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The extended tanh method for abundant solitary wave solutions of nonlinear wave equations
Applied Mathematics and Computation 187 (2007) 1131–1142 www.elsevier.com/locate/amc
The extended tanh method for abundant solitary wave solutions of nonlinear wave equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA
Abstract The extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations. The obtained solutions include solitons, kinks and plane periodic solutions. The study is an extension to the remarkable development by Malfliet [1]. The extended tanh method presents a wider applicability for handling nonlinear wave equations. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Extended tanh method; Solitons; Kinks; Nonlinear wave equations; Dispersion; Dissipation
1. Introduction Nonlinear wave equations have a significant role in several scientific and engineering fields. These equations appear in solid state physics, fluid mechanics, chemical kinetics, plasma physics, population models, nonlinear optics, propagation of fluxons in Josephson junctions, and many others. The pioneer work of Malfliet in [1] introduced the powerful hyperbolic tangent (tanh) method for a reliable treatment of the nonlinear wave equations. The useful tanh method is widely used by many such as in [2–11] and by the references therein. The method introduces a unifying method that one can find exact as well as approximate solutions in a straightforward and systematic way [1–4]. The tanh method has been subjected to many modifications that mainly depend on the Riccati equation and the solutions of well-known equations. The standard tanh method and the proposed modifications all depend on the balance method, where the linear terms of highest order are balanced with the highest order nonlinear terms of the reduced equation. It is the aim of this work to further complement the studies of [1–11]. It is interesting to point out that in these works, only one solitary wave solution was obtained for each investigated equation. We aim to use the extended tanh method [12,13] to obtain many solutions to these equations instead of one. We also want to confirm the wider applicability of the extended tanh method. In what follows, we highlight the main steps of these two methods briefly. More details can be found in [14–22] and the references therein. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.013
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2. The methods A PDE P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0
ð1Þ
can be converted to an ODE Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð2Þ
upon using a wave variable n = (x ct). Eq. (2) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. 2.1. The tanh method The tanh method developed by Malfliet in [1], and used in [2–13] among many others, introduces a new independent variable Y ¼ tanhðlnÞ;
n ¼ x ct;
ð3Þ
that 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
ð4Þ
d4 d d2 4 2 2 4 2 2 2 þ 4l ¼ 8l Y ð1 Y Þð3Y 2Þ ð1 Y Þ ð9Y 2Þ dY dY 2 dn4 12l4 Y ð1 Y 2 Þ3
d3 d4 þ l4 ð1 Y 2 Þ4 4 : 3 dY dY
The solution is therefore expressed in a finite series expansion uðlnÞ ¼ SðY Þ ¼
M X
ak Y k ;
ð5Þ
k¼0
where M will be obtained by balancing the linear term of highest order with nonlinear terms. 2.2. The extended tanh method The extended tanh method [12,13] follows the assumptions made in (3) and (4), and then admits the use of the finite expansion uðlnÞ ¼ SðY Þ ¼
M X k¼0
ak Y k þ
M X
bk Y k ;
ð6Þ
k¼1
where M is a positive integer, for both methods, that will be determined. Expansion (6) reduces to the standard tanh method [1] for bk = 0, 1 6 k 6 M. The parameter M is usually obtained, as stated before, by balancing the linear terms of highest order in the resulting equation with the highest order nonlinear terms. If M is not an integer, then a transformation formula should be used to overcome this difficulty. Substituting (6) into the ODE results in an algebraic system of equations in powers of Y that will lead to the determination of the parameters ak (k = 0, . . . , M), l, and c.
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3. The Burgers equation The Burgers equation is characterized by a dissipative term and models fluid turbulence. It reads ut þ uux uxx ¼ 0:
ð7Þ
Using the wave variable n = x ct carries Eq. (7) into the ODE 1 cu þ u2 u0 ¼ 0 2
ð8Þ
obtained after integrating the ODE once and setting the constant of integration equal to zero. Balancing uu 0 with u2 in (8) gives M þ 1 ¼ 2M;
ð9Þ
so that M ¼ 1:
ð10Þ
The extended tanh method (6) admits the use of the finite expansion uðnÞ ¼ a0 þ a1 Y þ
b1 : Y
ð11Þ
Substituting (11) into (8), and collecting the coefficients of Y we obtain a system of algebraic equations for a0, a1, b1, and l, and by solving this system we obtain the three sets of solutions (i) The first set: a0 ¼ c;
b1 ¼ 0;
c l¼ : 2
ð12Þ
b1 ¼ c;
c l¼ : 2
ð13Þ
a1 ¼ c;
(ii) The second set: a0 ¼ c;
a1 ¼ 0;
(iii) The third set: a0 ¼ c;
c a1 ¼ ; 2
c b1 ¼ ; 2
c l¼ : 4
ð14Þ
In view of this we obtain the following kinks solutions hc i u1 ðx; tÞ ¼ c 1 tanh ðx ctÞ ; h c2 i u2 ðx; tÞ ¼ c 1 coth ðx ctÞ 2
ð15Þ ð16Þ
and hc i 1 hc i 1 u3 ðx; tÞ ¼ c 1 tanh ðx ctÞ coth ðx ctÞ ; 2 4 2 4
ð17Þ
where c is left as a free parameter. It is obvious that three pairs of solutions were obtained by using the extended tanh method, whereas only one solution was obtained in [1–4]. 4. The KdV equation The KdV equation is characterized by the convection term uux and the dispersion term uxxx. It reads ut þ 6uux þ uxxx ¼ 0:
ð18Þ
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Using the wave variable n = x ct carries Eq. (18) into the ODE cu þ 3u2 þ u00 ¼ 0
ð19Þ
obtained after integrating the ODE once and setting the constant of integration equal to zero. Balancing u00 with u2 in (19) gives M þ 2 ¼ 2M;
ð20Þ
so that M ¼ 2:
ð21Þ
The extended tanh method (6) admits the use of the finite expansion uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 þ
b1 b2 þ : Y Y2
ð22Þ
Substituting (22) into (19), collecting the coefficients of Y we obtain the following system of algebraic equations for a0, a1, a2, b1, b2 and l, and solving this system we obtain the six sets of solutions (i) The first set: c a0 ¼ ; 6
a1 ¼ b1 ¼ b2 ¼ 0;
c a2 ¼ ; 2
l¼
1 pffiffiffiffiffiffi c: 2
ð23Þ
c b2 ¼ ; 2
l¼
1 pffiffiffiffiffiffi c: 2
ð24Þ
c a2 ¼ ; 2
l¼
1 pffiffiffi c: 2
ð25Þ
c b2 ¼ ; 2
l¼
1 pffiffiffi c: 2
ð26Þ
(ii) The second set: c a0 ¼ ; 6
a1 ¼ b1 ¼ a2 ¼ 0;
(iii) The third set: c a0 ¼ ; 2
a1 ¼ b1 ¼ b2 ¼ 0;
(iv) The fourth set: c a0 ¼ ; a1 ¼ b1 ¼ a2 ¼ 0; 2 (v) The fifth set: a0 ¼
c ; 12
a1 ¼ b1 ¼ 0;
c a2 ¼ b2 ¼ ; 8
l¼
1 pffiffiffi c: 4
ð27Þ
(vi) The sixth set: c a0 ¼ ; 4
a1 ¼ b1 ¼ 0;
c a2 ¼ b2 ¼ ; 8
l¼
1 pffiffiffi c; 4
ð28Þ
where c is left as a free parameter. The first set gives the soliton solution c 2 1 pffiffiffiffiffiffi 1 3tanh cðx ctÞ ; c < 0 u1 ðx; tÞ ¼ 6 2 and the periodic solution c 1 pffiffiffi u2 ðx; tÞ ¼ 1 þ 3 tan2 cðx ctÞ ; c > 0: 6 2 The second set gives the soliton solution c 2 1 pffiffiffiffiffiffi u3 ðx; tÞ ¼ 1 3coth cðx ctÞ ; c < 0 6 2
ð29Þ
ð30Þ
ð31Þ
A.-M. Wazwaz / Applied Mathematics and Computation 187 (2007) 1131–1142
and the periodic solution pffiffiffi c 2 1 1 þ 3cot cðx ctÞ ; u4 ðx; tÞ ¼ 6 2
ð32Þ
c > 0:
The third set gives the soliton solution c 2 1 pffiffiffi u5 ðx; tÞ ¼ sech cðx ctÞ ; c > 0 2 2 and the periodic solution c 1 pffiffiffiffiffiffi u6 ðx; tÞ ¼ sec2 cðx ctÞ ; 2 2
ð33Þ
ð34Þ
c < 0:
The fourth set gives the soliton solution c 2 1 pffiffiffi u7 ðx; tÞ ¼ csch cðx ctÞ ; c > 0 2 2 and the periodic solution c 2 1 pffiffiffiffiffiffi u8 ðx; tÞ ¼ csc cðx ctÞ ; 2 2
ð35Þ
ð36Þ
c < 0:
The fifth set gives the soliton solution c 3 2 1 pffiffiffiffiffiffi 2 1 pffiffiffiffiffiffi u9 ðx; tÞ ¼ 1þ tanh cðx ctÞ þ coth cðx ctÞ ; 12 2 4 4 and the periodic solution c 3 1 pffiffiffi 1 pffiffiffi u10 ðx; tÞ ¼ 1 tan2 cðx ctÞ þ cot2 cðx ctÞ ; 12 2 4 4 Finally, the last set gives the soliton solution c 1 pffiffiffi 1 pffiffiffi u11 ðx; tÞ ¼ 2 tanh2 cðx ctÞ þ coth2 cðx ctÞ ; 8 4 4 and the periodic solution pffiffiffiffiffiffi pffiffiffiffiffiffi c 2 1 2 1 u12 ðx; tÞ ¼ 2 þ tan cðx ctÞ cot cðx ctÞ ; 8 4 4
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c<0
ð37Þ
c > 0:
ð38Þ
c>0
ð39Þ
c < 0:
ð40Þ
Although one bell-shaped soliton solution is obtained in [1], six pairs of solutions were obtained by using the extended tanh method. 5. The mKdV equation The modified KdV (mKdV) equation serves as a model of solitons in a multicomponent plasma and phonons in anharmonic lattice [1]. The mKdV reads ut þ 6u2 ux þ uxxx ¼ 0:
ð41Þ
Proceeding as before, the wave variable n = x ct carries Eq. (41) into the ODE cu þ 2u3 þ u00 ¼ 0:
ð42Þ
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Balancing u00 with u3 in (42) gives M þ 2 ¼ 3M; ð43Þ so that M ¼ 1: ð44Þ It is interesting to point out that in [1], it is stated that for M = 1 the tanh method does not lead to a real solution. Similarly, the extended tanh method gives only complex solutions. Moreover, a specific ansatz of the form 1
SðY Þ ¼ cð1 Y 2 Þ2
ð45Þ
was used to obtain the soliton solution pffiffiffi pffiffiffi uðx; tÞ ¼ c sechð cðx ctÞÞ
ð46Þ
for c > 0. 6. The Fisher equation The Fisher equation describes the process of interaction between diffusion and reaction. This equation is encountered in chemical kinetics and population dynamics which includes problems such as nonlinear evolution of a population in one-dimensional habitual, neutron population in a nuclear reaction [1]. It reads ut uxx uð1 uÞ ¼ 0:
ð47Þ
Using the wave variable n = x ct carries Eq. (47) into the ODE cu0 u00 uð1 uÞ ¼ 0:
ð48Þ
2
00
Balancing u with u in (48) gives M þ 2 ¼ 2M; so that M ¼ 2:
ð49Þ ð50Þ
The extended tanh method (6) applies the finite expansion b1 b2 uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 þ þ 2 : ð51Þ Y Y Substituting (51) into (48), collecting the coefficients of Y we obtain the following system of algebraic equations for a0, a1, a2, b1, b2 and l, and solving this system we obtain real and complex sets of solutions. In what follows, we list only the three sets of real solutions (i) The first set: 1 a0 ¼ ; 4
1 a1 ¼ ; 2
1 a2 ¼ ; 4
b1 ¼ b2 ¼ 0;
5 c ¼ pffiffiffi ; 6
1 l ¼ pffiffiffi : 2 6
ð52Þ
5 c ¼ pffiffiffi ; 6
1 l ¼ pffiffiffi : 2 6
ð53Þ
(ii) The second set: 1 a0 ¼ ; 4
a1 ¼ a2 ¼ 0;
1 b1 ¼ ; 2
1 b2 ¼ ; 4
(iii) The third set: 3 a0 ¼ ; 8
1 a1 ¼ ; 4
a2 ¼
1 ; 16
1 b1 ¼ ; 4
The first set gives the soliton solution 2 1 1 5 1 tanh2 pffiffiffi x pffiffiffi t u1 ðx; tÞ ¼ : 4 2 6 6
b2 ¼
1 ; 16
5 c ¼ pffiffiffi ; 6
1 l ¼ pffiffiffi : 4 6
ð54Þ
ð55Þ
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1137
The second set gives the soliton solution 2 1 1 5 2 1 coth pffiffiffi x pffiffiffi t u2 ðx; tÞ ¼ : 4 2 6 6
ð56Þ
The third set gives the soliton solution 1 1 5 1 5 2 p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi 6 4 tanh u3 ðx; tÞ ¼ x t þ tanh x t 16 4 6 6 2 6 6 1 1 5 1 5 2 4 coth pffiffiffi x pffiffiffi t þ coth pffiffiffi x pffiffiffi t þ : 16 4 6 6 2 6 6
ð57Þ
7. The Burgers–Fisher equation The Burgers–Fisher equation reads ut þ uux þ uxx þ uð1 uÞ ¼ 0;
ð58Þ
that will be carried into the ODE cu0 þ uu0 þ u00 þ uð1 uÞ ¼ 0
ð59Þ 0
00
obtained after using the wave variable n = x ct. Balancing uu with u gives M þ 2 ¼ 2M þ 1;
ð60Þ
so that M ¼ 1:
ð61Þ
Accordingly, the extended tanh method (6) is of the form uðnÞ ¼ a0 þ a1 Y þ
b1 : Y
ð62Þ
Substituting (62) into (59), and proceeding as before we obtain the three sets of solutions (i) The first set: 1 1 a0 ¼ ; a1 ¼ ; 2 2 (ii) The second set: 1 a0 ¼ ; 2
a1 ¼ 0;
b1 ¼ 0;
5 c¼ ; 2
1 l¼ : 4
ð63Þ
1 b1 ¼ ; 2
5 c¼ ; 2
1 l¼ : 4
ð64Þ
1 b1 ¼ ; 4
5 c¼ ; 2
1 l¼ : 8
ð65Þ
(iii) The third set: 1 a0 ¼ ; 2
1 a1 ¼ ; 4
This in turn gives the following kinks solutions 1 1 5 1 þ tanh x t ; 2 4 2 1 1 5 1 þ coth x t ; u2 ðx; tÞ ¼ 2 4 2
u1 ðx; tÞ ¼
ð66Þ ð67Þ
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and u3 ðx; tÞ ¼
1 1 5 1 5 2 þ tanh x t þ coth x t : 4 8 2 8 2
ð68Þ
8. The Huxley equation The Huxley equation reads ut auxx uðk un Þðun 1Þ ¼ 0;
ð69Þ
n > 1:
This equation is used for nerve propagation in neurophysics and wall propagation in liquid crystals [10]. The wave variable n = x ct carries out Eq. (69) into the ODE cu0 au00 ðk þ 1Þunþ1 þ u2nþ1 þ ku ¼ 0:
ð70Þ
Balancing u00 with u2n+1 gives M þ 2 ¼ ð2n þ 1ÞM;
ð71Þ
so that 1 M¼ : n
ð72Þ
To obtain a closed form solution, M should be an integer, therefore we use the transformation 1
uðx; tÞ ¼ vn and as a result Eq. (70) becomes
ð73Þ 2
cnvv0 anvv00 að1 nÞðv0 Þ ðk þ 1Þn2 v3 þ n2 v4 þ kn2 v2 ¼ 0:
ð74Þ
4
Balancing vv00 with v gives M = 1. Consequently, the extended tanh method (6) becomes b1 vðnÞ ¼ a0 þ a1 Y þ : Y Substituting (75) into (74), and proceeding as before we obtain the sets of solutions
ð75Þ
(i) The first set: 1 a0 ¼ ; 2
1 a1 ¼ ; 2
b1 ¼ 0;
c ¼ c;
n l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð76Þ
1 b1 ¼ ; 2
c ¼ c;
n l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð77Þ
(ii) The second set: 1 a0 ¼ ; 2
a1 ¼ 0;
(iii) The third set: 1 a0 ¼ ; 2
1 a1 ¼ ; 4
1 b1 ¼ ; 4
c ¼ c;
n l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð78Þ
(iv) The fourth set: k a0 ¼ ; 2
k a1 ¼ ; 2
b1 ¼ 0;
c ¼ k;
kn l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð79Þ
k b1 ¼ ; 2
c ¼ k;
kn l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð80Þ
(v) The fifth set: k a0 ¼ ; 2
a1 ¼ 0;
A.-M. Wazwaz / Applied Mathematics and Computation 187 (2007) 1131–1142
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(vi) The sixth set: k a0 ¼ ; 2
k a1 ¼ ; 4
k b1 ¼ ; 4
c ¼ k;
kn l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 aðn þ 1Þ
ð81Þ
where
rffiffiffiffiffiffiffiffiffiffiffi a ; c ¼ ðkðn þ 1Þ 1Þ nþ1 rffiffiffiffiffiffiffiffiffiffiffi a : k ¼ ðn k þ 1Þ nþ1
ð82Þ
1
Recall that u ¼ vn . Based on this, the first three sets give the following kinks solutions: ( " #!)1n 1 n 1 tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ctÞ u1 ðx; tÞ ¼ ; 2 2 aðn þ 1Þ ( u2 ðx; tÞ ¼
ð83Þ
" #!)1n 1 n 1 coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ctÞ 2 2 aðn þ 1Þ
ð84Þ
" # " #!)1n 1 n n 2 tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ctÞ coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ctÞ : 4 4 aðn þ 1Þ 4 aðn þ 1Þ
ð85Þ
and ( u3 ðx; tÞ ¼
The last three sets lead to the kinks solutions ( " #!)1n k kn 1 tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ktÞ ; u4 ðx; tÞ ¼ 2 2 aðn þ 1Þ ( " #!)1n 1 kn 1 coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ktÞ u5 ðx; tÞ ¼ 2 2 aðn þ 1Þ and
( u6 ðx; tÞ ¼
" # " #!)1n k kn kn 2 tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ktÞ coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ktÞ : 4 4 aðn þ 1Þ 4 aðn þ 1Þ
ð86Þ
ð87Þ
ð88Þ
9. The nonlinear reaction–diffusion equations The nonlinear reaction–diffusion equation appear in many forms [1,10]. In this section we will focus our work on two of these reaction–diffusion equations, namely ut ðu2 Þxx uð1 uÞ ¼ 0
ð89Þ
ut ðu3 Þxx uð1 u2 Þ ¼ 0:
ð90Þ
and
9.1. First type Using the wave variable n = x ct carries Eq. (89) into the ODE 00
cu0 ðu2 Þ uð1 uÞ ¼ 0
ð91Þ
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or equivalently 2
cu0 2uu00 2ðu0 Þ u þ u2 ¼ 0: 00
ð92Þ
0
Balancing uu with u gives 2M þ 2 ¼ M þ 1;
ð93Þ
so that M ¼ 1:
ð94Þ
In [1], an alternative series expansion given by uðnÞ ¼
1 A ¼ a0 þ a1 Y 1 þ aY
ð95Þ
was used to obtain one singular solution that blows up as x ! 1. However, to obtain more solutions, the extended tanh method (6) can be used in the form uðnÞ ¼
1 A ¼ : b1 1 þ aY þ Yb a0 þ a1 Y þ Y
ð96Þ
Substituting (96) into (92), collecting the coefficients of Y and solving the resulting system we obtain the following solutions: (i) The first set: A ¼ 2;
b ¼ 0;
c ¼ 1;
1 l¼ : 4
ð97Þ
b ¼ 1;
c ¼ 1;
1 l¼ : 4
ð98Þ
a ¼ 1;
(ii) The second set: A ¼ 2;
a ¼ 0;
(iii) The third set: A ¼ 2;
1 a¼ ; 2
1 b¼ ; 2
c ¼ 1;
1 l¼ : 8
ð99Þ
In view of these results, we obtain the three pairs of singular solutions u1 ðx; tÞ ¼
1 tanh
2 1 4
ðx tÞ
:
ð100Þ
The second set gives the soliton solution u2 ðx; tÞ ¼
1 coth
2 1 4
ðx tÞ
ð101Þ
and u3 ðx; tÞ ¼
2
: 1 tanh 8 ðx tÞ 12 coth 18 ðx tÞ 1 2
1
ð102Þ
9.2. Second type Using the wave variable n = x ct carries Eq. (90) into the ODE 00
cu0 ðu3 Þ uð1 u2 Þ ¼ 0
ð103Þ
A.-M. Wazwaz / Applied Mathematics and Computation 187 (2007) 1131–1142
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or equivalently 2
cu0 3u2 u00 6uðu0 Þ u þ u3 ¼ 0: 2 00
ð104Þ
0
Balancing u u with u gives 3M þ 2 ¼ M þ 1;
ð105Þ
so that 1 M ¼ : 2
ð106Þ
Proceeding as before, the extended tanh method (6) can be used in the form !12 12 1 A uðnÞ ¼ ¼ : 1 þ aY þ Yb a0 þ a1 Y þ bY1
ð107Þ
Substituting (107) into (104), collecting the coefficients of Y and solving the resulting system we obtain the following solutions (i) The first set: A ¼ 2;
b ¼ 0;
c ¼ 1;
1 l¼ : 3
ð108Þ
b ¼ 1;
c ¼ 1;
1 l¼ : 3
ð109Þ
a ¼ 1;
(ii) The second set: A ¼ 2;
a ¼ 0;
(iii) The third set: A ¼ 2;
1 a¼ ; 2
1 b¼ ; 2
c ¼ 1;
1 l¼ : 6
In view of these results, we obtain the three pairs of singular solutions !12 2
: u1 ðx; tÞ ¼ 1 tanh 13 ðx tÞ
ð110Þ
ð111Þ
The second set gives the soliton solution u2 ðx; tÞ ¼
1 coth
2 1 3
!12 ðx tÞ
ð112Þ
and u3 ðx; tÞ ¼
!12 2
: 1 12 tanh 16 ðx tÞ 12 coth 16 ðx tÞ
ð113Þ
10. Concluding remarks The extended tanh method [12,13] was successfully used to establish abundant solitary wave solutions, mostly solitons and kinks solutions. Many well known nonlinear wave equations were handled by this method to show the new solutions compared to the solutions obtained in [1–4]. The performance of the extended tanh method is reliable and effective and gives more solutions. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave equations.
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References [1] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (7) (1992) 650–654. [2] W. Malfliet, W. Hereman, The tanh method. I: Exact solutions of nonlinear evolution and wave equations, Phys. Scr. 54 (1996) 563– 568. [3] W. Malfliet, W. Hereman, The tanh method. II: Perturbation technique for conservative systems, Phys. Scr. 54 (1996) 569–575. [4] W. Hereman, W. Malfliet, in: The tanh method: a tool to solve nonlinear partial differential equations with symbolic software, Proceedings 9th World Multi-Conference on Systemics, Cybernetics and Informatics, Orlando, FL, 2005, pp. 165–168. [5] A.M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput. 154 (3) (2004) 713–723. [6] A.M. Wazwaz, The tanh method: exact solutions of the Sine-Gordon and the Sinh-Gordon equations, Appl. Math. Comput. 49 (2005) 565–574. [7] A.M. Wazwaz, The tanh and the sine–cosine methods for a reliable treatment of the modified equal width equation and its variants, Comm. Nonlinear Sci. Numer. Simul. 11 (2) (2006) 148–160. [8] A.M. Wazwaz, The tanh and the sine–cosine methods for compact and noncompact solutions of the nonlinear Klein–Gordon equation, Appl. Math. Comput. 167 (2) (2005) 1179–1195. [9] A.M. Wazwaz, The tanh method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd– Bullough equations, Chaos, Solitons Fractals 25 (1) (2005) 55–63. [10] A.M. Wazwaz, The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations, Appl. Math. Comput. 169 (2005) 321–338. [11] A.M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations, Appl. Math. Comput. 169 (2005) 639–656. [12] A.M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.07.002. [13] A.M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis–Procesi and Camass–Holm equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.07.092. [14] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [15] A.M. Wazwaz, Solitary wave solutions for modified forms of the Degasperis–Procesi and Camassa–Holm and equations, Phys. Lett. A 352 (2006) 500–504. [16] A.M. Wazwaz, A study on nonlinear dispersive partial differential equations of compact and noncompact solutions, Appl. Math. Comput. 135 (2–3) (2003) 399–409. [17] A.M. Wazwaz, A construction of compact and noncompact solutions of nonlinear dispersive equations of even order, Appl. Math. Comput. 135 (2–3) (2003) 324–411. [18] A.M. Wazwaz, Compactons in a class of nonlinear dispersive equations, Math. Comput. Modell. 37 (3–4) (2003) 333–341. [19] A.M. Wazwaz, Variants of the two-dimensional Boussinesq equations with compactons, solitons and periodic solutions, Comput. Math. Appl. 49 (2005) 295–301. [20] A.M. Wazwaz, New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations, Chaos, Solitons Compact. 22 (1) (2004) 249–260. [21] A.M. Wazwaz, Special types of the nonlinear dispersive Zakharov–Kuznetsov equation with compactons, solitons and periodic solutions, Int. J. Comput. Math. 81 (9) (2004) 1107–1119. [22] A.M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa–Holm equation with compact and noncompact solutions, Appl. Math. Comput. 165 (2005) 485–501.
The extended tanh method for abundant solitary wave solutions of nonlinear wave equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA
Abstract The extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations. The obtained solutions include solitons, kinks and plane periodic solutions. The study is an extension to the remarkable development by Malfliet [1]. The extended tanh method presents a wider applicability for handling nonlinear wave equations. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Extended tanh method; Solitons; Kinks; Nonlinear wave equations; Dispersion; Dissipation
1. Introduction Nonlinear wave equations have a significant role in several scientific and engineering fields. These equations appear in solid state physics, fluid mechanics, chemical kinetics, plasma physics, population models, nonlinear optics, propagation of fluxons in Josephson junctions, and many others. The pioneer work of Malfliet in [1] introduced the powerful hyperbolic tangent (tanh) method for a reliable treatment of the nonlinear wave equations. The useful tanh method is widely used by many such as in [2–11] and by the references therein. The method introduces a unifying method that one can find exact as well as approximate solutions in a straightforward and systematic way [1–4]. The tanh method has been subjected to many modifications that mainly depend on the Riccati equation and the solutions of well-known equations. The standard tanh method and the proposed modifications all depend on the balance method, where the linear terms of highest order are balanced with the highest order nonlinear terms of the reduced equation. It is the aim of this work to further complement the studies of [1–11]. It is interesting to point out that in these works, only one solitary wave solution was obtained for each investigated equation. We aim to use the extended tanh method [12,13] to obtain many solutions to these equations instead of one. We also want to confirm the wider applicability of the extended tanh method. In what follows, we highlight the main steps of these two methods briefly. More details can be found in [14–22] and the references therein. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.013
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2. The methods A PDE P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0
ð1Þ
can be converted to an ODE Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð2Þ
upon using a wave variable n = (x ct). Eq. (2) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. 2.1. The tanh method The tanh method developed by Malfliet in [1], and used in [2–13] among many others, introduces a new independent variable Y ¼ tanhðlnÞ;
n ¼ x ct;
ð3Þ
that 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
ð4Þ
d4 d d2 4 2 2 4 2 2 2 þ 4l ¼ 8l Y ð1 Y Þð3Y 2Þ ð1 Y Þ ð9Y 2Þ dY dY 2 dn4 12l4 Y ð1 Y 2 Þ3
d3 d4 þ l4 ð1 Y 2 Þ4 4 : 3 dY dY
The solution is therefore expressed in a finite series expansion uðlnÞ ¼ SðY Þ ¼
M X
ak Y k ;
ð5Þ
k¼0
where M will be obtained by balancing the linear term of highest order with nonlinear terms. 2.2. The extended tanh method The extended tanh method [12,13] follows the assumptions made in (3) and (4), and then admits the use of the finite expansion uðlnÞ ¼ SðY Þ ¼
M X k¼0
ak Y k þ
M X
bk Y k ;
ð6Þ
k¼1
where M is a positive integer, for both methods, that will be determined. Expansion (6) reduces to the standard tanh method [1] for bk = 0, 1 6 k 6 M. The parameter M is usually obtained, as stated before, by balancing the linear terms of highest order in the resulting equation with the highest order nonlinear terms. If M is not an integer, then a transformation formula should be used to overcome this difficulty. Substituting (6) into the ODE results in an algebraic system of equations in powers of Y that will lead to the determination of the parameters ak (k = 0, . . . , M), l, and c.
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3. The Burgers equation The Burgers equation is characterized by a dissipative term and models fluid turbulence. It reads ut þ uux uxx ¼ 0:
ð7Þ
Using the wave variable n = x ct carries Eq. (7) into the ODE 1 cu þ u2 u0 ¼ 0 2
ð8Þ
obtained after integrating the ODE once and setting the constant of integration equal to zero. Balancing uu 0 with u2 in (8) gives M þ 1 ¼ 2M;
ð9Þ
so that M ¼ 1:
ð10Þ
The extended tanh method (6) admits the use of the finite expansion uðnÞ ¼ a0 þ a1 Y þ
b1 : Y
ð11Þ
Substituting (11) into (8), and collecting the coefficients of Y we obtain a system of algebraic equations for a0, a1, b1, and l, and by solving this system we obtain the three sets of solutions (i) The first set: a0 ¼ c;
b1 ¼ 0;
c l¼ : 2
ð12Þ
b1 ¼ c;
c l¼ : 2
ð13Þ
a1 ¼ c;
(ii) The second set: a0 ¼ c;
a1 ¼ 0;
(iii) The third set: a0 ¼ c;
c a1 ¼ ; 2
c b1 ¼ ; 2
c l¼ : 4
ð14Þ
In view of this we obtain the following kinks solutions hc i u1 ðx; tÞ ¼ c 1 tanh ðx ctÞ ; h c2 i u2 ðx; tÞ ¼ c 1 coth ðx ctÞ 2
ð15Þ ð16Þ
and hc i 1 hc i 1 u3 ðx; tÞ ¼ c 1 tanh ðx ctÞ coth ðx ctÞ ; 2 4 2 4
ð17Þ
where c is left as a free parameter. It is obvious that three pairs of solutions were obtained by using the extended tanh method, whereas only one solution was obtained in [1–4]. 4. The KdV equation The KdV equation is characterized by the convection term uux and the dispersion term uxxx. It reads ut þ 6uux þ uxxx ¼ 0:
ð18Þ
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Using the wave variable n = x ct carries Eq. (18) into the ODE cu þ 3u2 þ u00 ¼ 0
ð19Þ
obtained after integrating the ODE once and setting the constant of integration equal to zero. Balancing u00 with u2 in (19) gives M þ 2 ¼ 2M;
ð20Þ
so that M ¼ 2:
ð21Þ
The extended tanh method (6) admits the use of the finite expansion uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 þ
b1 b2 þ : Y Y2
ð22Þ
Substituting (22) into (19), collecting the coefficients of Y we obtain the following system of algebraic equations for a0, a1, a2, b1, b2 and l, and solving this system we obtain the six sets of solutions (i) The first set: c a0 ¼ ; 6
a1 ¼ b1 ¼ b2 ¼ 0;
c a2 ¼ ; 2
l¼
1 pffiffiffiffiffiffi c: 2
ð23Þ
c b2 ¼ ; 2
l¼
1 pffiffiffiffiffiffi c: 2
ð24Þ
c a2 ¼ ; 2
l¼
1 pffiffiffi c: 2
ð25Þ
c b2 ¼ ; 2
l¼
1 pffiffiffi c: 2
ð26Þ
(ii) The second set: c a0 ¼ ; 6
a1 ¼ b1 ¼ a2 ¼ 0;
(iii) The third set: c a0 ¼ ; 2
a1 ¼ b1 ¼ b2 ¼ 0;
(iv) The fourth set: c a0 ¼ ; a1 ¼ b1 ¼ a2 ¼ 0; 2 (v) The fifth set: a0 ¼
c ; 12
a1 ¼ b1 ¼ 0;
c a2 ¼ b2 ¼ ; 8
l¼
1 pffiffiffi c: 4
ð27Þ
(vi) The sixth set: c a0 ¼ ; 4
a1 ¼ b1 ¼ 0;
c a2 ¼ b2 ¼ ; 8
l¼
1 pffiffiffi c; 4
ð28Þ
where c is left as a free parameter. The first set gives the soliton solution c 2 1 pffiffiffiffiffiffi 1 3tanh cðx ctÞ ; c < 0 u1 ðx; tÞ ¼ 6 2 and the periodic solution c 1 pffiffiffi u2 ðx; tÞ ¼ 1 þ 3 tan2 cðx ctÞ ; c > 0: 6 2 The second set gives the soliton solution c 2 1 pffiffiffiffiffiffi u3 ðx; tÞ ¼ 1 3coth cðx ctÞ ; c < 0 6 2
ð29Þ
ð30Þ
ð31Þ
A.-M. Wazwaz / Applied Mathematics and Computation 187 (2007) 1131–1142
and the periodic solution pffiffiffi c 2 1 1 þ 3cot cðx ctÞ ; u4 ðx; tÞ ¼ 6 2
ð32Þ
c > 0:
The third set gives the soliton solution c 2 1 pffiffiffi u5 ðx; tÞ ¼ sech cðx ctÞ ; c > 0 2 2 and the periodic solution c 1 pffiffiffiffiffiffi u6 ðx; tÞ ¼ sec2 cðx ctÞ ; 2 2
ð33Þ
ð34Þ
c < 0:
The fourth set gives the soliton solution c 2 1 pffiffiffi u7 ðx; tÞ ¼ csch cðx ctÞ ; c > 0 2 2 and the periodic solution c 2 1 pffiffiffiffiffiffi u8 ðx; tÞ ¼ csc cðx ctÞ ; 2 2
ð35Þ
ð36Þ
c < 0:
The fifth set gives the soliton solution c 3 2 1 pffiffiffiffiffiffi 2 1 pffiffiffiffiffiffi u9 ðx; tÞ ¼ 1þ tanh cðx ctÞ þ coth cðx ctÞ ; 12 2 4 4 and the periodic solution c 3 1 pffiffiffi 1 pffiffiffi u10 ðx; tÞ ¼ 1 tan2 cðx ctÞ þ cot2 cðx ctÞ ; 12 2 4 4 Finally, the last set gives the soliton solution c 1 pffiffiffi 1 pffiffiffi u11 ðx; tÞ ¼ 2 tanh2 cðx ctÞ þ coth2 cðx ctÞ ; 8 4 4 and the periodic solution pffiffiffiffiffiffi pffiffiffiffiffiffi c 2 1 2 1 u12 ðx; tÞ ¼ 2 þ tan cðx ctÞ cot cðx ctÞ ; 8 4 4
1135
c<0
ð37Þ
c > 0:
ð38Þ
c>0
ð39Þ
c < 0:
ð40Þ
Although one bell-shaped soliton solution is obtained in [1], six pairs of solutions were obtained by using the extended tanh method. 5. The mKdV equation The modified KdV (mKdV) equation serves as a model of solitons in a multicomponent plasma and phonons in anharmonic lattice [1]. The mKdV reads ut þ 6u2 ux þ uxxx ¼ 0:
ð41Þ
Proceeding as before, the wave variable n = x ct carries Eq. (41) into the ODE cu þ 2u3 þ u00 ¼ 0:
ð42Þ
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Balancing u00 with u3 in (42) gives M þ 2 ¼ 3M; ð43Þ so that M ¼ 1: ð44Þ It is interesting to point out that in [1], it is stated that for M = 1 the tanh method does not lead to a real solution. Similarly, the extended tanh method gives only complex solutions. Moreover, a specific ansatz of the form 1
SðY Þ ¼ cð1 Y 2 Þ2
ð45Þ
was used to obtain the soliton solution pffiffiffi pffiffiffi uðx; tÞ ¼ c sechð cðx ctÞÞ
ð46Þ
for c > 0. 6. The Fisher equation The Fisher equation describes the process of interaction between diffusion and reaction. This equation is encountered in chemical kinetics and population dynamics which includes problems such as nonlinear evolution of a population in one-dimensional habitual, neutron population in a nuclear reaction [1]. It reads ut uxx uð1 uÞ ¼ 0:
ð47Þ
Using the wave variable n = x ct carries Eq. (47) into the ODE cu0 u00 uð1 uÞ ¼ 0:
ð48Þ
2
00
Balancing u with u in (48) gives M þ 2 ¼ 2M; so that M ¼ 2:
ð49Þ ð50Þ
The extended tanh method (6) applies the finite expansion b1 b2 uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 þ þ 2 : ð51Þ Y Y Substituting (51) into (48), collecting the coefficients of Y we obtain the following system of algebraic equations for a0, a1, a2, b1, b2 and l, and solving this system we obtain real and complex sets of solutions. In what follows, we list only the three sets of real solutions (i) The first set: 1 a0 ¼ ; 4
1 a1 ¼ ; 2
1 a2 ¼ ; 4
b1 ¼ b2 ¼ 0;
5 c ¼ pffiffiffi ; 6
1 l ¼ pffiffiffi : 2 6
ð52Þ
5 c ¼ pffiffiffi ; 6
1 l ¼ pffiffiffi : 2 6
ð53Þ
(ii) The second set: 1 a0 ¼ ; 4
a1 ¼ a2 ¼ 0;
1 b1 ¼ ; 2
1 b2 ¼ ; 4
(iii) The third set: 3 a0 ¼ ; 8
1 a1 ¼ ; 4
a2 ¼
1 ; 16
1 b1 ¼ ; 4
The first set gives the soliton solution 2 1 1 5 1 tanh2 pffiffiffi x pffiffiffi t u1 ðx; tÞ ¼ : 4 2 6 6
b2 ¼
1 ; 16
5 c ¼ pffiffiffi ; 6
1 l ¼ pffiffiffi : 4 6
ð54Þ
ð55Þ
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The second set gives the soliton solution 2 1 1 5 2 1 coth pffiffiffi x pffiffiffi t u2 ðx; tÞ ¼ : 4 2 6 6
ð56Þ
The third set gives the soliton solution 1 1 5 1 5 2 p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi 6 4 tanh u3 ðx; tÞ ¼ x t þ tanh x t 16 4 6 6 2 6 6 1 1 5 1 5 2 4 coth pffiffiffi x pffiffiffi t þ coth pffiffiffi x pffiffiffi t þ : 16 4 6 6 2 6 6
ð57Þ
7. The Burgers–Fisher equation The Burgers–Fisher equation reads ut þ uux þ uxx þ uð1 uÞ ¼ 0;
ð58Þ
that will be carried into the ODE cu0 þ uu0 þ u00 þ uð1 uÞ ¼ 0
ð59Þ 0
00
obtained after using the wave variable n = x ct. Balancing uu with u gives M þ 2 ¼ 2M þ 1;
ð60Þ
so that M ¼ 1:
ð61Þ
Accordingly, the extended tanh method (6) is of the form uðnÞ ¼ a0 þ a1 Y þ
b1 : Y
ð62Þ
Substituting (62) into (59), and proceeding as before we obtain the three sets of solutions (i) The first set: 1 1 a0 ¼ ; a1 ¼ ; 2 2 (ii) The second set: 1 a0 ¼ ; 2
a1 ¼ 0;
b1 ¼ 0;
5 c¼ ; 2
1 l¼ : 4
ð63Þ
1 b1 ¼ ; 2
5 c¼ ; 2
1 l¼ : 4
ð64Þ
1 b1 ¼ ; 4
5 c¼ ; 2
1 l¼ : 8
ð65Þ
(iii) The third set: 1 a0 ¼ ; 2
1 a1 ¼ ; 4
This in turn gives the following kinks solutions 1 1 5 1 þ tanh x t ; 2 4 2 1 1 5 1 þ coth x t ; u2 ðx; tÞ ¼ 2 4 2
u1 ðx; tÞ ¼
ð66Þ ð67Þ
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and u3 ðx; tÞ ¼
1 1 5 1 5 2 þ tanh x t þ coth x t : 4 8 2 8 2
ð68Þ
8. The Huxley equation The Huxley equation reads ut auxx uðk un Þðun 1Þ ¼ 0;
ð69Þ
n > 1:
This equation is used for nerve propagation in neurophysics and wall propagation in liquid crystals [10]. The wave variable n = x ct carries out Eq. (69) into the ODE cu0 au00 ðk þ 1Þunþ1 þ u2nþ1 þ ku ¼ 0:
ð70Þ
Balancing u00 with u2n+1 gives M þ 2 ¼ ð2n þ 1ÞM;
ð71Þ
so that 1 M¼ : n
ð72Þ
To obtain a closed form solution, M should be an integer, therefore we use the transformation 1
uðx; tÞ ¼ vn and as a result Eq. (70) becomes
ð73Þ 2
cnvv0 anvv00 að1 nÞðv0 Þ ðk þ 1Þn2 v3 þ n2 v4 þ kn2 v2 ¼ 0:
ð74Þ
4
Balancing vv00 with v gives M = 1. Consequently, the extended tanh method (6) becomes b1 vðnÞ ¼ a0 þ a1 Y þ : Y Substituting (75) into (74), and proceeding as before we obtain the sets of solutions
ð75Þ
(i) The first set: 1 a0 ¼ ; 2
1 a1 ¼ ; 2
b1 ¼ 0;
c ¼ c;
n l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð76Þ
1 b1 ¼ ; 2
c ¼ c;
n l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð77Þ
(ii) The second set: 1 a0 ¼ ; 2
a1 ¼ 0;
(iii) The third set: 1 a0 ¼ ; 2
1 a1 ¼ ; 4
1 b1 ¼ ; 4
c ¼ c;
n l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð78Þ
(iv) The fourth set: k a0 ¼ ; 2
k a1 ¼ ; 2
b1 ¼ 0;
c ¼ k;
kn l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð79Þ
k b1 ¼ ; 2
c ¼ k;
kn l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 aðn þ 1Þ
ð80Þ
(v) The fifth set: k a0 ¼ ; 2
a1 ¼ 0;
A.-M. Wazwaz / Applied Mathematics and Computation 187 (2007) 1131–1142
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(vi) The sixth set: k a0 ¼ ; 2
k a1 ¼ ; 4
k b1 ¼ ; 4
c ¼ k;
kn l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4 aðn þ 1Þ
ð81Þ
where
rffiffiffiffiffiffiffiffiffiffiffi a ; c ¼ ðkðn þ 1Þ 1Þ nþ1 rffiffiffiffiffiffiffiffiffiffiffi a : k ¼ ðn k þ 1Þ nþ1
ð82Þ
1
Recall that u ¼ vn . Based on this, the first three sets give the following kinks solutions: ( " #!)1n 1 n 1 tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ctÞ u1 ðx; tÞ ¼ ; 2 2 aðn þ 1Þ ( u2 ðx; tÞ ¼
ð83Þ
" #!)1n 1 n 1 coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ctÞ 2 2 aðn þ 1Þ
ð84Þ
" # " #!)1n 1 n n 2 tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ctÞ coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ctÞ : 4 4 aðn þ 1Þ 4 aðn þ 1Þ
ð85Þ
and ( u3 ðx; tÞ ¼
The last three sets lead to the kinks solutions ( " #!)1n k kn 1 tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ktÞ ; u4 ðx; tÞ ¼ 2 2 aðn þ 1Þ ( " #!)1n 1 kn 1 coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ktÞ u5 ðx; tÞ ¼ 2 2 aðn þ 1Þ and
( u6 ðx; tÞ ¼
" # " #!)1n k kn kn 2 tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ktÞ coth pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ktÞ : 4 4 aðn þ 1Þ 4 aðn þ 1Þ
ð86Þ
ð87Þ
ð88Þ
9. The nonlinear reaction–diffusion equations The nonlinear reaction–diffusion equation appear in many forms [1,10]. In this section we will focus our work on two of these reaction–diffusion equations, namely ut ðu2 Þxx uð1 uÞ ¼ 0
ð89Þ
ut ðu3 Þxx uð1 u2 Þ ¼ 0:
ð90Þ
and
9.1. First type Using the wave variable n = x ct carries Eq. (89) into the ODE 00
cu0 ðu2 Þ uð1 uÞ ¼ 0
ð91Þ
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or equivalently 2
cu0 2uu00 2ðu0 Þ u þ u2 ¼ 0: 00
ð92Þ
0
Balancing uu with u gives 2M þ 2 ¼ M þ 1;
ð93Þ
so that M ¼ 1:
ð94Þ
In [1], an alternative series expansion given by uðnÞ ¼
1 A ¼ a0 þ a1 Y 1 þ aY
ð95Þ
was used to obtain one singular solution that blows up as x ! 1. However, to obtain more solutions, the extended tanh method (6) can be used in the form uðnÞ ¼
1 A ¼ : b1 1 þ aY þ Yb a0 þ a1 Y þ Y
ð96Þ
Substituting (96) into (92), collecting the coefficients of Y and solving the resulting system we obtain the following solutions: (i) The first set: A ¼ 2;
b ¼ 0;
c ¼ 1;
1 l¼ : 4
ð97Þ
b ¼ 1;
c ¼ 1;
1 l¼ : 4
ð98Þ
a ¼ 1;
(ii) The second set: A ¼ 2;
a ¼ 0;
(iii) The third set: A ¼ 2;
1 a¼ ; 2
1 b¼ ; 2
c ¼ 1;
1 l¼ : 8
ð99Þ
In view of these results, we obtain the three pairs of singular solutions u1 ðx; tÞ ¼
1 tanh
2 1 4
ðx tÞ
:
ð100Þ
The second set gives the soliton solution u2 ðx; tÞ ¼
1 coth
2 1 4
ðx tÞ
ð101Þ
and u3 ðx; tÞ ¼
2
: 1 tanh 8 ðx tÞ 12 coth 18 ðx tÞ 1 2
1
ð102Þ
9.2. Second type Using the wave variable n = x ct carries Eq. (90) into the ODE 00
cu0 ðu3 Þ uð1 u2 Þ ¼ 0
ð103Þ
A.-M. Wazwaz / Applied Mathematics and Computation 187 (2007) 1131–1142
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or equivalently 2
cu0 3u2 u00 6uðu0 Þ u þ u3 ¼ 0: 2 00
ð104Þ
0
Balancing u u with u gives 3M þ 2 ¼ M þ 1;
ð105Þ
so that 1 M ¼ : 2
ð106Þ
Proceeding as before, the extended tanh method (6) can be used in the form !12 12 1 A uðnÞ ¼ ¼ : 1 þ aY þ Yb a0 þ a1 Y þ bY1
ð107Þ
Substituting (107) into (104), collecting the coefficients of Y and solving the resulting system we obtain the following solutions (i) The first set: A ¼ 2;
b ¼ 0;
c ¼ 1;
1 l¼ : 3
ð108Þ
b ¼ 1;
c ¼ 1;
1 l¼ : 3
ð109Þ
a ¼ 1;
(ii) The second set: A ¼ 2;
a ¼ 0;
(iii) The third set: A ¼ 2;
1 a¼ ; 2
1 b¼ ; 2
c ¼ 1;
1 l¼ : 6
In view of these results, we obtain the three pairs of singular solutions !12 2
: u1 ðx; tÞ ¼ 1 tanh 13 ðx tÞ
ð110Þ
ð111Þ
The second set gives the soliton solution u2 ðx; tÞ ¼
1 coth
2 1 3
!12 ðx tÞ
ð112Þ
and u3 ðx; tÞ ¼
!12 2
: 1 12 tanh 16 ðx tÞ 12 coth 16 ðx tÞ
ð113Þ
10. Concluding remarks The extended tanh method [12,13] was successfully used to establish abundant solitary wave solutions, mostly solitons and kinks solutions. Many well known nonlinear wave equations were handled by this method to show the new solutions compared to the solutions obtained in [1–4]. The performance of the extended tanh method is reliable and effective and gives more solutions. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave equations.
1142
A.-M. Wazwaz / Applied Mathematics and Computation 187 (2007) 1131–1142
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