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Explicit travelling wave solutions of variants of the K(n, n) and the ZK(n, n) equations with compact and noncompact structures
Applied Mathematics and Computation 173 (2006) 213–230 www.elsevier.com/locate/amc
Explicit travelling wave solutions of variants of the K(n, n) and the ZK(n, n) equations with compact and noncompact structures Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, United States
Abstract In this work two powerful schemes, that use the reliable ideas of the sine–cosine method and the tanh method, are presented. Variants of the K(n, n) and the ZK(n, n) are selected to illustrate the two methods and to derive compact and noncompact solutions for these nonlinear variants with dispersive effects. The coefficients of the derivatives of the equation play a major role in change of the physical structures of the solutions. Ó 2005 Elsevier Inc. All rights reserved. Keywords: K(n, n) equation; The KP equation; The ZK equation; Compactons; Sine–cosine method; The tanh method
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.02.050
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1. Introduction The balance between the nonlinear convection uux and the linear dispersion uxxx in the integrable nonlinear KdV equation ut þ auux þ buxxx ¼ 0;
ð1Þ
gives rise to solitons: waves with infinite support. Solitons are defined as localized waves that propagate without change of its shape and velocity properties and stable against mutual collisions [1–7]. Two well-known generalizations of the KdV equations, namely the integrable Kadomtsov–Petviashivilli (KP) equation, and the nonintegrable Zakharov– Kuznetsov (ZK) equation, given by fut þ auux þ uxxx gx þ kuyy ¼ 0
ð2Þ
ut þ auux þ ðr2 uÞx ¼ 0;
ð3Þ
and
respectively, were developed in [8,9], respectively, where r2 ¼ o2x þ o2y þ o2z is the isotropic Laplacian [9–12]. The delicate interaction between nonlinear convection (un)x with the genuine nonlinear dispersion (un)xxx in the well-known K(n, n) equation [13] ut þ aðun Þx þ ðun Þxxx ¼ 0; n > 1;
ð4Þ
generates the so termed compactons: solitary waves with exact compact support. Compactons are defined as solitons with finite wavelengths or solitons free of exponential tails [13–22]. The solitary wave with compact support is called compacton to indicate that it has the property of a particle, such as phonon, photon, and soliton. The stability analysis has shown that compacton solutions are stable, where the stability condition is satisfied for arbitrary values of the nonlinearity parameter. The stability of the compactons solutions was investigated by means of both linear stability and by Lyapunov stability criteria as well. It is the objective of this work to further complement our studies in [20,21] on the K(n, n) equation. Our first interest in the present work being in implementing the tanh method [22–24] to stress its power in handling nonlinear equations so that one can apply it to models of various types of nonlinearity. The next interest is the determination of exact travelling wave solutions with distinct physical structures to the K(n, n, 2n) equation given by ut þ aðun Þx þ b½un ðu2n Þxx x ¼ 0;
ð5Þ
and the K(n, 2n, n) given by ut þ aðun Þx þ b½u2n ðun Þxx x ¼ 0.
ð6Þ
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
215
In addition, we will extend our analysis to the related equations ZK(n, n, 2n) and ZK(n, 2n, n) given by ut þ aðun Þx þ ½bun ðu2n Þxx þ kðun Þyy x ¼ 0
ð7Þ
ut þ aðun Þx þ ½bu2n ðun Þxx þ kðun Þyy x ¼ 0;
ð8Þ
and
respectively. The aforementioned variants (7) and (8) are developed similarly to the ZK equation (3) from the relevant Eqs. (5) and (6), respectively. It is thus normal to call Eqs. (7) and (8) the ZK(n, n, 2n), and the ZK(n, 2n, n) equations. Our approach depends mainly on the sine–cosine method [14–21] and the tanh method [22–24] that have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be solved by using Mathematica or Maple. In what follows, we highlight the main steps of the proposed methods.
2. Analysis of the two methods For both methods, we first use the wave variable n = x ct to carry a PDE in two independent variables P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0;
ð9Þ
into an ODE Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0.
ð10Þ
Eq. (10) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. 2.1. The sine–cosine method The sine–cosine method admits the use of the solution in the form kcosb ðlnÞ; jlnj < p2 ; uðx; tÞ ¼ 0; otherwise; or in the form uðx; tÞ ¼
(
ksinb ðlnÞ; jlnj < p; 0;
otherwise.
ð11Þ
ð12Þ
The parameters k, l, and b will be determined, and l and c are the wave number and the wave speed, respectively. Consequently, we set
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uðnÞ ¼ kcosb ðlnÞ; 00
ðun Þ ðnÞ ¼ n2 l2 b2 kn cosnb ðlnÞ þ nl2 kn bðnb 1Þcosnb2 ðlnÞ;
ð13Þ
and for (12) we use uðnÞ ¼ ksinb ðlnÞ; 00
ðun Þ ðnÞ ¼ n2 l2 b2 kn sinnb ðlnÞ þ nl2 kn bðnb 1Þsinnb2 ðlnÞ.
ð14Þ
Substituting (13) or (14) into the integrated ODE gives a trigonometric equation of cosb ðlnÞ or sinb ðlnÞ terms. The parameters b, k, and l, are then obtained by equating the exponents of each pair of cosine or sine, and by collecting all coefficients of the same power in cosk ðlnÞ or sink ðlnÞ, and set it equal to zero. 2.2. The tanh method The tanh method is developed by Malfliet and coworkers [23,24] where the tanh is introduced as a new variable, since all derivatives of a tanh are represented by a tanh itself. We use a new independent variable Y ¼ tanhðlnÞ;
ð15Þ
that leads to the change of derivatives: d d ¼ lð1 Y 2 Þ ; dn dY 2 d2 d 2 2 2 d þ ð1 Y ¼ l ð1 Y Þ 2Y Þ . dY dY 2 dn2
ð16Þ
We then apply the following finite expansion: uðlnÞ ¼ SðY Þ ¼
M X
ak Y k ;
ð17Þ
k¼0
where M is a positive integer that will be determined to derive a closed form analytic solution. However, if M is not an integer, a transformation formula is usually used. Substituting (16) and (17) into the simplified ODE (10) results in an 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 (17) we obtain an analytic solution u(x, t) in a closed form.
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217
3. Using the sine–cosine method 3.1. The K(n, n, 2n) equation The K(n, n, 2n) is given by ut þ aðun Þx þ b½un ðu2n Þxx x ¼ 0.
ð18Þ
We use the travelling wave solutions of (18) in the form u(x, t) = u(n), where the wave variable is n = x ct to carry out Eq. (18) to the ODE 0
00 0
cu0 þ aðun Þ þ b½un ðu2n Þ ¼ 0.
ð19Þ
Integrating (19), setting the constant of integration to be zero, we find 00
cu þ aun þ bun ðu2n Þ ¼ 0.
ð20Þ
Substituting (11) into (20) yields ckcosb ðlnÞ þ akn cosnb ðlnÞ; 4bn2 l2 b2 kn cosnb ðlnÞ þ 2bnkn l2 bð2nb 1Þcosnb2 ðlnÞ ¼ 0.
ð21Þ
Equating the exponents of the first and the last cosine functions, collecting the coefficients of each pair of cosine functions of like exponents, and setting it equal to zero, we obtain the following system of algebraic equations: 2nb 1 6¼ 0; nb 2 ¼ b; 4bn2 l2 b2 ¼ a;
ð22Þ
2bnk2n l2 bð2nb 1Þ ¼ ck. Solving this system yields 2 ; n 1 rffiffiffi n1 a l¼ ; b > 0; 4n b 1 n1 4nc ; a > 0; n > 1; k¼ að3n þ 1Þ b¼
ð23Þ
that can also be obtained by using the sine ansatz (12). Consequently, for ab > 0 we obtain a family of compactons solutions 8n 1 h ion1 < 2 n1 pffiffi 4nc a sin ; jlnj < p; ðx ctÞ að3nþ1Þ 4n b uðx; tÞ ¼ ð24Þ : 0; otherwise;
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and uðx; tÞ ¼
8n < :
h
4nc cos2 n1 að3nþ1Þ 4n
1 ion1 pffiffia ; jlnj < p2 ; ðx ctÞ b
0;
ð25Þ
otherwise.
However, for ab < 0 we obtain solitary patterns solutions uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 4nc a 2 n1 sinh ðx ctÞ að3n þ 1Þ 4n b
ð26Þ
and uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 4nc a 2 n1 cosh . ðx ctÞ að3n þ 1Þ 4n b
ð27Þ
3.2. The K(n, 2n, n) equation The ZK(n, n, 2n) is given by ut þ aðun Þx þ b½u2n ðun Þxx x ¼ 0.
ð28Þ
Using the wave variable n = x ct carries out Eq. (28) to the ODE 0
00 0
cu0 þ aðun Þ þ b½u2n ðun Þ ¼ 0.
ð29Þ
Integrating (29), setting the constant of integration to be zero, we find 00
cu þ aun þ bu2n ðun Þ ¼ 0.
ð30Þ
Substituting the cosine ansatz (11) into (30) yields ckcosb ðlnÞ þ akn cosnb ðlnÞ bn2 l2 b2 kn cosnb ðlnÞ þ bnkn l2 bðnb þ 1Þcosnb2 ðlnÞ ¼ 0.
ð31Þ
Balancing the exponents of the first and the last cosine functions, collecting the coefficients of each pair of cosine functions of like exponents, and setting it equal to zero, we obtain the following system: nb þ 1 6¼ 0; b ¼ nb 2; bn2 l2 b2 ¼ a; bnkn l2 bðnb þ 1Þ ¼ ck; from which we obtain
ð32Þ
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
2 ; n 1 rffiffiffi n1 a l¼ ; b > 0; 2n b 1 2nc k¼ n 1; að3n 1Þ
219
b¼
ð33Þ a > 0; n > 1;
that can also be obtained by using the sine assumption (12). Consequently, for > 0 we obtain a family of compactons solutions 8n 1 h ion1 < 2 n1 pffiffi 2nc a ðx ctÞ sin ; jlnj < p; að3n1Þ 2n b uðx; tÞ ¼ ð34Þ : 0; otherwise;
a b
and uðx; tÞ ¼
8n < :
h
2nc cos2 n1 að3n1Þ 2n
1 ion1 pffiffia ; jlnj < p2 ; ðx ctÞ b
0;
ð35Þ
otherwise.
a b
However, for < 0 we obtain solitary patterns solutions 1 rffiffiffiffiffiffiffi n1 2nc a 2 n1 sinh uðx; tÞ ¼ ðx ctÞ að3n 1Þ 2n b
ð36Þ
and uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 2nc a 2 n1 cosh . ðx ctÞ að3n 1Þ 2n b
ð37Þ
3.3. The ZK(n, n, 2n) equation The ZK(n, n, 2n) is given by ut þ aðun Þx þ ½bun ðu2n Þxx þ kðun Þyy x ¼ 0.
ð38Þ
The wave variable n = x + y ct carries (38) to the ODE 0
00
00
cu0 þ aðun Þ þ ½bun ðu2n Þ þ kðun Þ 0.
ð39Þ
Integrating (39), setting the constant of integration to be zero, we find 00
00
cu þ aun þ bun ðu2n Þ þ kðun Þ ¼ 0.
ð40Þ
Substituting (11) into (40) yields ckcosb ðlnÞ þ kn ða 4bn2 l2 b2 kn2 l2 b2 Þcosnb ðlnÞ þ nkn l2 bð2bð2nb 1Þ þ kðnb 1ÞÞcosnb2 ðlnÞ ¼ 0.
ð41Þ
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Equating the exponents of the first and the last cosine functions and proceeding as before we obtain the following system of algebraic equations: 2nb 1 6¼ 0; nb 1 6¼ 0; nb 2 ¼ b; 2 2 2
ð42Þ 2 2 2
4bn l b þ kn l b ¼ a; 2bnkn l2 bð2nb 1Þ þ knkn l2 ðnb 1Þ ¼ ck. Solving this system yields 2 ; n 1 rffiffiffiffiffiffiffiffiffiffiffiffiffi n1 a ; 4b þ k > 0; l¼ 2n 4b þ k 1 n1 2ncð4b þ kÞ k¼ ; a½2bð3n þ 1Þ þ kðn þ 1Þ b¼
ð43Þ a > 0; n > 1;
that can also be obtained by using the sine ansatz (12). a Consequently, for 4bþk > 0 we obtain a family of compactons solutions 8n 1 h ion1 ffi < 2ncð4bþkÞ 2 n1 pffiffiffiffiffiffiffi a sin ; jlnj < p; ðx þ y ctÞ 2n 4bþk a½2bð3nþ1Þþkðnþ1Þ uðx; y; tÞ ¼ : 0; otherwise; ð44Þ and uðx; y; tÞ ¼
8n < :
h
pffiffiffiffiffiffiffi ffi 2ncð4bþkÞ a cos2 n1 ðx 2n 4bþk a½2bð3nþ1Þþkðnþ1Þ
0;
þ y ctÞ
1 ion1
;
jlnj < p2 ; otherwise. ð45Þ
a 4bþk
However, for < 0 we obtain solitary patterns solutions 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2ncð4b þ kÞ a 2 n1 sinh ðx þ y ctÞ uðx;y;tÞ ¼ a½2bð3n þ 1Þ þ kðn þ 1Þ 2n 4b þ k ð46Þ and 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2ncð4b þ kÞ a 2 n1 cosh ðx þ y ctÞ uðx; y;tÞ ¼ . a½2bð3n þ 1Þ þ kðn þ 1Þ 2n 4b þ k
ð47Þ
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
221
3.4. The ZK(n, 2n, n) equation The ZK(n, 2n, n) is given by ut þ aðun Þx þ ½bu2n ðun Þxx þ kðun Þyy x ¼ 0.
ð48Þ
The wave variable n = x + y ct carries (48) to the ODE 0
00
00
cu0 þ aðun Þ þ ½bu2n ðun Þ þ kðun Þ ¼ 0.
ð49Þ
Integrating (49), setting the constant of integration to be zero, we find cu þ aun þ bu2n ðun Þ00 þ kðun Þ00 ¼ 0.
ð50Þ
Substituting (11) into (50) yields ckcosb ðlnÞ þ kn ða bn2 l2 b2 kn2 l2 b2 Þcosnb ðlnÞ þ nkn l2 bðbðnb þ 1Þ þ kðnb 1ÞÞcosnb2 ðlnÞ ¼ 0.
ð51Þ
Equating the exponents of the first and the last cosine functions and proceeding as before we obtain the following system of algebraic equations: nb 1 6¼ 0; nb þ 1 6¼ 0; nb 2 ¼ b;
ð52Þ
bn2 l2 b2 þ kn2 l2 b2 ¼ a; nkn l2 bðnb þ 1Þ þ knkn l2 bðnb 1Þ ¼ ck. Solving this system yields 2 ; n 1 rffiffiffiffiffiffiffiffiffiffiffi n1 a ; 4b þ k > 0; l¼ 2n bþk 1 2ncðb þ kÞ k¼ n 1; a½bð3n 1Þ þ kðn þ 1Þ b¼
ð53Þ a > 0; n > 1;
that can also be obtained by using the sine ansatz (12). a Consequently, for 4bþk > 0 we obtain a family of compactons solutions 8n 1 h ion1 < 2ncðbþkÞ 2 n1 pffiffiffiffiffiffi a sin ; jlnj < p; ðx ctÞ 2n bþk a½bð3n1Þþkðnþ1Þ uðx; y; tÞ ¼ : 0; otherwise; ð54Þ
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A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
and uðx; y; tÞ ¼
8n < :
h
pffiffiffiffiffiffi 2ncðbþkÞ a cos2 n1 ðx 2n bþk a½bð3n1Þþkðnþ1Þ
þ y ctÞ
1 ion1
0;
; jlnj < p2 ; otherwise. ð55Þ
However, for
a 4bþk
< 0 we obtain solitary patterns solutions
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2ncðb þ kÞ a 2 n1 sinh ðx þ y ctÞ uðx; y; tÞ ¼ a½bð3n 1Þ þ kðn þ 1Þ 2n bþk
ð56Þ and 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2ncðb þ kÞ a 2 n1 cosh ðx þ y ctÞ uðx; y; tÞ ¼ . a½bð3n 1Þ þ kðn þ 1Þ 2n bþk
ð57Þ
4. Using the tanh method 4.1. The K(n, n, 2n) equation The K(n, n, 2n) is given before by ut þ aðun Þx þ b½un ðu2n Þxx x ¼ 0.
ð58Þ
We use the wave variable n = x ct to carry out Eq. (18) to the ODE 00
cu þ aun þ b½un ðu2n Þ ¼ 0;
ð59Þ
or equivalently 2
cu þ aun þ 2bnun1 u00 þ 2bnð2n 1Þun2 ðu0 Þ ¼ 0; upon integrating once. Balancing u(x, t) with u M ¼ ðn 1ÞM þ 4 þ M 2;
ð60Þ
n1 00
u we find ð61Þ
so that M ¼
2 . n1
ð62Þ
To get a closed form analytic solution, the parameter M should be an integer. A transformation formula 1
u ¼ vn1 ;
ð63Þ
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
223
should be used to achieve our goal. This in turn transforms (60) to 2
2
2
cðn 1Þ v3 þ aðn 1Þ v2 2bnðn 1Þvv00 þ 2bnð3n 1Þðv0 Þ ¼ 0.
ð64Þ
Balancing vv00 and v3 gives M = 2. The tanh method allows us to use the substitution vðx; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 .
ð65Þ
Substituting (65) into (64), collecting the coefficients of each power of Y, and using Mathematica to solve the resulting system of algebraic equations we obtain að3n þ 1Þ ; 4cn a1 ¼ 0;
a0 ¼
ð66Þ
að3n þ 1Þ ; 4cn rffiffiffiffiffiffiffi n1 a M¼ ; 4n b
a2 ¼
1
where c is selected as a free parameter. Noting that uðx; tÞ ¼ vn1 , we find a family of solitary patterns solutions 1 rffiffiffiffiffiffiffi n1 4nc n1 a sinh2 uðx; tÞ ¼ ð67Þ ðx ctÞ að3n þ 1Þ 4n b and uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 4nc n1 a cosh2 ; ðx ctÞ að3n þ 1Þ 4n b
ð68Þ
valid for ab < 0. However, for ab > 0, we obtain a family of compactons solutions given by 8n 1 h pffiffi ion1 < 4nc a ðx ctÞ sin2 n1 ; jlnj < p; að3nþ1Þ 4n b uðx; tÞ ¼ ð69Þ : 0; otherwise; and uðx; tÞ ¼
8n < :
h
4nc cos2 n1 að3nþ1Þ 4n
0;
1 ion1 pffiffia ; jlnj < p2 ; ðx ctÞ b
ð70Þ
otherwise.
The results obtained above by using the tanh method are consistent with the results obtained by using the sine–cosine method.
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4.2. The K(n, 2n, n) equation The K(n, 2n, n) is given before by ut þ aðun Þx þ b½u2n ðun Þxx x ¼ 0.
ð71Þ
The wave variable n = x ct carries Eq. (71) to the ODE cu þ aun bnun1 u00 þ bnðn þ 1Þun2 ðu0 Þ2 ¼ 0; upon integrating once. Balancing u(x, t) with u
ð72Þ
n1 00
u we find
M ¼ ðn 1ÞM þ 4 þ M 2;
ð73Þ
so that M ¼
2 . n1
ð74Þ
It is normal to seek an integer value for the parameter M. Therefore, we set a transformation formula 1
u ¼ vn1 ;
ð75Þ
to achieve our goal. Consequently, (72) is reduced to 2
2
2
cðn 1Þ v3 þ aðn 1Þ v2 þ bnðn 1Þvv00 þ bnðv0 Þ ¼ 0.
ð76Þ
Balancing vv00 and v3 gives M = 2. The tanh method allows us to use the substitution vðx; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 .
ð77Þ
Substituting (77) into (76), collecting the coefficients of each power of Y, and using Mathematica to solve the resulting system of algebraic equations we obtain að3n 1Þ ; 2cn a1 ¼ 0; að3n 1Þ ; a2 ¼ 2cn rffiffiffiffiffiffiffi n1 a M¼ ; 2n b
a0 ¼
ð78Þ
1
where c is selected as a free parameter. Noting that uðx; tÞ ¼ vn1 , a family of solitary patterns solutions 1 rffiffiffiffiffiffiffi n1 2cn a 2 n1 sinh uðx; tÞ ¼ ð79Þ ðx ctÞ að3n 1Þ 2n b
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
225
and uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 2cn a 2 n1 cosh ; ðx ctÞ að3n 1Þ 2n b
ð80Þ
is readily obtained valid for ab < 0. However, for ab > 0, a family of compactons solutions given by 8n 1 h ion1 < 2 n1 pffiffi 2cn a ðx ctÞ sin ; jlnj < p; að3n1Þ 2n b uðx; tÞ ¼ ð81Þ : 0; otherwise; and uðx; tÞ ¼
8n < :
h
2nc cos2 n1 að3n1Þ 2n
1 ion1 pffiffia ; jlnj < p2 ; ðx ctÞ b
0;
ð82Þ
otherwise;
follows immediately. The results obtained above by using the tanh method are consistent with the results obtained by using the sine–cosine method. 4.3. The ZK(n, n, 2n) equation The ZK(n, n, 2n) is given by ut þ aðun Þx þ ½bun ðu2n Þxx þ kðun Þyy x ¼ 0.
ð83Þ
We use the travelling wave solutions of (83) in the form u(x, y, t) = u(ln), where the wave variable is n = x + y ct to carry out Eq. (83) to the ODE cu0 þ aðun Þ0 þ ½bun ðu2n Þ00 þ kðun Þ00 0 ¼ 0.
ð84Þ
Integrating once, setting the constant of integration to be zero, we obtain cu þ aun þ nð2b þ kÞun1 u00 þ nð2bð2n 1Þ þ kðn 1ÞÞun2 ðu0 Þ2 ¼ 0.
ð85Þ
Balancing u(x, y, t) with u
n1 00
u we find
M ¼ ðn 1ÞM þ 4 þ M 2;
ð86Þ
so that M ¼
2 . n1
ð87Þ
A closed form analytic solution is obtained if the parameter M is an integer. We therefore use a transformation 1
u ¼ vn1
ð88Þ
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A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
to achieve our goal. This in turn transforms (85) to 2
2
cðn 1Þ v3 þ aðn 1Þ v2 nð2b þ kÞðn 1Þvv00 þ nð2bð3n 1Þ 2
þ kð2n 1ÞÞðv0 Þ ¼ 0.
ð89Þ
3
00
Balancing vv and v gives M = 2. As presented before, the tanh method admits the use of the substitution vðx; y; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 .
ð90Þ
Substituting (90) into (89), collecting the coefficients of each power of Y and proceeding as before we find að2bð3n þ 1Þ þ kðn þ 1ÞÞ ; 2cnð4b þ kÞ a1 ¼ 0;
a0 ¼
ð91Þ
að2bð3n þ 1Þ þ kðn þ 1ÞÞ ; 2cnð4b þ kÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 a ; M¼ 2n 4b þ k
a2 ¼
1
where c is selected as a free parameter. Noting that uðx; y; tÞ ¼ vn1 , we find a family of solitary patterns solutions 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2cnðb þ kÞ n1 a sinh2 ðx ctÞ uðx; y; tÞ ¼ að2bð3n þ 1Þ þ kðn þ 1ÞÞ 2n 4b þ k ð92Þ and
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2cnðb þ kÞ a 2 n1 cosh ðx ctÞ uðx; y; tÞ ¼ ; að2bð3n þ 1Þ þ kðn þ 1ÞÞ 2n 4b þ k
a 4bþk
a 4bþk
ð93Þ > 0, we obtain a family of compactons
< 0. However, for valid for solutions given by 8n 1 h pffiffiffiffiffiffiffiffi ion1 < 2cnðbþkÞ a sin2 n1 ; jlnj < p; ðx ctÞ 4n 4bþk að2bð3nþ1Þþkðnþ1ÞÞ uðx; y; tÞ ¼ : 0; otherwise;
ð94Þ and uðx; y; tÞ ¼
8n < :
h
pffiffiffiffiffiffiffi ffi 2cnðbþkÞ a cos2 n1 ðx 4n 4bþk að2bð3nþ1Þþkðnþ1ÞÞ
0;
ctÞ
1 ion1
;
jlnj < p2 ; otherwise. ð95Þ
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
227
The results obtained above by using the tanh method are consistent with the results obtained by using the sine–cosine method. 4.4. The ZK(n, 2n, n) equation The ZK(n, 2n, n) is given by ut þ aðun Þx þ ½bu2n ðun Þxx þ kðun Þyy x ¼ 0.
ð96Þ
We use the travelling wave solutions u(x, y, t) = u(ln), where the wave variable is n = x + y ct to carry out Eq. (96) to the ODE 2
cu þ aun þ nð2b kÞun1 u00 þ nð2bðn þ 1Þ þ kðn 1ÞÞun2 ðu0 Þ ¼ 0. ð97Þ Balancing u(x, y, t) with u
n1 00
u we find
M ¼ ðn 1ÞM þ 4 þ M 2;
ð98Þ
so that 2 . n1
M ¼
ð99Þ
A closed form analytic solution is obtained if the parameter Mis an integer. We therefore use a transformation 1
u ¼ vn1 ;
ð100Þ
to achieve our goal. This in turn transforms (97) to cðn 1Þ2 v3 þ aðn 1Þ2 v2 þ nðb kÞðn 1Þvv00 þ nð2kn þ b kÞðv0 Þ2 ¼ 0. ð101Þ 00
3
Balancing vv and v gives M = 2. We therefore set vðx; y; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 .
ð102Þ
Substituting (102) into (101), collecting the coefficients of each power of Y and proceeding as before we get a0 ¼
aðbð3n 1Þ þ kðn þ 1ÞÞ ; 2cnðb þ kÞ
a1 ¼ 0; aðbð3n 1Þ þ kðn þ 1ÞÞ ; 2cnðb þ kÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 a ; M¼ 2n bþk
a2 ¼
ð103Þ
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A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230 1
where c is selected as a free parameter. Noting that uðx; y; tÞ ¼ vn1 , we find a family of solitary patterns solutions 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2cnðb þ kÞ a 2 n1 sinh ðx ctÞ uðx; y; tÞ ¼ aðbð3n 1Þ þ kðn þ 1ÞÞ 2n bþk ð104Þ and uðx; y; tÞ ¼
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2cnðb þ kÞ n1 a cosh2 ðx ctÞ ; aðbð3n 1Þ þ kðn þ 1ÞÞ 2n bþk
ð105Þ a bþk
a bþk
< 0. However, for > 0, we obtain a family of compactons soluvalid for tions given by 8n 1 h ion1 < 2cnðbþkÞ 2 n1 pffiffiffiffiffiffi a sin ; jlnj < p; ðx ctÞ 2n bþk aðbð3n1Þþkðnþ1ÞÞ uðx; y; tÞ ¼ : 0; otherwise; ð106Þ and uðx; y; tÞ ¼
8n < :
h
pffiffiffiffiffi ffi 2cnðbþkÞ a cos2 n1 ðx 2n bþk aðbð3n1Þþkðnþ1ÞÞ
0;
ctÞ
1 ion1
; jlnj < p2 ; otherwise. ð107Þ
The results obtained above by using the tanh method are consistent with the results obtained by using the sine–cosine method.
5. Discussion The sine–cosine method and the tanh method were used to investigate variants of the K(n, n) and the ZK(n, n) equations. The study revealed compactons solutions and solitary patterns solutions for all examined variants. The study emphasized the fact that the two methods are reliable and one method complements the other in handling nonlinear problems. The obtained results clearly demonstrate the efficiency of the two methods used in this work. Moreover, the methods are capable of greatly minimizing the size of computational work compared to other existing techniques. Although the two methods provided the same solutions, but the sine–cosine method minimizes the size of calculation compared to the tanh method. In addition, specific restriction is usually required in that the value of M must
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
229
be an integer to get closed form analytic solutions, therefore transformation formula is required to overcome this difficulty. The two methods worked successfully in handling nonlinear dispersive equations. This emphasizes the fact that the two methods are applicable to a wide variety of nonlinear problems.
References [1] M. Wadati, Introduction to solitons, Pramana: J. Phys. 57 (5–6) (2001) 841–847. [2] M. Wadati, The modified Kortweg–de Vries equation, J. Phys. Soc. Jpn. 34 (1973) 1289–1296. [3] N.J. Zabusky, M.D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965) 240–243. [4] R.T. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R Soc. London 272 (1972) 47–78. [5] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. [6] W. Hereman, M. Takaoka, Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA, J. Phys. A 23 (1990) 4805–4822. [7] W. Hereman, A. Korpel, P.P. Banerjee, A general physical approach to solitary wave construction from linear solutions, Wave Motion 7 (1985) 283–289. [8] B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (1970) 539–541. [9] V.E. Zakharov, E.A. Kuznetsov, On three-dimensional solitons, Sov. Phys. 39 (1974) 285–288. [10] S. Monro, E.J. Parkes, The derivation of a modified Zakharov–Kuznetsov equation and the stability of its solutions, J. Plasma Phys. 62 (3) (1999) 305–317. [11] S. Monro, E.J. Parkes, Stability of solitary-wave solutions to a modified Zakharov–Kuznetsov equation, J. Plasma Phys. 64 (3) (2000) 411–426. [12] E.J. Parkes, B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun. 98 (1996) 288–300. [13] P. Rosenau, J.M. Hyman, Compactons: solitons with finite wavelengths, Phys. Rev. Lett. 70 (5) (1993) 564–567. [14] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [15] A.M. Wazwaz, M.A. Helal, variants of the generalized fifth-order KdV equation with compact and noncompact structures, Chaos Soliton Fract. 21 (2004) 579–589. [16] A.M. Wazwaz, New solitary-wave special solutions with compact support for the nonlinear dispersive K(m, n) equations, Chaos Soliton Fract. 13 (2) (2002) 321–330. [17] A.M. Wazwaz, General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive K(n, n) equations in higher dimensional spaces, Appl. Math. Comput. 133 (2/3) (2002) 229–244. [18] A.M. Wazwaz, A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simul. 56 (2001) 269–276. [19] A.M. Wazwaz, Compactons dispersive structures for variants of the K(n, n) and the KP equations, Chaos Soliton Fract. 13 (5) (2002) 1053–1062. [20] A.M. Wazwaz, Compactons and solitary patterns structures for variants of the KdV and the KP equations, Appl. Math. Comput. 139 (1) (2003) 37–54. [21] A.M. Wazwaz, An analytic study of Compactons structures in a class of nonlinear dispersive equations, Math. Comput. Simul. 63 (1) (2003) 35–44. [22] A.M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput. 154 (3) (2004) 713–723.
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[23] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (7) (1992) 650–654. [24] W. Malfliet, W. Hereman, The tanh method: II. Perturbation technique for conservative systems, Phys. Scr. 54 (1996) 569–575.
Explicit travelling wave solutions of variants of the K(n, n) and the ZK(n, n) equations with compact and noncompact structures Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, United States
Abstract In this work two powerful schemes, that use the reliable ideas of the sine–cosine method and the tanh method, are presented. Variants of the K(n, n) and the ZK(n, n) are selected to illustrate the two methods and to derive compact and noncompact solutions for these nonlinear variants with dispersive effects. The coefficients of the derivatives of the equation play a major role in change of the physical structures of the solutions. Ó 2005 Elsevier Inc. All rights reserved. Keywords: K(n, n) equation; The KP equation; The ZK equation; Compactons; Sine–cosine method; The tanh method
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.02.050
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1. Introduction The balance between the nonlinear convection uux and the linear dispersion uxxx in the integrable nonlinear KdV equation ut þ auux þ buxxx ¼ 0;
ð1Þ
gives rise to solitons: waves with infinite support. Solitons are defined as localized waves that propagate without change of its shape and velocity properties and stable against mutual collisions [1–7]. Two well-known generalizations of the KdV equations, namely the integrable Kadomtsov–Petviashivilli (KP) equation, and the nonintegrable Zakharov– Kuznetsov (ZK) equation, given by fut þ auux þ uxxx gx þ kuyy ¼ 0
ð2Þ
ut þ auux þ ðr2 uÞx ¼ 0;
ð3Þ
and
respectively, were developed in [8,9], respectively, where r2 ¼ o2x þ o2y þ o2z is the isotropic Laplacian [9–12]. The delicate interaction between nonlinear convection (un)x with the genuine nonlinear dispersion (un)xxx in the well-known K(n, n) equation [13] ut þ aðun Þx þ ðun Þxxx ¼ 0; n > 1;
ð4Þ
generates the so termed compactons: solitary waves with exact compact support. Compactons are defined as solitons with finite wavelengths or solitons free of exponential tails [13–22]. The solitary wave with compact support is called compacton to indicate that it has the property of a particle, such as phonon, photon, and soliton. The stability analysis has shown that compacton solutions are stable, where the stability condition is satisfied for arbitrary values of the nonlinearity parameter. The stability of the compactons solutions was investigated by means of both linear stability and by Lyapunov stability criteria as well. It is the objective of this work to further complement our studies in [20,21] on the K(n, n) equation. Our first interest in the present work being in implementing the tanh method [22–24] to stress its power in handling nonlinear equations so that one can apply it to models of various types of nonlinearity. The next interest is the determination of exact travelling wave solutions with distinct physical structures to the K(n, n, 2n) equation given by ut þ aðun Þx þ b½un ðu2n Þxx x ¼ 0;
ð5Þ
and the K(n, 2n, n) given by ut þ aðun Þx þ b½u2n ðun Þxx x ¼ 0.
ð6Þ
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
215
In addition, we will extend our analysis to the related equations ZK(n, n, 2n) and ZK(n, 2n, n) given by ut þ aðun Þx þ ½bun ðu2n Þxx þ kðun Þyy x ¼ 0
ð7Þ
ut þ aðun Þx þ ½bu2n ðun Þxx þ kðun Þyy x ¼ 0;
ð8Þ
and
respectively. The aforementioned variants (7) and (8) are developed similarly to the ZK equation (3) from the relevant Eqs. (5) and (6), respectively. It is thus normal to call Eqs. (7) and (8) the ZK(n, n, 2n), and the ZK(n, 2n, n) equations. Our approach depends mainly on the sine–cosine method [14–21] and the tanh method [22–24] that have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be solved by using Mathematica or Maple. In what follows, we highlight the main steps of the proposed methods.
2. Analysis of the two methods For both methods, we first use the wave variable n = x ct to carry a PDE in two independent variables P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0;
ð9Þ
into an ODE Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0.
ð10Þ
Eq. (10) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. 2.1. The sine–cosine method The sine–cosine method admits the use of the solution in the form kcosb ðlnÞ; jlnj < p2 ; uðx; tÞ ¼ 0; otherwise; or in the form uðx; tÞ ¼
(
ksinb ðlnÞ; jlnj < p; 0;
otherwise.
ð11Þ
ð12Þ
The parameters k, l, and b will be determined, and l and c are the wave number and the wave speed, respectively. Consequently, we set
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A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
uðnÞ ¼ kcosb ðlnÞ; 00
ðun Þ ðnÞ ¼ n2 l2 b2 kn cosnb ðlnÞ þ nl2 kn bðnb 1Þcosnb2 ðlnÞ;
ð13Þ
and for (12) we use uðnÞ ¼ ksinb ðlnÞ; 00
ðun Þ ðnÞ ¼ n2 l2 b2 kn sinnb ðlnÞ þ nl2 kn bðnb 1Þsinnb2 ðlnÞ.
ð14Þ
Substituting (13) or (14) into the integrated ODE gives a trigonometric equation of cosb ðlnÞ or sinb ðlnÞ terms. The parameters b, k, and l, are then obtained by equating the exponents of each pair of cosine or sine, and by collecting all coefficients of the same power in cosk ðlnÞ or sink ðlnÞ, and set it equal to zero. 2.2. The tanh method The tanh method is developed by Malfliet and coworkers [23,24] where the tanh is introduced as a new variable, since all derivatives of a tanh are represented by a tanh itself. We use a new independent variable Y ¼ tanhðlnÞ;
ð15Þ
that leads to the change of derivatives: d d ¼ lð1 Y 2 Þ ; dn dY 2 d2 d 2 2 2 d þ ð1 Y ¼ l ð1 Y Þ 2Y Þ . dY dY 2 dn2
ð16Þ
We then apply the following finite expansion: uðlnÞ ¼ SðY Þ ¼
M X
ak Y k ;
ð17Þ
k¼0
where M is a positive integer that will be determined to derive a closed form analytic solution. However, if M is not an integer, a transformation formula is usually used. Substituting (16) and (17) into the simplified ODE (10) results in an 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 (17) we obtain an analytic solution u(x, t) in a closed form.
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
217
3. Using the sine–cosine method 3.1. The K(n, n, 2n) equation The K(n, n, 2n) is given by ut þ aðun Þx þ b½un ðu2n Þxx x ¼ 0.
ð18Þ
We use the travelling wave solutions of (18) in the form u(x, t) = u(n), where the wave variable is n = x ct to carry out Eq. (18) to the ODE 0
00 0
cu0 þ aðun Þ þ b½un ðu2n Þ ¼ 0.
ð19Þ
Integrating (19), setting the constant of integration to be zero, we find 00
cu þ aun þ bun ðu2n Þ ¼ 0.
ð20Þ
Substituting (11) into (20) yields ckcosb ðlnÞ þ akn cosnb ðlnÞ; 4bn2 l2 b2 kn cosnb ðlnÞ þ 2bnkn l2 bð2nb 1Þcosnb2 ðlnÞ ¼ 0.
ð21Þ
Equating the exponents of the first and the last cosine functions, collecting the coefficients of each pair of cosine functions of like exponents, and setting it equal to zero, we obtain the following system of algebraic equations: 2nb 1 6¼ 0; nb 2 ¼ b; 4bn2 l2 b2 ¼ a;
ð22Þ
2bnk2n l2 bð2nb 1Þ ¼ ck. Solving this system yields 2 ; n 1 rffiffiffi n1 a l¼ ; b > 0; 4n b 1 n1 4nc ; a > 0; n > 1; k¼ að3n þ 1Þ b¼
ð23Þ
that can also be obtained by using the sine ansatz (12). Consequently, for ab > 0 we obtain a family of compactons solutions 8n 1 h ion1 < 2 n1 pffiffi 4nc a sin ; jlnj < p; ðx ctÞ að3nþ1Þ 4n b uðx; tÞ ¼ ð24Þ : 0; otherwise;
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and uðx; tÞ ¼
8n < :
h
4nc cos2 n1 að3nþ1Þ 4n
1 ion1 pffiffia ; jlnj < p2 ; ðx ctÞ b
0;
ð25Þ
otherwise.
However, for ab < 0 we obtain solitary patterns solutions uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 4nc a 2 n1 sinh ðx ctÞ að3n þ 1Þ 4n b
ð26Þ
and uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 4nc a 2 n1 cosh . ðx ctÞ að3n þ 1Þ 4n b
ð27Þ
3.2. The K(n, 2n, n) equation The ZK(n, n, 2n) is given by ut þ aðun Þx þ b½u2n ðun Þxx x ¼ 0.
ð28Þ
Using the wave variable n = x ct carries out Eq. (28) to the ODE 0
00 0
cu0 þ aðun Þ þ b½u2n ðun Þ ¼ 0.
ð29Þ
Integrating (29), setting the constant of integration to be zero, we find 00
cu þ aun þ bu2n ðun Þ ¼ 0.
ð30Þ
Substituting the cosine ansatz (11) into (30) yields ckcosb ðlnÞ þ akn cosnb ðlnÞ bn2 l2 b2 kn cosnb ðlnÞ þ bnkn l2 bðnb þ 1Þcosnb2 ðlnÞ ¼ 0.
ð31Þ
Balancing the exponents of the first and the last cosine functions, collecting the coefficients of each pair of cosine functions of like exponents, and setting it equal to zero, we obtain the following system: nb þ 1 6¼ 0; b ¼ nb 2; bn2 l2 b2 ¼ a; bnkn l2 bðnb þ 1Þ ¼ ck; from which we obtain
ð32Þ
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
2 ; n 1 rffiffiffi n1 a l¼ ; b > 0; 2n b 1 2nc k¼ n 1; að3n 1Þ
219
b¼
ð33Þ a > 0; n > 1;
that can also be obtained by using the sine assumption (12). Consequently, for > 0 we obtain a family of compactons solutions 8n 1 h ion1 < 2 n1 pffiffi 2nc a ðx ctÞ sin ; jlnj < p; að3n1Þ 2n b uðx; tÞ ¼ ð34Þ : 0; otherwise;
a b
and uðx; tÞ ¼
8n < :
h
2nc cos2 n1 að3n1Þ 2n
1 ion1 pffiffia ; jlnj < p2 ; ðx ctÞ b
0;
ð35Þ
otherwise.
a b
However, for < 0 we obtain solitary patterns solutions 1 rffiffiffiffiffiffiffi n1 2nc a 2 n1 sinh uðx; tÞ ¼ ðx ctÞ að3n 1Þ 2n b
ð36Þ
and uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 2nc a 2 n1 cosh . ðx ctÞ að3n 1Þ 2n b
ð37Þ
3.3. The ZK(n, n, 2n) equation The ZK(n, n, 2n) is given by ut þ aðun Þx þ ½bun ðu2n Þxx þ kðun Þyy x ¼ 0.
ð38Þ
The wave variable n = x + y ct carries (38) to the ODE 0
00
00
cu0 þ aðun Þ þ ½bun ðu2n Þ þ kðun Þ 0.
ð39Þ
Integrating (39), setting the constant of integration to be zero, we find 00
00
cu þ aun þ bun ðu2n Þ þ kðun Þ ¼ 0.
ð40Þ
Substituting (11) into (40) yields ckcosb ðlnÞ þ kn ða 4bn2 l2 b2 kn2 l2 b2 Þcosnb ðlnÞ þ nkn l2 bð2bð2nb 1Þ þ kðnb 1ÞÞcosnb2 ðlnÞ ¼ 0.
ð41Þ
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Equating the exponents of the first and the last cosine functions and proceeding as before we obtain the following system of algebraic equations: 2nb 1 6¼ 0; nb 1 6¼ 0; nb 2 ¼ b; 2 2 2
ð42Þ 2 2 2
4bn l b þ kn l b ¼ a; 2bnkn l2 bð2nb 1Þ þ knkn l2 ðnb 1Þ ¼ ck. Solving this system yields 2 ; n 1 rffiffiffiffiffiffiffiffiffiffiffiffiffi n1 a ; 4b þ k > 0; l¼ 2n 4b þ k 1 n1 2ncð4b þ kÞ k¼ ; a½2bð3n þ 1Þ þ kðn þ 1Þ b¼
ð43Þ a > 0; n > 1;
that can also be obtained by using the sine ansatz (12). a Consequently, for 4bþk > 0 we obtain a family of compactons solutions 8n 1 h ion1 ffi < 2ncð4bþkÞ 2 n1 pffiffiffiffiffiffiffi a sin ; jlnj < p; ðx þ y ctÞ 2n 4bþk a½2bð3nþ1Þþkðnþ1Þ uðx; y; tÞ ¼ : 0; otherwise; ð44Þ and uðx; y; tÞ ¼
8n < :
h
pffiffiffiffiffiffiffi ffi 2ncð4bþkÞ a cos2 n1 ðx 2n 4bþk a½2bð3nþ1Þþkðnþ1Þ
0;
þ y ctÞ
1 ion1
;
jlnj < p2 ; otherwise. ð45Þ
a 4bþk
However, for < 0 we obtain solitary patterns solutions 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2ncð4b þ kÞ a 2 n1 sinh ðx þ y ctÞ uðx;y;tÞ ¼ a½2bð3n þ 1Þ þ kðn þ 1Þ 2n 4b þ k ð46Þ and 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2ncð4b þ kÞ a 2 n1 cosh ðx þ y ctÞ uðx; y;tÞ ¼ . a½2bð3n þ 1Þ þ kðn þ 1Þ 2n 4b þ k
ð47Þ
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
221
3.4. The ZK(n, 2n, n) equation The ZK(n, 2n, n) is given by ut þ aðun Þx þ ½bu2n ðun Þxx þ kðun Þyy x ¼ 0.
ð48Þ
The wave variable n = x + y ct carries (48) to the ODE 0
00
00
cu0 þ aðun Þ þ ½bu2n ðun Þ þ kðun Þ ¼ 0.
ð49Þ
Integrating (49), setting the constant of integration to be zero, we find cu þ aun þ bu2n ðun Þ00 þ kðun Þ00 ¼ 0.
ð50Þ
Substituting (11) into (50) yields ckcosb ðlnÞ þ kn ða bn2 l2 b2 kn2 l2 b2 Þcosnb ðlnÞ þ nkn l2 bðbðnb þ 1Þ þ kðnb 1ÞÞcosnb2 ðlnÞ ¼ 0.
ð51Þ
Equating the exponents of the first and the last cosine functions and proceeding as before we obtain the following system of algebraic equations: nb 1 6¼ 0; nb þ 1 6¼ 0; nb 2 ¼ b;
ð52Þ
bn2 l2 b2 þ kn2 l2 b2 ¼ a; nkn l2 bðnb þ 1Þ þ knkn l2 bðnb 1Þ ¼ ck. Solving this system yields 2 ; n 1 rffiffiffiffiffiffiffiffiffiffiffi n1 a ; 4b þ k > 0; l¼ 2n bþk 1 2ncðb þ kÞ k¼ n 1; a½bð3n 1Þ þ kðn þ 1Þ b¼
ð53Þ a > 0; n > 1;
that can also be obtained by using the sine ansatz (12). a Consequently, for 4bþk > 0 we obtain a family of compactons solutions 8n 1 h ion1 < 2ncðbþkÞ 2 n1 pffiffiffiffiffiffi a sin ; jlnj < p; ðx ctÞ 2n bþk a½bð3n1Þþkðnþ1Þ uðx; y; tÞ ¼ : 0; otherwise; ð54Þ
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A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
and uðx; y; tÞ ¼
8n < :
h
pffiffiffiffiffiffi 2ncðbþkÞ a cos2 n1 ðx 2n bþk a½bð3n1Þþkðnþ1Þ
þ y ctÞ
1 ion1
0;
; jlnj < p2 ; otherwise. ð55Þ
However, for
a 4bþk
< 0 we obtain solitary patterns solutions
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2ncðb þ kÞ a 2 n1 sinh ðx þ y ctÞ uðx; y; tÞ ¼ a½bð3n 1Þ þ kðn þ 1Þ 2n bþk
ð56Þ and 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2ncðb þ kÞ a 2 n1 cosh ðx þ y ctÞ uðx; y; tÞ ¼ . a½bð3n 1Þ þ kðn þ 1Þ 2n bþk
ð57Þ
4. Using the tanh method 4.1. The K(n, n, 2n) equation The K(n, n, 2n) is given before by ut þ aðun Þx þ b½un ðu2n Þxx x ¼ 0.
ð58Þ
We use the wave variable n = x ct to carry out Eq. (18) to the ODE 00
cu þ aun þ b½un ðu2n Þ ¼ 0;
ð59Þ
or equivalently 2
cu þ aun þ 2bnun1 u00 þ 2bnð2n 1Þun2 ðu0 Þ ¼ 0; upon integrating once. Balancing u(x, t) with u M ¼ ðn 1ÞM þ 4 þ M 2;
ð60Þ
n1 00
u we find ð61Þ
so that M ¼
2 . n1
ð62Þ
To get a closed form analytic solution, the parameter M should be an integer. A transformation formula 1
u ¼ vn1 ;
ð63Þ
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223
should be used to achieve our goal. This in turn transforms (60) to 2
2
2
cðn 1Þ v3 þ aðn 1Þ v2 2bnðn 1Þvv00 þ 2bnð3n 1Þðv0 Þ ¼ 0.
ð64Þ
Balancing vv00 and v3 gives M = 2. The tanh method allows us to use the substitution vðx; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 .
ð65Þ
Substituting (65) into (64), collecting the coefficients of each power of Y, and using Mathematica to solve the resulting system of algebraic equations we obtain að3n þ 1Þ ; 4cn a1 ¼ 0;
a0 ¼
ð66Þ
að3n þ 1Þ ; 4cn rffiffiffiffiffiffiffi n1 a M¼ ; 4n b
a2 ¼
1
where c is selected as a free parameter. Noting that uðx; tÞ ¼ vn1 , we find a family of solitary patterns solutions 1 rffiffiffiffiffiffiffi n1 4nc n1 a sinh2 uðx; tÞ ¼ ð67Þ ðx ctÞ að3n þ 1Þ 4n b and uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 4nc n1 a cosh2 ; ðx ctÞ að3n þ 1Þ 4n b
ð68Þ
valid for ab < 0. However, for ab > 0, we obtain a family of compactons solutions given by 8n 1 h pffiffi ion1 < 4nc a ðx ctÞ sin2 n1 ; jlnj < p; að3nþ1Þ 4n b uðx; tÞ ¼ ð69Þ : 0; otherwise; and uðx; tÞ ¼
8n < :
h
4nc cos2 n1 að3nþ1Þ 4n
0;
1 ion1 pffiffia ; jlnj < p2 ; ðx ctÞ b
ð70Þ
otherwise.
The results obtained above by using the tanh method are consistent with the results obtained by using the sine–cosine method.
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4.2. The K(n, 2n, n) equation The K(n, 2n, n) is given before by ut þ aðun Þx þ b½u2n ðun Þxx x ¼ 0.
ð71Þ
The wave variable n = x ct carries Eq. (71) to the ODE cu þ aun bnun1 u00 þ bnðn þ 1Þun2 ðu0 Þ2 ¼ 0; upon integrating once. Balancing u(x, t) with u
ð72Þ
n1 00
u we find
M ¼ ðn 1ÞM þ 4 þ M 2;
ð73Þ
so that M ¼
2 . n1
ð74Þ
It is normal to seek an integer value for the parameter M. Therefore, we set a transformation formula 1
u ¼ vn1 ;
ð75Þ
to achieve our goal. Consequently, (72) is reduced to 2
2
2
cðn 1Þ v3 þ aðn 1Þ v2 þ bnðn 1Þvv00 þ bnðv0 Þ ¼ 0.
ð76Þ
Balancing vv00 and v3 gives M = 2. The tanh method allows us to use the substitution vðx; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 .
ð77Þ
Substituting (77) into (76), collecting the coefficients of each power of Y, and using Mathematica to solve the resulting system of algebraic equations we obtain að3n 1Þ ; 2cn a1 ¼ 0; að3n 1Þ ; a2 ¼ 2cn rffiffiffiffiffiffiffi n1 a M¼ ; 2n b
a0 ¼
ð78Þ
1
where c is selected as a free parameter. Noting that uðx; tÞ ¼ vn1 , a family of solitary patterns solutions 1 rffiffiffiffiffiffiffi n1 2cn a 2 n1 sinh uðx; tÞ ¼ ð79Þ ðx ctÞ að3n 1Þ 2n b
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230
225
and uðx; tÞ ¼
1 rffiffiffiffiffiffiffi n1 2cn a 2 n1 cosh ; ðx ctÞ að3n 1Þ 2n b
ð80Þ
is readily obtained valid for ab < 0. However, for ab > 0, a family of compactons solutions given by 8n 1 h ion1 < 2 n1 pffiffi 2cn a ðx ctÞ sin ; jlnj < p; að3n1Þ 2n b uðx; tÞ ¼ ð81Þ : 0; otherwise; and uðx; tÞ ¼
8n < :
h
2nc cos2 n1 að3n1Þ 2n
1 ion1 pffiffia ; jlnj < p2 ; ðx ctÞ b
0;
ð82Þ
otherwise;
follows immediately. The results obtained above by using the tanh method are consistent with the results obtained by using the sine–cosine method. 4.3. The ZK(n, n, 2n) equation The ZK(n, n, 2n) is given by ut þ aðun Þx þ ½bun ðu2n Þxx þ kðun Þyy x ¼ 0.
ð83Þ
We use the travelling wave solutions of (83) in the form u(x, y, t) = u(ln), where the wave variable is n = x + y ct to carry out Eq. (83) to the ODE cu0 þ aðun Þ0 þ ½bun ðu2n Þ00 þ kðun Þ00 0 ¼ 0.
ð84Þ
Integrating once, setting the constant of integration to be zero, we obtain cu þ aun þ nð2b þ kÞun1 u00 þ nð2bð2n 1Þ þ kðn 1ÞÞun2 ðu0 Þ2 ¼ 0.
ð85Þ
Balancing u(x, y, t) with u
n1 00
u we find
M ¼ ðn 1ÞM þ 4 þ M 2;
ð86Þ
so that M ¼
2 . n1
ð87Þ
A closed form analytic solution is obtained if the parameter M is an integer. We therefore use a transformation 1
u ¼ vn1
ð88Þ
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to achieve our goal. This in turn transforms (85) to 2
2
cðn 1Þ v3 þ aðn 1Þ v2 nð2b þ kÞðn 1Þvv00 þ nð2bð3n 1Þ 2
þ kð2n 1ÞÞðv0 Þ ¼ 0.
ð89Þ
3
00
Balancing vv and v gives M = 2. As presented before, the tanh method admits the use of the substitution vðx; y; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 .
ð90Þ
Substituting (90) into (89), collecting the coefficients of each power of Y and proceeding as before we find að2bð3n þ 1Þ þ kðn þ 1ÞÞ ; 2cnð4b þ kÞ a1 ¼ 0;
a0 ¼
ð91Þ
að2bð3n þ 1Þ þ kðn þ 1ÞÞ ; 2cnð4b þ kÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 a ; M¼ 2n 4b þ k
a2 ¼
1
where c is selected as a free parameter. Noting that uðx; y; tÞ ¼ vn1 , we find a family of solitary patterns solutions 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2cnðb þ kÞ n1 a sinh2 ðx ctÞ uðx; y; tÞ ¼ að2bð3n þ 1Þ þ kðn þ 1ÞÞ 2n 4b þ k ð92Þ and
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2cnðb þ kÞ a 2 n1 cosh ðx ctÞ uðx; y; tÞ ¼ ; að2bð3n þ 1Þ þ kðn þ 1ÞÞ 2n 4b þ k
a 4bþk
a 4bþk
ð93Þ > 0, we obtain a family of compactons
< 0. However, for valid for solutions given by 8n 1 h pffiffiffiffiffiffiffiffi ion1 < 2cnðbþkÞ a sin2 n1 ; jlnj < p; ðx ctÞ 4n 4bþk að2bð3nþ1Þþkðnþ1ÞÞ uðx; y; tÞ ¼ : 0; otherwise;
ð94Þ and uðx; y; tÞ ¼
8n < :
h
pffiffiffiffiffiffiffi ffi 2cnðbþkÞ a cos2 n1 ðx 4n 4bþk að2bð3nþ1Þþkðnþ1ÞÞ
0;
ctÞ
1 ion1
;
jlnj < p2 ; otherwise. ð95Þ
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227
The results obtained above by using the tanh method are consistent with the results obtained by using the sine–cosine method. 4.4. The ZK(n, 2n, n) equation The ZK(n, 2n, n) is given by ut þ aðun Þx þ ½bu2n ðun Þxx þ kðun Þyy x ¼ 0.
ð96Þ
We use the travelling wave solutions u(x, y, t) = u(ln), where the wave variable is n = x + y ct to carry out Eq. (96) to the ODE 2
cu þ aun þ nð2b kÞun1 u00 þ nð2bðn þ 1Þ þ kðn 1ÞÞun2 ðu0 Þ ¼ 0. ð97Þ Balancing u(x, y, t) with u
n1 00
u we find
M ¼ ðn 1ÞM þ 4 þ M 2;
ð98Þ
so that 2 . n1
M ¼
ð99Þ
A closed form analytic solution is obtained if the parameter Mis an integer. We therefore use a transformation 1
u ¼ vn1 ;
ð100Þ
to achieve our goal. This in turn transforms (97) to cðn 1Þ2 v3 þ aðn 1Þ2 v2 þ nðb kÞðn 1Þvv00 þ nð2kn þ b kÞðv0 Þ2 ¼ 0. ð101Þ 00
3
Balancing vv and v gives M = 2. We therefore set vðx; y; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 .
ð102Þ
Substituting (102) into (101), collecting the coefficients of each power of Y and proceeding as before we get a0 ¼
aðbð3n 1Þ þ kðn þ 1ÞÞ ; 2cnðb þ kÞ
a1 ¼ 0; aðbð3n 1Þ þ kðn þ 1ÞÞ ; 2cnðb þ kÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 a ; M¼ 2n bþk
a2 ¼
ð103Þ
228
A.-M. Wazwaz / Appl. Math. Comput. 173 (2006) 213–230 1
where c is selected as a free parameter. Noting that uðx; y; tÞ ¼ vn1 , we find a family of solitary patterns solutions 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2cnðb þ kÞ a 2 n1 sinh ðx ctÞ uðx; y; tÞ ¼ aðbð3n 1Þ þ kðn þ 1ÞÞ 2n bþk ð104Þ and uðx; y; tÞ ¼
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 2cnðb þ kÞ n1 a cosh2 ðx ctÞ ; aðbð3n 1Þ þ kðn þ 1ÞÞ 2n bþk
ð105Þ a bþk
a bþk
< 0. However, for > 0, we obtain a family of compactons soluvalid for tions given by 8n 1 h ion1 < 2cnðbþkÞ 2 n1 pffiffiffiffiffiffi a sin ; jlnj < p; ðx ctÞ 2n bþk aðbð3n1Þþkðnþ1ÞÞ uðx; y; tÞ ¼ : 0; otherwise; ð106Þ and uðx; y; tÞ ¼
8n < :
h
pffiffiffiffiffi ffi 2cnðbþkÞ a cos2 n1 ðx 2n bþk aðbð3n1Þþkðnþ1ÞÞ
0;
ctÞ
1 ion1
; jlnj < p2 ; otherwise. ð107Þ
The results obtained above by using the tanh method are consistent with the results obtained by using the sine–cosine method.
5. Discussion The sine–cosine method and the tanh method were used to investigate variants of the K(n, n) and the ZK(n, n) equations. The study revealed compactons solutions and solitary patterns solutions for all examined variants. The study emphasized the fact that the two methods are reliable and one method complements the other in handling nonlinear problems. The obtained results clearly demonstrate the efficiency of the two methods used in this work. Moreover, the methods are capable of greatly minimizing the size of computational work compared to other existing techniques. Although the two methods provided the same solutions, but the sine–cosine method minimizes the size of calculation compared to the tanh method. In addition, specific restriction is usually required in that the value of M must
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229
be an integer to get closed form analytic solutions, therefore transformation formula is required to overcome this difficulty. The two methods worked successfully in handling nonlinear dispersive equations. This emphasizes the fact that the two methods are applicable to a wide variety of nonlinear problems.
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