Modified extended tanh-function method and its applications to nonlinear equations

Applied Mathematics and Computation 161 (2005) 403–412 www.elsevier.com/locate/amc

Modified extended tanh-function method and its applications to nonlinear equations S.A. Elwakil a, S.K. El-Labany b, M.A. Zahran R. Sabry b,* a

b

a,*

,

Theoretical Physics Group, Physics Department, Faculty of Science, Mansoura University, Mansoura, Egypt Theoretical Physics Group, Physics Department, Faculty of Science, Mansoura University, New Damietta, Damietta 34517, Egypt

Abstract New exact travelling wave solutions for the generalized shallow water wave equation, the improved Boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh-function method. The obtained results include rational, periodic, singular and solitary wave solutions. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Nonlinear evolution equations; Travelling wave solutions; Solitary wave solutions; Symbolic computation

1. Introduction Recently, based on an extended tanh-function method [1,2], we have presented the modified extended tanh-function (METF) method [3] to obtain more other kinds of exact solutions to nonlinear partial differential equations (PDEs ). In this paper, using the METF method, we will investigate three nonlinear physical models. The first one is the generalized shallow water wave (GSWW) equation

*

Corresponding authors. E-mail addresses: [email protected] (M.A. Zahran), [email protected] (R. Sabry). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.035

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uxxxt þ aux uxt þ but uxx  uxt  uxx ¼ 0;

ð1Þ

where a and b are arbirtrary, nonzero, constants. This equation, together with several variants, can be derived from the classical shallow water theory in the so-called Boussinesq approximation [4,5]. Two special cases of Eq. (1) have been discussed in the literature, a ¼ b and a ¼ 2b [5]. The GSWW equation is discussed by Hietarinta [6] who shows that it can be expressed in HirotaÕs bilinear form [7] if and only if either a ¼ b or a ¼ 2b. Clarkson and Mansfield [5] showed that Eq. (1) is completely integrable if and only if a ¼ b or a ¼ 2b. Also, they have shown that Eq. (1) is solvable by inverse scattering technique in these two special cases mentioned above. The second one is the improved Boussinesq (IB) equation 2

utt  uxx  uuxx  ðux Þ  uxxtt ¼ 0;

ð2Þ

which describes the vibrations in elastic rods [8–10] and DNA dynamics [11,12]. The third one is the coupled system of the approximate equations for long water waves 1 ut  uux  vx þ uxx ¼ 0; 2 1 vt  ðuvÞx  vxx ¼ 0; 2

ð3Þ

which was found by Whitham [13] and Broer [14]. The symmetries and conservation laws of the system (3) were discussed by Kupershmidt [15]. Wang [16], using the homogeneous balance method, found a soliton solution for the system (3). Now, let us recall the main steps of the modified extended tanh-function. Consider a given PDE, say in two independent variables H ðu; ux ; ut ; uxx ; . . .Þ ¼ 0:

ð4Þ

We first consider its travelling solutions uðx; tÞ ¼ uðfÞ, f ¼ x þ kt or f ¼ x  kt, then Eq. (4) becomes an ordinary differential equation. The next crucial step is that the solution we are looking for is expressed in the form uðfÞ ¼ a0 þ i ¼ 1

n X

xi ðai þ bi x2i Þ;

ð5Þ

and x0 ¼ b þ x2 ;

ð6Þ

where b is a parameter to be determined, x ¼ xðfÞ, x0 ¼ dx . The parameter m df can be found by balancing the highest-order linear term with the nonlinear terms [1,17–21]. Substitute (5) and (6) into the relevant ordinary differential equation will yield a system of algebraic equations with respect to ai , bi , b, k

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405

(where i ¼ 1; . . . ; m) because all the coefficients of xi have to vanish. With the aid of Mathematica, one can determine a0 , ai , bi , b, and k. The Riccati Eq. (6) has the general solutions  pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi bffi tanh½pffiffiffiffiffiffi bffi f ; with b < 0; x¼ ð7Þ  b coth½ bf ; with b < 0; 1 x¼ ; f

with b ¼ 0;

ð8Þ

and  pffiffiffi pffiffiffi b tan½ bf ; p ffiffi ffi p ffiffiffi x¼  b cot½ bf ;

with b > 0; with b > 0:

ð9Þ

2. The generalized shallow water wave equation To look for the travelling wave solution of Eq. (1), we use the transformation uðx; tÞ ¼ uðfÞ, f ¼ x  kt. Then Eq. (1) is reduced to the following ordinary differential equation 0000

kðu þ au0 u00 þ bu0 u00  u00 Þ þ u00 ¼ 0:

ð10Þ

0000

Balancing u with u0 u00 yields m ¼ 1. Therefore, we have u ¼ a0 þ a1 x þ b1 x1 :

ð11Þ

Substitute Eq. (11) into Eq. (10) and making use of Eq. (6), with the help of Mathematica, we get a system of algebraic equations, for a0 , a1 , b1 , b, and k, ba1 ½1 þ kð1 þ 8bÞ þ kða þ bÞðba1  b1 Þ ¼ 0; a1 ½1 þ kð1 þ 20bÞ þ kða þ bÞð2ba1  b1 Þ ¼ 0; ka1 ½12 þ a1 ða þ bÞ ¼ 0; bb1 ½1 þ kð1 þ 8bÞ þ kða þ bÞðba1  b1 Þ ¼ 0; b2 b1 ½1 þ kð1 þ 20bÞ þ kða þ bÞðba1  2b1 Þ ¼ 0; and b3 kb1 ½12b  ða þ bÞb1 ¼ 0: From which, we find b ¼ 0;

a1 ¼ 

12 ; ða þ bÞ

b1 ¼

1k ; kða þ bÞ

ð12Þ

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 1 1 12b ; b¼ 1þ ; a1 ¼ 0; b1 ¼ 4 k ða þ bÞ   1 12 ; b1 ¼ 0; b¼ 1þ ; a1 ¼  k ða þ bÞ

ð13Þ

ð14Þ

and 1 b¼ 16



 1 1þ ; k

a1 ¼ 

12 ; ða þ bÞ

b1 ¼

12b : ða þ bÞ

ð15Þ

According to Eq. (12), the solution to Eq. (1) reads 2

uðx; tÞ ¼

12k þ ð1 þ kÞðx  ktÞ þ c; kða þ bÞðx  ktÞ

where c, equal to a0 , is an arbitrary constant. Due to Eq. (13), for ð1 þ 1kÞ < 0, the solution to Eq. (1) reads "rffiffiffiffiffiffiffiffiffiffiffi # rffiffiffiffiffiffiffiffiffiffiffi 6 k1 k1 coth ðx  ktÞ þ c; uðx; tÞ ¼ ða þ bÞ k 4k while, for ð1 þ 1kÞ > 0 it is "rffiffiffiffiffiffiffiffiffiffiffi # rffiffiffiffiffiffiffiffiffiffiffi 6 1k 1k cot ðx  ktÞ þ c: uðx; tÞ ¼ ða þ bÞ k 4k From Eq. (14), it is clear that for the case ð1 þ 1kÞ < 0, we get "rffiffiffiffiffiffiffiffiffiffiffi # rffiffiffiffiffiffiffiffiffiffiffi 6 k1 k1 tanh ðx  ktÞ þ c; uðx; tÞ ¼ ða þ bÞ k 4k while, for ð1 þ 1kÞ > 0 it is "rffiffiffiffiffiffiffiffiffiffiffi # rffiffiffiffiffiffiffiffiffiffiffi 6 1k 1k tan ðx  ktÞ þ c: uðx; tÞ ¼  ða þ bÞ k 4k Eq. (15) leads to, for ð1 þ 1kÞ < 0 "rffiffiffiffiffiffiffiffiffiffiffi # rffiffiffiffiffiffiffiffiffiffiffi( 3 k1 k1 coth ðx  ktÞ uðx; tÞ ¼ ða þ bÞ k 16k "rffiffiffiffiffiffiffiffiffiffiffi #) k1 þ tanh ðx  ktÞ þ c; 16k

ð16Þ

ð17Þ

ð18Þ

ð19Þ

ð20Þ

ð21Þ

S.A. Elwakil et al. / Appl. Math. Comput. 161 (2005) 403–412

and

"rffiffiffiffiffiffiffiffiffiffiffi # rffiffiffiffiffiffiffiffiffiffiffi( 1k 1k cot ðx  ktÞ k 16k "rffiffiffiffiffiffiffiffiffiffiffi #) 1k ðx  ktÞ þ c;  tan 16k

407

3 uðx; tÞ ¼ ða þ bÞ

ð22Þ

for ð1 þ 1kÞ > 0. The solution in (16) is a rational-type solution. The solutions (19) and (20) are guaranteed by replacing the coth and cot-types solution of the Ricatti equation by tanh and tan-types in Eqs. (17) and (18), respectively. The solution (19) is a kink-type solitary wave solution. It is clear that all the obtained solutions (16)–(22) are valid for all values of a and b except for a 6¼ b. 3. The improved Boussinesq equation Let uðx; tÞ ¼ uðfÞ and f ¼ x þ kt, then Eq. (2) reduces to 2

0000

ðk2  u  1Þu00  ðu0 Þ  k2 u ¼ 0:

ð23Þ

It is clear that m ¼ 2, hence u ¼ a0 þ a1 x þ a2 x þ b1 x1 þ b2 x2 :

ð24Þ

Substitute Eq. (24) into Eq. (23) and making use of Eq. (6), with the help of Mathematica, we get a system of algebraic equations, for a0 , a1 , a2 , b1 , b2 , b, and k, b½a1 ð1 þ k2 ð1 þ 8kÞ þ a0 þ 3ba2 Þ þ a2 b1 ¼ 0; b½2a21 þ a2 ð4a0 þ 3ba2 þ 4 þ 4k2 ð1 þ 17bÞÞ ¼ 0; a1 ½1 þ a0 þ 9ba2 þ k2 ð1 þ 20bÞ þ a2 b1 ¼ 0; 3a21 þ 2a2 ½3 þ 3a0 þ 8ba2 þ 3k2 ð1 þ 40bÞ ¼ 0; a1 ð2k2 þ a2 Þ ¼ 0; a2 ð12k2 þ a2 Þ ¼ 0; bb1 ½1 þ k2 ð1 þ 8bÞ þ a0 þ b2 ðba1 þ 3b1 Þ ¼ 0; 2bb21 þ b2 ½4bð1 þ k2 ð1 þ 17bÞ þ a0 Þ þ 3b2 ¼ 0; b2 b1 ½1 þ k2 ð1 þ 20bÞ þ a0 þ bb2 ðba1 þ 9b1 Þ ¼ 0; b½3bb21 þ 6bb2 ð1 þ k2 ð1 þ 40bÞ þ a0 Þ þ 16b22 ¼ 0; b2 b1 ð2b2 k2 þ b2 Þ ¼ 0;

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and b2 b2 ð12b2 k2 þ b2 Þ ¼ 0: These equations lead to a0 ¼ 1 þ k2  8bk2 ; 2

2

a0 ¼ 1 þ k  8bk ;

a1 ¼ 0; a1 ¼ 0;

a2 ¼ 12k2 ; 2

a2 ¼ 12k ;

b1 ¼ 0;

b2 ¼ 0;

ð25Þ

b1 ¼ 0;

b2 ¼ 12b2 k2 ; ð26Þ

and a0 ¼ 1 þ k2  8bk2 ; 2 2

b2 ¼ 12b k ;

a1 ¼ 0;

a2 ¼ 0;

b1 ¼ 0;

where b 6¼ 0:

ð27Þ

Eq. (25) leads to three different solutions for Eq. (2). The first one is " # 12 uðx; tÞ ¼ 1 þ 1  ð28Þ k2 ; 2 ðx þ ktÞ for the case b ¼ 0. The second solution, for the case b < 0, reads pffiffiffiffiffiffiffi uðx; tÞ ¼ 1 þ ð1  8bÞk2 þ 12bk2 tanh2 ½ bðx þ ktÞ ;

ð29Þ

while, the third solution, for the case b > 0, is pffiffiffi uðx; tÞ ¼ 1 þ ð1  8bÞk2  12bk2 tan2 ½ bðx þ ktÞ :

ð30Þ

Due to Eq. (26), it is clear that for the case b < 0, we get pffiffiffiffiffiffiffi uðx; tÞ ¼ 1 þ k2  8bk2 þ 12bk2 coth2 ½ bðx þ ktÞ pffiffiffiffiffiffiffi þ 12bk2 tanh2 ½ bðx þ ktÞ ;

ð31Þ

while, for b > 0 it is pffiffiffi uðx; tÞ ¼ 1 þ k2  8bk2  12bk2 cot2 ½ bðx þ ktÞ pffiffiffi  12bk2 tan2 ½ bðx þ ktÞ : Finally, Eq. (27) leads to, for the case b < 0, pffiffiffiffiffiffiffi uðx; tÞ ¼ 1 þ ð1  8bÞk2 þ 12bk2 coth2 ½ bðx þ ktÞ ; while for the case b > 0, the solution to Eq. (2) reads pffiffiffi uðx; tÞ ¼ 1 þ ð1  8bÞk2  12bk2 cot2 ½ bðx þ ktÞ :

ð32Þ

ð33Þ

ð34Þ

The solution in (28) is a rational-type solution. The solution given in (29) is a bell shaped solitary wave solution. The solutions (33) and (34) are guaranteed

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409

by replacing the tanh and tan-types solution of the Ricatti equation by coth and cot-types in Eqs. (29) and (30), respectively.

4. The approximate equations for long water waves Substitute uðx; tÞ ¼ uðfÞ and vðx; tÞ ¼ vðfÞ where f ¼ x  kt into Eq. (3) results in 1 00 u  ðk þ uÞu0  v0 ¼ 0; 2 1 00 v þ ðu þ kÞv0 þ vu0 ¼ 0: 2

ð35Þ

Now, let us take the ans€ atz u ¼ a0 þ xða1 þ b1 x2 Þ; v ¼ s0 þ xðs1 þ r1 x2 Þ þ x2 ðs2 þ r2 x4 Þ;

ð36Þ

where x satisfies Eq. (6). Substitute Eq. (36) into Eq. (35) and making use of Eq. (6), with the help of Mathematica, we get a system of algebraic equations, for a0 , a1 , b1 , s0 , s1 , s2 , r1 , r2 , b, and k, bða1  a21  2s2 Þ ¼ 0; ka1  a0 a1  s1 ¼ 0; a1  a21  2s2 ¼ 0; bb1 þ b21 þ 2r2 ¼ 0; bðkb1 þ a0 b1 þ r1 Þ ¼ 0; bðbb1 þ b21 þ 2r2 Þ ¼ 0; ðk þ a0 Þðba1  b1 Þ þ r1  bs1 ¼ 0; bðs1 þ 2a1 s1 þ 2ks2 þ 2a0 s2 Þ ¼ 0; a1 s0 þ ks1 þ a0 s1 þ 4bs2 þ 3ba1 s2 þ b1 s2 ¼ 0; s1 þ 2a1 s1 þ 2ks2 þ 2a0 s2 ¼ 0; ð1 þ a1 Þs2 ¼ 0; br1 þ 2b1 r1 þ 2kr2 þ 2a0 r2 ¼ 0; ðkr1 þ a0 r1  4r2 þ a1 r2 þ b1 s0 Þb þ 3b1 r2 ¼ 0; bðbr1 þ 2b1 r1 þ 2kr2 þ 2a0 r2 Þ ¼ 0; ðb  b1 Þbr2 ¼ 0;

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and a1 ½r2  bs0 þ ðk þ a0 Þðr1  bs1 Þ þ b1 ðs0  bs2 Þ  r2  b2 s2 ¼ 0: Which with the help of Mathematica yields five different cases Case 1: a0 ¼ k; s1 ¼ 0;

a1 ¼ 1;

b1 ¼ 0;

r1 ¼ 0;

r2 ¼ 0;

s0 ¼ b;

s2 ¼ 1;

ð37Þ

Case 2: a0 ¼ k; s1 ¼ 0;

a1 ¼ 1;

b1 ¼ b;

r1 ¼ 0;

r2 ¼ 0;

s0 ¼ 0;

s2 ¼ 1;

ð38Þ

Case 3: a0 ¼ k; a1 ¼ 1; b1 ¼ b; s1 ¼ 0; s2 ¼ 1; s0 ¼ 2b;

r1 ¼ 0;

r2 ¼ b2 ; ð39Þ

Case 4: a0 ¼ k; s2 ¼ 0;

a1 ¼ 0; s0 ¼ b;

b1 ¼ b;

r1 ¼ 0;

r2 ¼ b2 ;

s1 ¼ 0;

b 6¼ 0;

ð40Þ

Case 5: a0 ¼ k; a1 ¼ 1; b1 ¼ b; s1 ¼ 0; s2 ¼ 0; b 6¼ 0:

r1 ¼ 0;

r2 ¼ b2 ;

s0 ¼ 0; ð41Þ

According to case 1, we have three different types of travelling wave solutions for u and v Type 1: for b ¼ 0 1 ; x  kt 1 vðx; tÞ ¼  ; 2 ðx  ktÞ uðx; tÞ ¼ k þ

ð42Þ

Type 2: for b < 0 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi b tanh½ bðx  ktÞ ; pffiffiffiffiffiffiffi vðx; tÞ ¼ b sech2 ½ bðx  ktÞ ;

uðx; tÞ ¼ k þ

ð43Þ

Type 3: for b > 0 pffiffiffi pffiffiffi b tan½ bðx  ktÞ ; pffiffiffi vðx; tÞ ¼ b sec2 ½ bðx  ktÞ ;

uðx; tÞ ¼ k 

ð44Þ

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411

Due to case 2, we have three different types of solutions. The first one, for b ¼ 0, will lead to the same solution obtained in Eq. (42). The second one, where b < 0, gives pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðx; tÞ ¼ k  b coth½ bðx  ktÞ þ b tanh½ bðx  ktÞ ; ð45Þ pffiffiffiffiffiffiffi vðx; tÞ ¼ b tanh2 ½ bðx  ktÞ ; while the third one, where b > 0, is pffiffiffi pffiffiffi pffiffiffi pffiffiffi uðx; tÞ ¼ k  b cot½ bðx  ktÞ  b tan½ bðx  ktÞ ; pffiffiffi vðx; tÞ ¼ b tan2 ½ bðx  ktÞ :

ð46Þ

Also, case 3 will lead to three different types of solutions. The first one, for b ¼ 0, will lead to the same solution obtained in Eq. (42). The second one, where b < 0, gives pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi uðx; tÞ ¼ k þ b coth½ bðx  ktÞ þ b tanh½ bðx  ktÞ ; ð47Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðx; tÞ ¼ b csch2 ½ bðx  ktÞ sech2 ½ bðx  ktÞ ; while the third one, where b > 0, is pffiffiffi pffiffiffi pffiffiffi pffiffiffi uðx; tÞ ¼ k þ b cot½ bðx  ktÞ  b tan½ bðx  ktÞ ; pffiffiffi pffiffiffi vðx; tÞ ¼ b csc2 ½ bðx  ktÞ sec2 ½ bðx  ktÞ :

ð48Þ

The solutions obtained from case 4 can be obtained directly from Eqs. (43) and (44) by replacing tanh and tan-functions by coth and cot-functions, respectively, so there is no need to list them here. Also, the solutions due to case 5 can be obtained directly from Eqs. (45) and (46) by replacing tanh and tanfunctions by coth and cot-functions, respectively, so there is no need to list them here. If we make the transformations k ! ða=2Þ  c and b ! ða2 =4Þ into Eq. (42), it will lead directly to the solitary wave solution obtained by Wang [16]. The remaining solutions include rational, periodical and singular solutions which appear to be new.

5. Conclusion We have obtained new travelling wave solutions for the generalized shallow water wave equation, the improved Boussinesq equation and the coupled system for the approximate equations for water waves. The obtained solutions include rational, periodical, singular and solitary wave solutions. Rational solutions may be helpful to explain certain physical phenomena. Because a rational solution is a disjoint union of manifolds, particle systems describing

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the motion of a pole of rational solutions for a kdV equation were analyzed in [22–24]. Triangle-type periodical solutions (i.e. Singular solitary wave solutions [25]) develop singularity at a finite point, i.e. for any fixed t ¼ t0 there exist an x0 at which these solutions blow up. There is much current interest in the formation of so called ‘‘hot-spots’’ or ‘‘blow-up’’ of solutions [25–27]. It appears that the singular solutions will model these physical phenomena.

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