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Solitary wave solutions and periodic solutions for higher-order nonlinear evolution equations
Applied Mathematics and Computation 181 (2006) 1683–1692 www.elsevier.com/locate/amc
Solitary wave solutions and periodic solutions for higher-order nonlinear evolution equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, United States
Abstract The sine–cosine and the tanh methods combined with the homogeneous balance method are used for analytic study for higher-order nonlinear evolution equations. Exact solitary wave solutions and periodic solutions are developed. A comparison between the two used methods is presented. Ó 2006 Elsevier Inc. All rights reserved. Keywords: The tanh method; The sine–cosine method; Higher-order PDEs; Solitons
1. Introduction This paper is concerned with the specific fourth-order nonlinear evolution equation [1,2] given by vtt þ avxt vxx þ bvxxxt ¼ 0;
ð1Þ
and the system of nonlinear partial differential equations [1,2] given by wxt þ avx vt ¼ 0;
vt þ vxxx þ ðvx Þ3 þ bvx wxx ¼ 0;
ð2Þ
where v = v(x, t) and w = w(x, t) are sufficiently-often differentiable functions. Attention has been focused on equations like (1) and (2) due to its appearance in many branches of physics and nonlinear sciences such as nonlinear optics and quantum field theory. The main focus of these works was the solitary wave solutions. It is well known in solitary wave theory that four types of travelling waves are of particular interest to researchers. Three of these types as given in [3] are: the solitary waves, which are localized travelling waves, asymptotically zero at large distances, the periodic solutions, the kink waves which rise or descend from one asymptotic state to another. The fourth type is the compactons [4], which are solitons with compact spatial support such that each compacton is a soliton confined to a finite core. Most recently, Rosenau and Pikovsky [5] introduced a fifth type of travelling wave, termed kovatons, which are compact formations of glued together kink–antikink pairs that may assume an arbitrary width. The discrete kovatons are hat-shaped [5] configuration kink–antikink formations. Compactons and kovatons collide elastically. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.03.021
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The objectives of this work are twofold. Firstly, we seek to establish exact solutions of distinct physical structures for higher-order nonlinear equations (1) and (2). Secondly, we aim to implement two strategies to achieve our goal, namely, the tanh method [6–10] and the sine–cosine method [11–14], and to emphasize the fact that the sine–cosine method facilitates the evaluation work. However, as will be shown later that the two methods do not necessarily provide the same solutions. The tanh method [6–10] and the sine–cosine method [11–14] have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be easily solved by using symbolic computation such as Mathematica or Maple. The power of the two methods, that will be used, is its ease of use to determine shock or solitary type of solutions. In what follows, the sine–cosine ansatz and the tanh method will be reviewed briefly. 2. Analysis of the two methods The features of this method can be summarized as follows. A PDE P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0;
ð3Þ
can be converted to an ODE Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð4Þ
upon using a wave variable n = (x ct). Eq. (4) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. 2.1. The sine–cosine method The solutions of the reduced ODE Eq. (4) can be expressed in the form p fk cosb ðlnÞg; jnj 6 2l ; uðx; tÞ ¼ 0; otherwise or in the form uðx; tÞ ¼
(
fk sinb ðlnÞg; 0;
jnj 6 pl ; otherwise,
ð5Þ
ð6Þ
where k, l, and b are parameters that will be determined, l and c are the wave number and the wave speed respectively. These assumptions give ðun Þ00 ¼ n2 l2 b2 kn cosnb ðlnÞ þ nl2 kn bðnb 1Þ cosnb2 ðlnÞ;
ð7Þ
and uðivÞ ¼ l4 b4 k cosb ðlnÞ 2l4 kbðb 1Þðb2 2b þ 2Þ cosb2 ðlnÞ þ l4 kbðb 1Þðb 2Þðb 3Þ cosb4 ðlnÞ;
ð8Þ
where similar equations can be obtained for the sine assumption. Using the sine–cosine assumptions and its derivatives into the reduced ODE gives a trigonometric equation of cosR(ln) or sinR(ln) terms. The parameters are then determined by first balancing the exponents of each pair of cosine or sine to determine R. We next collect all coefficients of the same power in cosk(ln) or sink(ln), where these coefficients have to vanish. This gives a system of algebraic equations among the unknowns b, k and l that will be determined. The solutions proposed in (5) and (6) follow immediately. 2.2. The tanh method The tanh method is developed by Malfliet [9–11] where the tanh is used as a new variable, since all derivatives of a tanh are represented by a tanh itself.
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Introducing a new independent variable Y ¼ tanhðlnÞ;
n ¼ x ct;
ð9Þ
leads to the change of derivatives: d d ¼ lð1 Y 2 Þ ; dn dY 2 d2 2 2 d 2 2 2 d þ l ¼ 2l Y ð1 Y Þ ð1 Y Þ ; dY dY 2 dn2 d3 d d2 d3 6l3 Y ð1 Y 2 Þ2 2 þ l3 ð1 Y 2 Þ3 3 ; ¼ 2l3 ð1 Y 2 Þð3Y 2 1Þ 3 dY dY dY dn 4 2 d d d þ 4l4 ð1 Y 2 Þ2 ð9Y 2 2Þ 2 ¼ 8l4 Y ð1 Y 2 Þð3Y 2 2Þ 4 dY dY dn 12l4 Y ð1 Y 2 Þ3
ð10Þ
4 d3 4 2 4 d þ l ð1 Y Þ . dY 3 dY 4
We then apply the following finite series expansion: uðlnÞ ¼ SðY Þ ¼
M X
ak Y k ;
ð11Þ
k¼0
where M is a positive integer, in most cases, that will be determined. However, if M is not an integer, a transformation formula is usually used to overcome this difficulty. Substituting (10) and (11) into the ODE (4) results in an algebraic equation in powers of Y. To determine the parameter M, we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms. With M determined, we collect all coefficients of powers of Y in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters ak (k = 0, . . . , M), l, and c. Having determined these parameters, knowing that M is a positive integer in most cases, and using (11) we obtain an analytic solution u(x, t) in a closed form. 3. The fourth-order nonlinear equation 3.1. Using the sine–cosine method We first consider the fourth-order nonlinear equation vtt þ avxt vxx þ bvxxxt ¼ 0;
ð12Þ
can be converted to the ODE 2
cv00 aðv00 Þ bvðivÞ ¼ 0;
ð13Þ
upon using the wave variable n = x ct. Following [1] we set u ¼ v00 ;
ð14Þ
that will carry (13) into the nonlinear ODE cu au2 bu00 ¼ 0.
ð15Þ
Using the sine–cosine ansatz (5) into (15) gives ðbl2 b2 k þ ckÞ cosb ðlnÞ ak2 cos2b ðlnÞ bkl2 bðb 1Þ cosb2 ðlnÞ ¼ 0.
ð16Þ
j
Balancing the exponents and setting the coefficients of each cos (ln) to zero we find bðb 1Þ 6¼ 0;
2b ¼ b 2;
bl2 b2 ¼ c;
ak ¼ bl2 bðb 1Þ;
ð17Þ
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from which we find b 6¼ 0; 1;
b ¼ 2;
1 l¼ 2
rffiffiffiffiffiffiffi c c ; < 0; b b
k¼
3c . 2a
3.1.1. The periodic solutions Substituting the previous results into (5) gives the periodic solutions for u(x, t) for bc < 0 rffiffiffiffiffiffiffi 3c 1 c sec2 uðx; tÞ ¼ ðx ctÞ 2a 2 b
ð18Þ
ð19Þ
and uðx; tÞ ¼
rffiffiffiffiffiffiffi 3c 2 1 c csc ðx ctÞ . 2a 2 b
3.1.2. Solitons solutions However, for bc > 0, the solitons solutions for u(x, t) rffiffiffi 3c 1 c uðx; tÞ ¼ sech2 ðx ctÞ 2a 2 b
ð20Þ
ð21Þ
and rffiffiffi 3c c 2 1 uðx; tÞ ¼ csch ðx ctÞ ; 2a 2 b are readily obtained. Recall that u(n) = v00 (n). This in turn gives the periodic solutions for bc < 0 rffiffiffiffiffiffiffi 6b 1 c ln cos vðx; tÞ ¼ ðx ctÞ a 2 b
ð22Þ
ð23Þ
and vðx; tÞ ¼
rffiffiffiffiffiffiffi 6b 1 c ln sin ðx ctÞ . a 2 b
However, for bc > 0 we obtain the solitons solutions rffiffiffi 6b 1 c ln cosh vðx; tÞ ¼ ðx ctÞ a 2 b
ð24Þ
ð25Þ
and rffiffiffi 6b 1 c ln sinh vðx; tÞ ¼ ðx ctÞ . a 2 b
ð26Þ
The last four solutions for v(x, t) are obtained by considering the constants of integration to be zeros. 3.2. Using the tanh method In this section, we will use the tanh method to handle the reduced equation (15) of the fourth-order equation (12) given by cu au2 bu00 ¼ 0.
ð27Þ
Balancing u00 with u2 in (27) by using (10) we find M ¼ 2.
ð28Þ
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The tanh method presents the finite expansion uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 .
ð29Þ
Substituting (29) into (27), collecting the coefficients of Y, and solving the resulting system we find the following two sets of solutions, where the first set is given by rffiffiffi 3c 3c 1 c c a1 ¼ 0; l¼ ð30Þ a0 ¼ ; a2 ¼ ; ; > 0; 2a 2a 2 b b and the second set is a0 ¼
c ; 2a
a1 ¼ 0;
a2 ¼
3c ; 2a
l¼
1 2
rffiffiffiffiffiffiffi c c ; < 0. b b
ð31Þ
It is obvious that the first set (30) of parameters provides the solutions obtained above by using the sine–cosine method. However, the second set (31) of parameters gives the following solutions: 3.2.1. The solitons solutions For bc < 0, we obtain the solitons solutions rffiffiffiffiffiffiffi c c 2 1 1 3 tanh uðx; tÞ ¼ ðx ctÞ 2a 2 b
ð32Þ
and uðx; tÞ ¼
rffiffiffiffiffiffiffi c 1 c 1 3 coth2 ðx ctÞ . 2a 2 b
3.2.2. The periodic solutions For bc > 0, we obtain the periodic solutions rffiffiffi c 1 c 1 þ 3 tan2 uðx; tÞ ¼ ðx ctÞ 2a 2 b
ð33Þ
ð34Þ
and rffiffiffi c c 2 1 1 þ 3cot ðx ctÞ . uðx; tÞ ¼ 2a 2 b
ð35Þ
Recall that u(n) = v00 (n). 3.2.3. Travelling wave solutions Based on the previous results, we obtain the following solutions related to the solitons solutions (32) and (33): ! !! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! bn bn cn2 3b vðnÞ ¼ ln tanh 1 þ ln tanh þ1 a 2 2 4a ! ! ! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! bn bn bn 3b 1 1 3b ln tanh tanh ln tanh þ 1 ln þ 1 2a 2 2 2a 2 2 2 ! ! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! bn bn 3b 3b ln 2 1 1 ln tanh ln tanh ð36Þ þ1 þ 2a 2a 2 2 2 2
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! pffiffiffiffiffifficffi ! bn cn2 3b ln coth vðnÞ ¼ 1 þ ln a 2 4a ! pffiffiffiffiffifficffi ! bn 3b 1 1 ln ln coth coth þ 2 2a 2 ! pffiffiffiffiffifficffi ! bn 3b 3b ln 2 ln coth ln þ1 þ 2a 2a 2
!! pffiffiffiffiffifficffi ! bn coth þ1 2 ! ! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! bn bn 1 3b þ 1 ln coth 2 2 2 2a ! pffiffiffiffiffifficffi ! bn 1 1 coth ; 2 2 2
where n = x ct. Moreover, we also obtain the following solutions related to the periodic solutions (34) and (35): rffiffiffi rffiffiffi cn2 6b 1 c 3b 1 c ln sec arctan tan2 vðnÞ ¼ n þ n a 2 b a 2 b 4a and
rffiffiffi rffiffiffi cn2 6b 1 c 3b c 3p pffiffiffiffiffi bp 2 1 ln csc bc þ n þ arccot cot n . vðnÞ ¼ a 2 b a 2 b 2a 2 4a
ð37Þ
ð38Þ
ð39Þ
4. The system of nonlinear PDEs 4.1. Using the sine–cosine method We next consider the system of nonlinear partial differential equations [1,2] given by 3
awxt þ avx vt ¼ 0;
vt þ vxxx þ ðvx Þ þ bvx wxx ¼ 0;
ð40Þ
where v = v(x, t) and w = w(x, t) are sufficiently often differentiable functions. This system can be converted to the system of ODEs 2
cw00 acðv0 Þ ¼ 0;
3
cv0 þ v000 þ ðv0 Þ þ bv0 w00 ¼ 0;
ð41Þ 0 2
upon using the wave variable n = x ct. Substituting w00 = a(v ) from the first equation into the second equation gives v000 cv0 þ ð1 abÞðv0 Þ3 ¼ 0;
ab 6¼ 1;
ð42Þ
that can be converted to u00 cu þ ð1 abÞu3 ¼ 0;
ð43Þ
0
upon substituting u = v . Using the cosine ansatz (5) into (43) gives ðl2 b2 k ckÞ cosb ðlnÞ þ ð1 abÞk3 cos3b ðlnÞ þ kl2 bðb 1Þ cosb2 ðlnÞ ¼ 0.
ð44Þ
j
Balancing the exponents and setting the coefficients of each cos (ln) to zero we find bðb 1Þ 6¼ 0;
3b ¼ b 2;
l2 b2 ¼ c;
hence we obtain b 6¼ 0; 1;
b ¼ 1;
pffiffiffiffiffiffi l ¼ c; c < 0;
ð1 abÞk2 ¼ l2 bðb 1Þ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c k¼ . 1 ab
4.1.1. The periodic solutions Using the previous results in (5) gives the periodic solutions for c < 0 and ab > 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2c uðx; tÞ ¼ sec½ cðx ctÞ; 1 ab
ð45Þ
ð46Þ
ð47Þ
A.-M. Wazwaz / Applied Mathematics and Computation 181 (2006) 1683–1692
and
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2c csc½ cðx ctÞ. uðx; tÞ ¼ 1 ab
1689
ð48Þ
4.1.2. The solitons solutions For c > 0, the solitons solutions for u(x, t) read rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2c uðx; tÞ ¼ sech½ cðx ctÞ; ab < 1 1 ab and rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2c csch½ cðx ctÞ; ab > 1. uðx; tÞ ¼ ab 1
ð49Þ
ð50Þ
are readily obtained. 4.1.3. Travelling wave solutions Recall that u(n) = v 0 (n) and w00 = a(v 0 )2. This in turn gives the periodic solutions for c < 0, ab > 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 vðnÞ ¼ lnðsecð cnÞ þ tanð cnÞÞ ab 1 and rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 lnðcscð cnÞ þ cotð cnÞÞ . vðnÞ ¼ ab 1 However, for c > 0 we obtain the solitons solutions rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 arctanðsinh½ cnÞ; ab < 1 vðnÞ ¼ 1 ab and rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 pffiffiffi cn ; ab > 1. vðnÞ ¼ ln tanh ab 1 2
ð51Þ
ð52Þ
ð53Þ
ð54Þ
The last four solutions for v(x, t) are obtained by considering the constants of integration to be zeros. Recall that w00 = a(v 0 )2. Using the previous results for v we obtain the following travelling wave solutions for c < 0: pffiffiffiffiffiffi pffiffiffiffiffiffi ac 2ac tan2 ð cnÞ lnðcosð cnÞÞ; ð55Þ wðnÞ ¼ 3ð1 abÞ 3ð1 abÞ pffiffiffiffiffiffi pffiffiffiffiffiffi ac 2ac wðnÞ ¼ cot2 ð cnÞ lnðsinð cnÞÞ. ð56Þ 3ð1 abÞ 3ð1 abÞ However, for c > 0, we obtain pffiffiffi pffiffiffi ac 2ac sech2 ð cnÞ lnðcoshð cnÞÞ; 3ð1 abÞ 3ð1 abÞ pffiffiffi pffiffiffi ac 2ac wðnÞ ¼ csch2 ð cnÞ þ lnðsinhð cnÞÞ. 3ð1 abÞ 3ð1 abÞ
wðnÞ ¼
ð57Þ ð58Þ
4.2. Using the tanh method In this section we will use the tanh method to handle the reduced system of (40) given by 2
cw00 acðv0 Þ ¼ 0;
3
cv0 þ v000 þ ðv0 Þ þ bv0 w00 ¼ 0;
ð59Þ
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A.-M. Wazwaz / Applied Mathematics and Computation 181 (2006) 1683–1692
upon using the wave variable n = x ct. Following [1], we substitute w00 = a(v 0 )2 from the first equation into the second equation to obtain 3
v000 cv0 þ ð1 abÞðv0 Þ ¼ 0;
ab 6¼ 1;
ð60Þ
that can be converted to u00 cu þ ð1 abÞu3 ¼ 0 ¼ 0;
ð61Þ
upon substituting u = v 0 . Balancing u00 with u3 in (61) by using (10) we find 3M ¼ M þ 2;
ð62Þ
so that M = 1. The tanh method presents the finite expansion uðnÞ ¼ a0 þ a1 Y .
ð63Þ
Substituting (63) into (61), collecting the coefficients of Y, and solving the resulting system we find one set of solutions rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c a1 ¼ a0 ¼ 0; ; l ¼ ; c < 0; ab > 1. ð64Þ 1 ab 2 It is obvious that we obtained only one set of parameter solutions. This means that we find: 4.2.1. The kink solutions For c < 0, ab > 1, we obtain the kink solutions rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi c c uðx; tÞ ¼ tanh ðx ctÞ 1 ab 2
ð65Þ
and uðx; tÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi c c coth ðx ctÞ . 1 ab 2
4.2.2. The periodic solutions For c > 0, ab > 1, we obtain the periodic solutions rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi c c tan ðx ctÞ uðx; tÞ ¼ ab 1 2
ð66Þ
ð67Þ
and uðx; tÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi c c cot ðx ctÞ . ab 1 2
4.2.3. Travelling wave solutions Recall that u(n) = v 0 (n). In view of this we obtain the travelling wave solutions rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi
1 c c vðnÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln tanh n 1 þ ln tanh n þ1 2 2 2ðab 1Þ
ð68Þ
ð69Þ
and rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi
1 c c vðnÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln coth n 1 þ ln coth n þ1 ; 2 2 2ðab 1Þ
ð70Þ
A.-M. Wazwaz / Applied Mathematics and Computation 181 (2006) 1683–1692
valid for c < 0, ab > 1. However, for c > 0, ab > 1, we find rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi 2 c vðnÞ ¼ ln sec n ab 1 2 and
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi 2 c vðnÞ ¼ ln csc n . ab 1 2
Recall that w00 = a(v 0 )2, therefore we find the following travelling wave solutions for w(n): rffiffiffiffiffiffi rffiffiffiffiffiffi 3a c 3a c a ln tanh ln tanh wðnÞ ¼ n 1 n þ1 2ðab 1Þ 2 2ðab 1Þ 2 2ðab 1Þ rffiffiffiffiffiffi rffiffiffiffiffiffi rffiffiffiffiffiffi c c 1 a ln 2 1 c 1 ln tanh ln tanh n 1 ln tanh n þ þ n 2 2 2 2ðab 1Þ 2 2 2
1691
ð71Þ
ð72Þ
ð73Þ
and rffiffiffiffiffiffi rffiffiffiffiffiffi 3a c 3a c a wðnÞ ¼ ln coth ln coth n 1 n þ1 2ðab 1Þ 2 2ðab 1Þ 2 2ðab 1Þ rffiffiffiffiffiffi rffiffiffiffiffiffi rffiffiffiffiffiffi c c 1 a ln 2 1 c 1 ln coth ln coth n 1 ln coth n þ þ n ; 2 2 2 2ðab 1Þ 2 2 2 and for c > 0, ab > 1 we find rffiffiffi rffiffiffi
a c c 2 ln sec wðnÞ ¼ n arctan tan2 n ab 1 2 2
ð74Þ
ð75Þ
and rffiffiffi rffiffiffi
rffiffiffi a c c ap c p 2 2 ln csc wðnÞ ¼ n arccot cot n nþ . ab 1 2 2 ab 1 2 4
ð76Þ
5. Concluding remarks The powerful sine–cosine and tanh methods were employed for analytic treatment for a fourth-order nonlinear PDE and a system of nonlinear PDEs. Solitary wave solutions, kinks, and periodic solutions were formally derived The performance of the two schemes show that the two methods are reliable and effective. The sine–cosine method can be used directly and in a straightforward manner. The tanh method requires transformation formula if the balancing process gives a fractional result for M. Unlike other works where the two methods provide the same set of travelling wave solutions, it is shown in this work that the two methods do not necessarily provide the same solutions. References [1] A. Huber, Solitary solutions of some nonlinear evolution equations, Appl. Math. Comput. 156 (2005) 464–474. [2] A. Ramani, Inverse scattering, ordinary differential equations of Painleve’-type and Hirota’s bilinear formalism, Ann. New York Acad. Sci. (1981) 54–64. [3] S. Kichenassamy, P. Olver, Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, SIAM J. Math. Anal. 23 (5) (1992) 1141–1166. [4] P. Rosenau, J.M. Hyman, Compactons: Solitons with finite wavelengths, Phys. Rev. Lett. 70 (5) (1993) 564–567. [5] P. Rosenau, A. Pikovsky, Phase compactons in chains of dispersively coupled oscillations, Phys. Rev. Lett. 94 (174102) (2005) 1–4. [6] D. Baldwin, U. Goktas, W. Hereman, L. Hong, R.S. Martino, J.C. Miller, Symbolic computation of exact solutions in hyperbolic and elliptic functions for nonlinear PDEs, J. Symbol. Comput. 37 (2004) 669–705. [7] U. Goktas, W. Hereman, Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symbol. Comput. 24 (1997) 591–621.
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W. Malfliet, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scr. 54 (1996) 563–568. W. Malfliet, The tanh method: II. Perturbation technique for conservative systems, Phys. Scr. 54 (1996) 569–575. A.M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput. 154 (3) (2004) 713–723. A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. A.M. Wazwaz, Compactons in a class of nonlinear dispersive equations, Math. Comput. Model. 37 (3/4) (2003) 333–341. A.M. Wazwaz, Distinct variants of the KdV equation with compact and noncompact structures, Appl. Math. Comput. 150 (2004) 365–377. [14] A.M. Wazwaz, Variants of the generalized KdV equation with compact and noncompact structures, Comput. Math. Appl. 47 (2004) 583–591.
Solitary wave solutions and periodic solutions for higher-order nonlinear evolution equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, United States
Abstract The sine–cosine and the tanh methods combined with the homogeneous balance method are used for analytic study for higher-order nonlinear evolution equations. Exact solitary wave solutions and periodic solutions are developed. A comparison between the two used methods is presented. Ó 2006 Elsevier Inc. All rights reserved. Keywords: The tanh method; The sine–cosine method; Higher-order PDEs; Solitons
1. Introduction This paper is concerned with the specific fourth-order nonlinear evolution equation [1,2] given by vtt þ avxt vxx þ bvxxxt ¼ 0;
ð1Þ
and the system of nonlinear partial differential equations [1,2] given by wxt þ avx vt ¼ 0;
vt þ vxxx þ ðvx Þ3 þ bvx wxx ¼ 0;
ð2Þ
where v = v(x, t) and w = w(x, t) are sufficiently-often differentiable functions. Attention has been focused on equations like (1) and (2) due to its appearance in many branches of physics and nonlinear sciences such as nonlinear optics and quantum field theory. The main focus of these works was the solitary wave solutions. It is well known in solitary wave theory that four types of travelling waves are of particular interest to researchers. Three of these types as given in [3] are: the solitary waves, which are localized travelling waves, asymptotically zero at large distances, the periodic solutions, the kink waves which rise or descend from one asymptotic state to another. The fourth type is the compactons [4], which are solitons with compact spatial support such that each compacton is a soliton confined to a finite core. Most recently, Rosenau and Pikovsky [5] introduced a fifth type of travelling wave, termed kovatons, which are compact formations of glued together kink–antikink pairs that may assume an arbitrary width. The discrete kovatons are hat-shaped [5] configuration kink–antikink formations. Compactons and kovatons collide elastically. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.03.021
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The objectives of this work are twofold. Firstly, we seek to establish exact solutions of distinct physical structures for higher-order nonlinear equations (1) and (2). Secondly, we aim to implement two strategies to achieve our goal, namely, the tanh method [6–10] and the sine–cosine method [11–14], and to emphasize the fact that the sine–cosine method facilitates the evaluation work. However, as will be shown later that the two methods do not necessarily provide the same solutions. The tanh method [6–10] and the sine–cosine method [11–14] have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be easily solved by using symbolic computation such as Mathematica or Maple. The power of the two methods, that will be used, is its ease of use to determine shock or solitary type of solutions. In what follows, the sine–cosine ansatz and the tanh method will be reviewed briefly. 2. Analysis of the two methods The features of this method can be summarized as follows. A PDE P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0;
ð3Þ
can be converted to an ODE Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð4Þ
upon using a wave variable n = (x ct). Eq. (4) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. 2.1. The sine–cosine method The solutions of the reduced ODE Eq. (4) can be expressed in the form p fk cosb ðlnÞg; jnj 6 2l ; uðx; tÞ ¼ 0; otherwise or in the form uðx; tÞ ¼
(
fk sinb ðlnÞg; 0;
jnj 6 pl ; otherwise,
ð5Þ
ð6Þ
where k, l, and b are parameters that will be determined, l and c are the wave number and the wave speed respectively. These assumptions give ðun Þ00 ¼ n2 l2 b2 kn cosnb ðlnÞ þ nl2 kn bðnb 1Þ cosnb2 ðlnÞ;
ð7Þ
and uðivÞ ¼ l4 b4 k cosb ðlnÞ 2l4 kbðb 1Þðb2 2b þ 2Þ cosb2 ðlnÞ þ l4 kbðb 1Þðb 2Þðb 3Þ cosb4 ðlnÞ;
ð8Þ
where similar equations can be obtained for the sine assumption. Using the sine–cosine assumptions and its derivatives into the reduced ODE gives a trigonometric equation of cosR(ln) or sinR(ln) terms. The parameters are then determined by first balancing the exponents of each pair of cosine or sine to determine R. We next collect all coefficients of the same power in cosk(ln) or sink(ln), where these coefficients have to vanish. This gives a system of algebraic equations among the unknowns b, k and l that will be determined. The solutions proposed in (5) and (6) follow immediately. 2.2. The tanh method The tanh method is developed by Malfliet [9–11] where the tanh is used as a new variable, since all derivatives of a tanh are represented by a tanh itself.
A.-M. Wazwaz / Applied Mathematics and Computation 181 (2006) 1683–1692
1685
Introducing a new independent variable Y ¼ tanhðlnÞ;
n ¼ x ct;
ð9Þ
leads to the change of derivatives: d d ¼ lð1 Y 2 Þ ; dn dY 2 d2 2 2 d 2 2 2 d þ l ¼ 2l Y ð1 Y Þ ð1 Y Þ ; dY dY 2 dn2 d3 d d2 d3 6l3 Y ð1 Y 2 Þ2 2 þ l3 ð1 Y 2 Þ3 3 ; ¼ 2l3 ð1 Y 2 Þð3Y 2 1Þ 3 dY dY dY dn 4 2 d d d þ 4l4 ð1 Y 2 Þ2 ð9Y 2 2Þ 2 ¼ 8l4 Y ð1 Y 2 Þð3Y 2 2Þ 4 dY dY dn 12l4 Y ð1 Y 2 Þ3
ð10Þ
4 d3 4 2 4 d þ l ð1 Y Þ . dY 3 dY 4
We then apply the following finite series expansion: uðlnÞ ¼ SðY Þ ¼
M X
ak Y k ;
ð11Þ
k¼0
where M is a positive integer, in most cases, that will be determined. However, if M is not an integer, a transformation formula is usually used to overcome this difficulty. Substituting (10) and (11) into the ODE (4) results in an algebraic equation in powers of Y. To determine the parameter M, we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms. With M determined, we collect all coefficients of powers of Y in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters ak (k = 0, . . . , M), l, and c. Having determined these parameters, knowing that M is a positive integer in most cases, and using (11) we obtain an analytic solution u(x, t) in a closed form. 3. The fourth-order nonlinear equation 3.1. Using the sine–cosine method We first consider the fourth-order nonlinear equation vtt þ avxt vxx þ bvxxxt ¼ 0;
ð12Þ
can be converted to the ODE 2
cv00 aðv00 Þ bvðivÞ ¼ 0;
ð13Þ
upon using the wave variable n = x ct. Following [1] we set u ¼ v00 ;
ð14Þ
that will carry (13) into the nonlinear ODE cu au2 bu00 ¼ 0.
ð15Þ
Using the sine–cosine ansatz (5) into (15) gives ðbl2 b2 k þ ckÞ cosb ðlnÞ ak2 cos2b ðlnÞ bkl2 bðb 1Þ cosb2 ðlnÞ ¼ 0.
ð16Þ
j
Balancing the exponents and setting the coefficients of each cos (ln) to zero we find bðb 1Þ 6¼ 0;
2b ¼ b 2;
bl2 b2 ¼ c;
ak ¼ bl2 bðb 1Þ;
ð17Þ
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from which we find b 6¼ 0; 1;
b ¼ 2;
1 l¼ 2
rffiffiffiffiffiffiffi c c ; < 0; b b
k¼
3c . 2a
3.1.1. The periodic solutions Substituting the previous results into (5) gives the periodic solutions for u(x, t) for bc < 0 rffiffiffiffiffiffiffi 3c 1 c sec2 uðx; tÞ ¼ ðx ctÞ 2a 2 b
ð18Þ
ð19Þ
and uðx; tÞ ¼
rffiffiffiffiffiffiffi 3c 2 1 c csc ðx ctÞ . 2a 2 b
3.1.2. Solitons solutions However, for bc > 0, the solitons solutions for u(x, t) rffiffiffi 3c 1 c uðx; tÞ ¼ sech2 ðx ctÞ 2a 2 b
ð20Þ
ð21Þ
and rffiffiffi 3c c 2 1 uðx; tÞ ¼ csch ðx ctÞ ; 2a 2 b are readily obtained. Recall that u(n) = v00 (n). This in turn gives the periodic solutions for bc < 0 rffiffiffiffiffiffiffi 6b 1 c ln cos vðx; tÞ ¼ ðx ctÞ a 2 b
ð22Þ
ð23Þ
and vðx; tÞ ¼
rffiffiffiffiffiffiffi 6b 1 c ln sin ðx ctÞ . a 2 b
However, for bc > 0 we obtain the solitons solutions rffiffiffi 6b 1 c ln cosh vðx; tÞ ¼ ðx ctÞ a 2 b
ð24Þ
ð25Þ
and rffiffiffi 6b 1 c ln sinh vðx; tÞ ¼ ðx ctÞ . a 2 b
ð26Þ
The last four solutions for v(x, t) are obtained by considering the constants of integration to be zeros. 3.2. Using the tanh method In this section, we will use the tanh method to handle the reduced equation (15) of the fourth-order equation (12) given by cu au2 bu00 ¼ 0.
ð27Þ
Balancing u00 with u2 in (27) by using (10) we find M ¼ 2.
ð28Þ
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The tanh method presents the finite expansion uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 .
ð29Þ
Substituting (29) into (27), collecting the coefficients of Y, and solving the resulting system we find the following two sets of solutions, where the first set is given by rffiffiffi 3c 3c 1 c c a1 ¼ 0; l¼ ð30Þ a0 ¼ ; a2 ¼ ; ; > 0; 2a 2a 2 b b and the second set is a0 ¼
c ; 2a
a1 ¼ 0;
a2 ¼
3c ; 2a
l¼
1 2
rffiffiffiffiffiffiffi c c ; < 0. b b
ð31Þ
It is obvious that the first set (30) of parameters provides the solutions obtained above by using the sine–cosine method. However, the second set (31) of parameters gives the following solutions: 3.2.1. The solitons solutions For bc < 0, we obtain the solitons solutions rffiffiffiffiffiffiffi c c 2 1 1 3 tanh uðx; tÞ ¼ ðx ctÞ 2a 2 b
ð32Þ
and uðx; tÞ ¼
rffiffiffiffiffiffiffi c 1 c 1 3 coth2 ðx ctÞ . 2a 2 b
3.2.2. The periodic solutions For bc > 0, we obtain the periodic solutions rffiffiffi c 1 c 1 þ 3 tan2 uðx; tÞ ¼ ðx ctÞ 2a 2 b
ð33Þ
ð34Þ
and rffiffiffi c c 2 1 1 þ 3cot ðx ctÞ . uðx; tÞ ¼ 2a 2 b
ð35Þ
Recall that u(n) = v00 (n). 3.2.3. Travelling wave solutions Based on the previous results, we obtain the following solutions related to the solitons solutions (32) and (33): ! !! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! bn bn cn2 3b vðnÞ ¼ ln tanh 1 þ ln tanh þ1 a 2 2 4a ! ! ! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! bn bn bn 3b 1 1 3b ln tanh tanh ln tanh þ 1 ln þ 1 2a 2 2 2a 2 2 2 ! ! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! bn bn 3b 3b ln 2 1 1 ln tanh ln tanh ð36Þ þ1 þ 2a 2a 2 2 2 2
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A.-M. Wazwaz / Applied Mathematics and Computation 181 (2006) 1683–1692
! pffiffiffiffiffifficffi ! bn cn2 3b ln coth vðnÞ ¼ 1 þ ln a 2 4a ! pffiffiffiffiffifficffi ! bn 3b 1 1 ln ln coth coth þ 2 2a 2 ! pffiffiffiffiffifficffi ! bn 3b 3b ln 2 ln coth ln þ1 þ 2a 2a 2
!! pffiffiffiffiffifficffi ! bn coth þ1 2 ! ! pffiffiffiffiffifficffi ! pffiffiffiffiffifficffi ! bn bn 1 3b þ 1 ln coth 2 2 2 2a ! pffiffiffiffiffifficffi ! bn 1 1 coth ; 2 2 2
where n = x ct. Moreover, we also obtain the following solutions related to the periodic solutions (34) and (35): rffiffiffi rffiffiffi cn2 6b 1 c 3b 1 c ln sec arctan tan2 vðnÞ ¼ n þ n a 2 b a 2 b 4a and
rffiffiffi rffiffiffi cn2 6b 1 c 3b c 3p pffiffiffiffiffi bp 2 1 ln csc bc þ n þ arccot cot n . vðnÞ ¼ a 2 b a 2 b 2a 2 4a
ð37Þ
ð38Þ
ð39Þ
4. The system of nonlinear PDEs 4.1. Using the sine–cosine method We next consider the system of nonlinear partial differential equations [1,2] given by 3
awxt þ avx vt ¼ 0;
vt þ vxxx þ ðvx Þ þ bvx wxx ¼ 0;
ð40Þ
where v = v(x, t) and w = w(x, t) are sufficiently often differentiable functions. This system can be converted to the system of ODEs 2
cw00 acðv0 Þ ¼ 0;
3
cv0 þ v000 þ ðv0 Þ þ bv0 w00 ¼ 0;
ð41Þ 0 2
upon using the wave variable n = x ct. Substituting w00 = a(v ) from the first equation into the second equation gives v000 cv0 þ ð1 abÞðv0 Þ3 ¼ 0;
ab 6¼ 1;
ð42Þ
that can be converted to u00 cu þ ð1 abÞu3 ¼ 0;
ð43Þ
0
upon substituting u = v . Using the cosine ansatz (5) into (43) gives ðl2 b2 k ckÞ cosb ðlnÞ þ ð1 abÞk3 cos3b ðlnÞ þ kl2 bðb 1Þ cosb2 ðlnÞ ¼ 0.
ð44Þ
j
Balancing the exponents and setting the coefficients of each cos (ln) to zero we find bðb 1Þ 6¼ 0;
3b ¼ b 2;
l2 b2 ¼ c;
hence we obtain b 6¼ 0; 1;
b ¼ 1;
pffiffiffiffiffiffi l ¼ c; c < 0;
ð1 abÞk2 ¼ l2 bðb 1Þ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c k¼ . 1 ab
4.1.1. The periodic solutions Using the previous results in (5) gives the periodic solutions for c < 0 and ab > 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2c uðx; tÞ ¼ sec½ cðx ctÞ; 1 ab
ð45Þ
ð46Þ
ð47Þ
A.-M. Wazwaz / Applied Mathematics and Computation 181 (2006) 1683–1692
and
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2c csc½ cðx ctÞ. uðx; tÞ ¼ 1 ab
1689
ð48Þ
4.1.2. The solitons solutions For c > 0, the solitons solutions for u(x, t) read rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2c uðx; tÞ ¼ sech½ cðx ctÞ; ab < 1 1 ab and rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2c csch½ cðx ctÞ; ab > 1. uðx; tÞ ¼ ab 1
ð49Þ
ð50Þ
are readily obtained. 4.1.3. Travelling wave solutions Recall that u(n) = v 0 (n) and w00 = a(v 0 )2. This in turn gives the periodic solutions for c < 0, ab > 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 vðnÞ ¼ lnðsecð cnÞ þ tanð cnÞÞ ab 1 and rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 lnðcscð cnÞ þ cotð cnÞÞ . vðnÞ ¼ ab 1 However, for c > 0 we obtain the solitons solutions rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 arctanðsinh½ cnÞ; ab < 1 vðnÞ ¼ 1 ab and rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 pffiffiffi cn ; ab > 1. vðnÞ ¼ ln tanh ab 1 2
ð51Þ
ð52Þ
ð53Þ
ð54Þ
The last four solutions for v(x, t) are obtained by considering the constants of integration to be zeros. Recall that w00 = a(v 0 )2. Using the previous results for v we obtain the following travelling wave solutions for c < 0: pffiffiffiffiffiffi pffiffiffiffiffiffi ac 2ac tan2 ð cnÞ lnðcosð cnÞÞ; ð55Þ wðnÞ ¼ 3ð1 abÞ 3ð1 abÞ pffiffiffiffiffiffi pffiffiffiffiffiffi ac 2ac wðnÞ ¼ cot2 ð cnÞ lnðsinð cnÞÞ. ð56Þ 3ð1 abÞ 3ð1 abÞ However, for c > 0, we obtain pffiffiffi pffiffiffi ac 2ac sech2 ð cnÞ lnðcoshð cnÞÞ; 3ð1 abÞ 3ð1 abÞ pffiffiffi pffiffiffi ac 2ac wðnÞ ¼ csch2 ð cnÞ þ lnðsinhð cnÞÞ. 3ð1 abÞ 3ð1 abÞ
wðnÞ ¼
ð57Þ ð58Þ
4.2. Using the tanh method In this section we will use the tanh method to handle the reduced system of (40) given by 2
cw00 acðv0 Þ ¼ 0;
3
cv0 þ v000 þ ðv0 Þ þ bv0 w00 ¼ 0;
ð59Þ
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A.-M. Wazwaz / Applied Mathematics and Computation 181 (2006) 1683–1692
upon using the wave variable n = x ct. Following [1], we substitute w00 = a(v 0 )2 from the first equation into the second equation to obtain 3
v000 cv0 þ ð1 abÞðv0 Þ ¼ 0;
ab 6¼ 1;
ð60Þ
that can be converted to u00 cu þ ð1 abÞu3 ¼ 0 ¼ 0;
ð61Þ
upon substituting u = v 0 . Balancing u00 with u3 in (61) by using (10) we find 3M ¼ M þ 2;
ð62Þ
so that M = 1. The tanh method presents the finite expansion uðnÞ ¼ a0 þ a1 Y .
ð63Þ
Substituting (63) into (61), collecting the coefficients of Y, and solving the resulting system we find one set of solutions rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi c c a1 ¼ a0 ¼ 0; ; l ¼ ; c < 0; ab > 1. ð64Þ 1 ab 2 It is obvious that we obtained only one set of parameter solutions. This means that we find: 4.2.1. The kink solutions For c < 0, ab > 1, we obtain the kink solutions rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi c c uðx; tÞ ¼ tanh ðx ctÞ 1 ab 2
ð65Þ
and uðx; tÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi c c coth ðx ctÞ . 1 ab 2
4.2.2. The periodic solutions For c > 0, ab > 1, we obtain the periodic solutions rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi c c tan ðx ctÞ uðx; tÞ ¼ ab 1 2
ð66Þ
ð67Þ
and uðx; tÞ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi c c cot ðx ctÞ . ab 1 2
4.2.3. Travelling wave solutions Recall that u(n) = v 0 (n). In view of this we obtain the travelling wave solutions rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi
1 c c vðnÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln tanh n 1 þ ln tanh n þ1 2 2 2ðab 1Þ
ð68Þ
ð69Þ
and rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi
1 c c vðnÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln coth n 1 þ ln coth n þ1 ; 2 2 2ðab 1Þ
ð70Þ
A.-M. Wazwaz / Applied Mathematics and Computation 181 (2006) 1683–1692
valid for c < 0, ab > 1. However, for c > 0, ab > 1, we find rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi 2 c vðnÞ ¼ ln sec n ab 1 2 and
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi 2 c vðnÞ ¼ ln csc n . ab 1 2
Recall that w00 = a(v 0 )2, therefore we find the following travelling wave solutions for w(n): rffiffiffiffiffiffi rffiffiffiffiffiffi 3a c 3a c a ln tanh ln tanh wðnÞ ¼ n 1 n þ1 2ðab 1Þ 2 2ðab 1Þ 2 2ðab 1Þ rffiffiffiffiffiffi rffiffiffiffiffiffi rffiffiffiffiffiffi c c 1 a ln 2 1 c 1 ln tanh ln tanh n 1 ln tanh n þ þ n 2 2 2 2ðab 1Þ 2 2 2
1691
ð71Þ
ð72Þ
ð73Þ
and rffiffiffiffiffiffi rffiffiffiffiffiffi 3a c 3a c a wðnÞ ¼ ln coth ln coth n 1 n þ1 2ðab 1Þ 2 2ðab 1Þ 2 2ðab 1Þ rffiffiffiffiffiffi rffiffiffiffiffiffi rffiffiffiffiffiffi c c 1 a ln 2 1 c 1 ln coth ln coth n 1 ln coth n þ þ n ; 2 2 2 2ðab 1Þ 2 2 2 and for c > 0, ab > 1 we find rffiffiffi rffiffiffi
a c c 2 ln sec wðnÞ ¼ n arctan tan2 n ab 1 2 2
ð74Þ
ð75Þ
and rffiffiffi rffiffiffi
rffiffiffi a c c ap c p 2 2 ln csc wðnÞ ¼ n arccot cot n nþ . ab 1 2 2 ab 1 2 4
ð76Þ
5. Concluding remarks The powerful sine–cosine and tanh methods were employed for analytic treatment for a fourth-order nonlinear PDE and a system of nonlinear PDEs. Solitary wave solutions, kinks, and periodic solutions were formally derived The performance of the two schemes show that the two methods are reliable and effective. The sine–cosine method can be used directly and in a straightforward manner. The tanh method requires transformation formula if the balancing process gives a fractional result for M. Unlike other works where the two methods provide the same set of travelling wave solutions, it is shown in this work that the two methods do not necessarily provide the same solutions. References [1] A. Huber, Solitary solutions of some nonlinear evolution equations, Appl. Math. Comput. 156 (2005) 464–474. [2] A. Ramani, Inverse scattering, ordinary differential equations of Painleve’-type and Hirota’s bilinear formalism, Ann. New York Acad. Sci. (1981) 54–64. [3] S. Kichenassamy, P. Olver, Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, SIAM J. Math. Anal. 23 (5) (1992) 1141–1166. [4] P. Rosenau, J.M. Hyman, Compactons: Solitons with finite wavelengths, Phys. Rev. Lett. 70 (5) (1993) 564–567. [5] P. Rosenau, A. Pikovsky, Phase compactons in chains of dispersively coupled oscillations, Phys. Rev. Lett. 94 (174102) (2005) 1–4. [6] D. Baldwin, U. Goktas, W. Hereman, L. Hong, R.S. Martino, J.C. Miller, Symbolic computation of exact solutions in hyperbolic and elliptic functions for nonlinear PDEs, J. Symbol. Comput. 37 (2004) 669–705. [7] U. Goktas, W. Hereman, Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symbol. Comput. 24 (1997) 591–621.
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W. Malfliet, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scr. 54 (1996) 563–568. W. Malfliet, The tanh method: II. Perturbation technique for conservative systems, Phys. Scr. 54 (1996) 569–575. A.M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput. 154 (3) (2004) 713–723. A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. A.M. Wazwaz, Compactons in a class of nonlinear dispersive equations, Math. Comput. Model. 37 (3/4) (2003) 333–341. A.M. Wazwaz, Distinct variants of the KdV equation with compact and noncompact structures, Appl. Math. Comput. 150 (2004) 365–377. [14] A.M. Wazwaz, Variants of the generalized KdV equation with compact and noncompact structures, Comput. Math. Appl. 47 (2004) 583–591.