl. Math. Comput. 203 (2008) 792–798. [21] S. Liang, D. Jeffrey, New travelling wave solutions to modified CH and DP equations, Comput. Phys. Commun. 180 (2009) 1429–1433. [22] B. G Zhang, Z.R. Liu, J.F. Mao, New exact solutions for mCH and mDP equations by auxiliary equation method, Appl. Math. Comput. 217 (2010) 1306– 1314.