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New solitary wave solutions with compact support and Jacobi elliptic function solutions for the nonlinearly dispersive Boussinesq equations
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Chaos, Solitons and Fractals 37 (2008) 792–798 www.elsevier.com/locate/chaos
New solitary wave solutions with compact support and Jacobi elliptic function solutions for the nonlinearly dispersive Boussinesq equations Mustafa Inc Department of Mathematics, Fırat University, 23119 Elazıg˘, Turkey Accepted 19 September 2006
Communicated by Prof. El Naschie
Abstract In this paper, we establish exact special solutions for the nonlinearly dispersive Boussinesq equations. A new extended sn–cn method is used for obtaining compacton, solitary patterns, solitary wave, periodic wave and Jacobi elliptic function solutions for this equation with minimal calculations. As a result, abundant new compactons:solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave, periodic wave solutions and Jacobi elliptic function solutions are obtained. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear wave phenomena appear in many fields, such as fluid mechanics, plasma physics, biology, hydrodynamics, solid state physics and optical fibers. These nonlinear phenomena are often related to nonlinear wave equations. In order to better understand these phenomena as well as further apply them in practical situations, it is important to seek their more exact solutions. Moreover, new exact solutions may help find new phenomena. Meantime powerful methods had been developed such as Backlund transformation [1,2], Darboux transformation [3], the inverse scattering transformation [4], the bilinear method [5], the tanh method [6,7], the sine–cosine method [8,9], the homogeneous balance method [10], the Riccati method [11] and the Jacobi elliptic function method [12–14], etc. In the well-known Korteweg–de Vries (KdV) equation ut auux þ uxxx ¼ 0;
ð1Þ
the nonlinear term uux causes the steepening of the wave form. On the other hand, the dispersion term uxxx in this equation makes the wave form spread [15]. The soliton is defined [15–17] as a nonlinear wave that has the following properties:
E-mail address: minc@firat.edu.tr 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.064
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(a) Some certain waves propogates which does not change its special behaviour, (b) Reginal waves does not loose their properties and also they are stable towards to the collisions. Another words it means that the solution has important properties of particle [18]. Solution is replaced the part of the region which is localized wave and also it has an infinite support or exponential term of wave equation. Indeed nonlinear scattering or dispersive equation K(m, n) is a special case of KdV equation can be expressed as ut þ aðum Þx þ ðun Þxxx ¼ 0;
ð2Þ
m; n > 1
by Rosenau and Hyman [19] and also sketch the graph of it is called compacton presented firstly by these authors. The solution of Eq. (2) is also very interesting because of the local nature in which can serve the reflaction of a wide range of extraordinary in nature [20]. Compactons were shown that it strikes elastically and also vanish in finite nuclei of exterior region. The compactons is a solitons which classifies not contain the infinite tail or carrying no wings and the extensiveness of compactons does not depends on amplitude. The investigation on compactons presents different proceses. For instance, internal and external fusion, deformed bundle in hydrodynamic models, deformed particle, the fussion of fluid drip in nuclear physics. In further studies at physical properties of compacton structures and the effection on nonlinear dispersion on pattern formation can be seen in Refs. [18–27]. To understand the role of nonlinear dispersions in the formation of patterns in liquid drops, we consider the following nonlinearly dispersive Boussinesq equations (shortly B(m, n) equation): utt uxx aðun Þxx þ bðum Þxxxx ¼ 0;
m; n > 1:
ð3Þ
If we take m = 1 and n = 2 in Eq. (3), then we get the Boussinesq equation [28] which has been proposed as a model of a large number of LC-circuits and as a model to describe vibrations of a single one-dimensional dense lattice. Recently, Yan and Bluman [29] studied Eq. (3) by the sine–cosine method and they obtained the following compacton and solitary pattern solutions: rffiffiffiffiffiffiffi 1=ðm1Þ 2mðk2 1Þ am 1 2 uðx; tÞ ¼ cos ðx ktÞ ; ð4Þ aðm þ 1Þ b 2m rffiffiffiffiffiffiffi 1=ðm1Þ 2mðk2 1Þ am 1 ð5Þ uðx; tÞ ¼ sin2 ðx ktÞ aðm þ 1Þ b 2m pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi for j ab m1 ðx ktÞj 6 p2 and u(x, t) = 0 for j ab m1 ðx ktÞj > p2. 2m 2m ffiffi ffi r 1=ðm1Þ 2mðk2 1Þ am 1 uðx; tÞ ¼ ; ð6Þ cosh2 ðx ktÞ aðm þ 1Þ b 2m r ffiffi ffi 1=ðm1Þ 2mð1 k2 Þ am 1 sinh2 ðx ktÞ : ð7Þ uðx; tÞ ¼ aðm þ 1Þ b 2m This paper is organized as follows: Section 1 is introduction. In Section 2, we present a new extended sn–cn method. In Section 3, we give a new type compacton, solitary pattern and Jacobi elliptic function solutions of Eq. (3). Finally, some conclusions are given in Section 4.
2. The extended sn–cn method In this section, we present a new general method, namely, the extended sn–cn method, to obtain exact special solutions for the genuinely nonlinear dispersive equations based on the methods given in [30,31] by Shang and in [32,33] by Inc. Given nonlinear partial differential equation, as follows: F ðu; ux ; ut ; uxx ; uxt ; . . .Þ ¼ 0;
ð8Þ
where F is a nonlinear function and the subscripts denote the partial derivatives. Let u(x, t) = u(n), n = x + ct, then Eq. (8) reduces to a nonlinear ordinary differential equation (ODE): F ðu; u0 ; u00 ; . . .Þ ¼ 0:
ð9Þ
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M. Inc / Chaos, Solitons and Fractals 37 (2008) 792–798
We suppose that the solution of Eq. (9) is of the form: n X ai ðvðnÞÞi ; uðx; tÞ ¼ uðnÞ ¼
ð10Þ
i¼0
where ai (i = 0, 1, 2, . . . , n) are constants to be determined and v = v(n) satisfies a nonlinear ODE of first order pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dv ¼ e a þ bv2 ; a; b 2 R; v0 ¼ dn
ð11Þ
where e = ± 1. Substituting (10) into (9) and using (11), then we obtain a set of nonlinear algebraic equations on ai (i = 0, 1, 2, . . . n), a, b, c. With the aid of Mathematica, we can solve the set of algebraic equations and obtain all the constants. So Eq. (11) has the following Jacobi elliptic function solutions: rffiffiffi a pffiffiffiffiffiffiffi ð12Þ vðnÞ ¼ e sn½ bðn þ n0 Þ; m; a; b > 0; 0 < m < 1; e ¼ 1; b rffiffiffiffiffiffiffi a pffiffiffiffiffiffiffi ð13Þ vðnÞ ¼ cn½ bðn þ n0 Þ; m; a < 0; b > 0; 0 < m < 1; e ¼ 1; b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ð14Þ vðnÞ ¼ c1 sn½ bðn þ n0 Þ; m þ c2 cn½ bðn þ n0 Þ; m; a ¼ 0; b < 0; e ¼ 1: If we take m ! 1, then we obtain solitary wave solutions rffiffiffi pffiffiffiffiffiffiffi a tanh½ bðx þ ct þ þn0 Þ; a; b > 0; e ¼ 1; vðnÞ ¼ e b rffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a vðnÞ ¼ sech½ bðx þ ct þ þn0 Þ; a < 0; b > 0; e ¼ 1; b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðnÞ ¼ c1 tanh½ bðx þ ct þ þn0 Þ þ c2 sech½ bðx þ ct þ þn0 Þ;
ð15Þ ð16Þ a ¼ 0; b < 0:
If we take m ! 0, then we get the solitary wave solutions with compact support rffiffiffi pffiffiffiffiffiffiffi a sin½ bðx þ ct þ þn0 Þ; a; b > 0; e ¼ 1; vðnÞ ¼ e b rffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a vðnÞ ¼ cos½ bðx þ ct þ þn0 Þ; a < 0; b > 0; e ¼ 1; b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðnÞ ¼ c1 sin½ bðx þ ctÞ þ c2 cos½ bðx þ ct þ þn0 Þ; a ¼ 0; b < 0; e ¼ 1;
ð17Þ
ð18Þ ð19Þ ð20Þ
where n0 is an arbitrary real parameter. Thus, many exact special solutions of nonlinear partial differential equation (8) are obtained by using (10), (12)–(20). Remark 1. Since when m ! 1, sn(n; m) ! tanh n and cn(n; m) ! sech n; while when m ! 0, sn(n; m) ! sinn and cn(n; m) ! cos n, thus it is easy to see that the present method is used to obtain both compacton solutions, solitary wave solutions and Jacobi elliptic function solutions. Therefore, it is easy to see that the solutions derived from the present method include both the results of the extend sine–cosine method [30,31], the sn–cn function method [32,33], the sine–cosine method [34], and new solutions.
3. Some illustrative examples In this section, we would like to choose two special equations, B(2, 2) and B(3, 3) to illustrate the concrete scheme of our method. Example 1. We first consider the B(2, 2) equation (a = b = 1 in Eq. (3)): utt uxx ðu2 Þxx þ ðu2 Þxxxx ¼ 0;
ð21Þ
where u(x, t) is an unknown function. Integrating twice (19) and setting the constants of integration to be zero, we have ðc2 1Þu u2 þ ðu2 Þ00 ¼ 0:
ð22Þ
M. Inc / Chaos, Solitons and Fractals 37 (2008) 792–798
We suppose that (22) has solutions in the form of n X ai ðvÞi : uðx; tÞ ¼ uðnÞ ¼
795
ð23Þ
i¼0
We select the balance constant n = 1. Thus we can assume that uðx; tÞ ¼ uðnÞ ¼ a0 þ a1 v:
ð24Þ
Substituting (24) into (22) and using (12)–(20), one gets a set of nonlinear algebraic equations c2 a0 a0 a20 þ 2a21 ¼ 0; c2 a1 2a0 a1 2a0 a1 b ¼ 0;
ð25Þ
a21 þ 4a21 b ¼ 0: To obtain a nontrivial solution, we need to suppose a1 5 0. Solving Eq. (25), we have 2 a0 ¼ c2 ; 3
1 b¼ ; 4
a1 ;
c 6¼ 0 arbitrary:
Thus we get the following new Jacobi elliptic function solutions: pffiffiffi 1 2 uðx; tÞ ¼ c2 þ 2ei asn iðx þ ct þ n0 Þ; m ; a > 0; 0 < m < 1; e ¼ 1; 3 2 pffiffiffi 2 2 1 uðx; tÞ ¼ c þ 2 acn ðx þ ct þ n0 Þ; m ; a < 0; 0 < m < 1; e ¼ 1; 3 2 2 2 1 1 uðx; tÞ ¼ c þ c1 sn ðx þ ct þ n0 Þ; m þ c2 cn ðx þ ct þ n0 Þ; m ; a ¼ 0; b < 0: 3 2 2 By combining Eqs. (15)–(17), (24) and (26), we obtain the following solitary wave solutions for Eq. (21) 2 1 pffiffiffi 1 uðx; tÞ ¼ c2 þ ei a tanh ðx þ ct þ n0 Þ ; a > 0; e ¼ 1; 3 2 2 2 2 1 pffiffiffi 1 uðx; tÞ ¼ c þ a sech ðx þ ct þ n0 Þ ; a < 0; e ¼ 1; 3 2 2 1 1 uðx; tÞ ¼ c1 tanh ðx þ ct þ n0 Þ þ c2 sech ðx þ ct þ n0 Þ ; a ¼ 0; b < 0: 2 2
ð26Þ
ð27Þ ð28Þ ð29Þ
ð30Þ ð31Þ ð32Þ
By combining Eqs. (18)–(20), (24) and (26), we obtain solitary wave solutions with compact support of the B(2, 2) given as ( pffiffiffi 2 2 c þ 1 ei a sin 12 ðx þ ct þ n0 Þ ; 0 6 12 ðx þ ct þ n0 Þ 6 2p; uðx; tÞ ¼ 32 2 2 ð33Þ c; Otherwise; 3 ( pffiffiffi 2 2 c þ 12 a cos 12 ðx þ ct þ n0 Þ ; 12 ðx þ ct þ n0 Þ 6 p; 3 ð34Þ uðx; tÞ ¼ 2 2 c; Otherwise; 3 ( 2 2 c þ c1 sin 12 ðx þ ctÞ þ c2 cos 12 ðx þ ctÞ ; 2h0 6 12 ðx þ ct þ n0 Þ 6 2ðp h0 Þ; ð35Þ uðx; tÞ ¼ 32 2 c; Otherwise; 3 1j ffi where h0 ¼ arccos pjcffiffiffiffiffiffiffiffi . 2 2
c1 þc2
Example 2. We now consider the B(3, 3) equation utt uxx ðu3 Þxx þ ðu3 Þxxxx ¼ 0;
ð36Þ
where u(x, t) is an unknown function. Integrating twice (36) and setting the constants of integration to be zero, we have ðc2 1Þu u3 þ ðu3 Þ00 ¼ 0: We select the balance constant n = 1. Thus we can assume that
ð37Þ
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uðx; tÞ ¼ uðnÞ ¼ a0 þ a1 v:
ð38Þ
Substituting (38) into (37) and, using (10) and (11), one gets a set of nonlinear algebraic equations ðc2 1Þa0 a30 þ 6a0 a21 a þ 6a31 a ¼ 0; ðc2 1Þa1 þ 3a20 a1 þ 3a20 a1 b ¼ 0; 3a0 a21 þ 12a0 a21 b þ 6a31 b ¼ 0;
ð39Þ
a31 þ 9a31 b ¼ 0: To obtain a nontrivial solution, we need to suppose a1 5 0. Solving Eq. (39), we have ffi ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 ¼ 3ð1 c2 Þ; b ¼ ; a1 ¼ 3ð1 c2 Þ; c 6¼ 0 arbitrary: 2 9 4 Thus we get the following new Jacobi elliptic function solutions: ffi 15ei pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ 3ð1 c2 Þ þ 3að1 c2 Þsn ðx þ ct þ n0 Þ; m ; a > 0; 0 < m < 1; e ¼ 1; 2 4 3 ffi 15e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3ð1 c2 Þ þ 3að1 c2 Þcn ðx þ ct þ n0 Þ; m ; a < 0; 0 < m < 1; e ¼ 1; uðx; tÞ ¼ 2 4 3 ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 uðx; tÞ ¼ 3ð1 c2 Þ þ c1 sn ðx þ ct þ n0 Þ; m þ c2 cn ðx þ ct þ n0 Þ; m ; a ¼ 0; b < 0: 2 3 3 By combining Eqs. (15)–(17), (39) and (40), we obtain the following solitary wave solutions for Eq. (21): ffi 15ei pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 uðx; tÞ ¼ 3ð1 c2 Þ þ 3að1 c2 Þ tanh ðx þ ct þ n0 Þ ; a > 0; e ¼ 1; 2 4 3 ffi 15e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 uðx; tÞ ¼ 3ð1 c2 Þ þ 3að1 c2 Þ sech ðx þ ct þ n0 Þ ; a < 0; e ¼ 1; 2 4 3 1 1 uðx; tÞ ¼ c1 tanh2 ðx þ ct þ n0 Þ þ c2 sech2 ðx þ ct þ n0 Þ ; a ¼ 0; b < 0: 3 3
ð40Þ
ð41Þ ð42Þ ð43Þ
ð44Þ ð45Þ ð46Þ
By combining Eqs. (18)–(20), (39) and (40), we obtain solitary wave solutions with compact support of the B(3, 3) given as ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3ð1 c2 Þ þ 15ei 3að1 c2 Þ sin 13 ðx þ ct þ n0 Þ ; 0 6 13 ðx þ ct þ n0 Þ 6 2p; 4 uðx; tÞ ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð47Þ 1 Otherwise; 3ð1 c2 Þ; 2 ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ; 3ð1 c2 Þ þ 15e 3að1 c2 Þ cos 13 ðx þ ct þ n0 Þ ; 13 ðx þ ct þ n0 Þ 6 3p 2 4 2 ð48Þ uðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Otherwise; 3ð1 c Þ; 2 ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3ð1 c2 Þ þ c1 sin 12 ðx þ ctÞ þ c2 cos 12 ðx þ ctÞ ; 12 ðx þ ct þ n0 Þ 6 p2 ; ð49Þ uðx; tÞ ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Otherwise: 3ð1 c2 Þ; 2
4. General formulas of the compacton solutions of B(m, n) equations In this section, we investigate a general formula for the exact special solutions of the B(m, n) equations with m = n > 1. We have obtained many types of solitary wave, Jacobi elliptic function and compacton solutions of B(2, 2) and B(3, 3) equations. We consider B(n, n) equations utt uxx ðun Þxx þ ðun Þxxxx ¼ 0
ð50Þ
for all integers n > 1. Substituting travelling wave solution u(x, t) = u(x + ct) = u(n) into (50) and integrating the resultant equation two with respect to n, thus we get the following a nonlinear ODE: ðc 1Þu un þ ðun Þ00 ¼ 0:
ð51Þ
In order to solve Eq. (51) we introduce an auxiliary function u by the following transformation: 1
uðnÞ ¼ ½uðnÞn1 :
ð52Þ
M. Inc / Chaos, Solitons and Fractals 37 (2008) 792–798
797
Thus, we have ðc 1ÞuðnÞ u2 ðnÞ þ
nðc 1Þ ðn 1Þ2
ðu0 Þ2 ðnÞ þ
nðc 1Þ uðnÞu00 ðnÞ ¼ 0: n1
ð53Þ
We first suppose that Eq. (53) has solutions of the form uðnÞ ¼ a0 þ a1 vðnÞ
ð54Þ
and v(n) is the solution of Eq. (11). Substituting (54) into (53) yields a0 ¼
nðc 1Þ ; nþ1
b¼
ðn 1Þ2 ; n2 ðc 1Þ
a¼
ðn 1Þ2 ðc 1Þ ðn þ 1Þ2 a21
;
a1 6¼ 0;
c 6¼ 1:
So Eq. (11) has the following another new Jacobi elliptic function solutions: 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 ðn þ n0 Þ; m sn ; vðnÞ ¼ ie a1 ðn þ 1Þ n c1 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 cn ; ðn þ n0 Þ; m vðnÞ ¼ a1 ðn þ 1Þ n c1 1 ( " # " #)n1 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi n1 1 n1 1 : ðn þ n0 Þ; m þ c2 cn ðn þ n0 Þ; m vðnÞ ¼ c1 sn n c1 n c1 If we take m ! 1, then we obtain solitary wave solutions 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 vðnÞ ¼ ie tanh ; ½ðx þ ct þ n0 Þ a1 ðn þ 1Þ n c1 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 sech ; vðnÞ ¼ ðx þ ct þ n0 Þ a1 ðn þ 1Þ n c1 1 ( " # " #)n1 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi n1 1 n1 1 : ðx þ ct þ n0 Þ þ c2 sech ðx þ ct þ n0 Þ vðnÞ ¼ c1 tanh n c1 n c1
ð55Þ
ð56Þ
ð57Þ
ð58Þ
ð59Þ
ð60Þ
ð61Þ
If we take m ! 0, then we get that the B(n, n) equation with n being odd integer >1 admits the solitary wave solutions with compact support 8n 1 qffiffiffiffiffiffi h qffiffiffiffiffiffi ion1 > > 1 1 np < ie anðc1Þ ðx þ ct þ n0 Þ ðx þ ct þ n0 Þ 6 2ðn1Þ sin n1 ; n1 ; n c1 n c1 1 ðnþ1Þ vðnÞ ¼
ð62Þ 1 > > : ie nðc1Þ n1 ; Otherwise; a1 ðnþ1Þ 8n 1 qffiffiffiffiffiffi h qffiffiffiffiffiffi ion1 > n1 c1Þ > n1 1 1 np < anððnþ1Þ ðx þ ct þ n ðx þ ct þ n cos Þ ; Þ ; 6 2ðn1Þ 0 0 n c1 n c1 1 vðnÞ ¼
ð63Þ 1 > nðc1Þ n1 > : ; Otherwise; a1 ðnþ1Þ 1 ( " # " #)n1 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi n1 1 n1 1 ðx þ ct þ n0 Þ þ c2 cos ðx þ ct þ n0 Þ ð64Þ vðnÞ ¼ c1 sin n c1 n c1 qffiffiffiffiffiffi 1 np for j n1 and v(n) = 0 for otherwise. ðx þ ct þ n0 Þj 6 2ðn1Þ n c1 In addition, we obtain that the B(n, n) equation with n being even integer >1 admits the multiple exact special solutions with solitary patterns 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 ðx þ ct þ n0 Þ vðnÞ ¼ ; ð65Þ sinh a1 ðn þ 1Þ n 1c
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M. Inc / Chaos, Solitons and Fractals 37 (2008) 792–798
(
1 " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 cosh vðnÞ ¼ ; ðx þ ct þ n0 Þ a1 ðn þ 1Þ n 1c 1 ( " # " #)n1 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi n1 1 n1 1 : ðx þ ct þ n0 Þ c2 cosh ðx þ ct þ n0 Þ vðnÞ ¼ c1 sinh n 1c n 1c
ð66Þ
ð67Þ
5. Conclusions In this paper we obtained several types of compacton, solitary patterns, Jacobi elliptic function and solitary wave solutions of the B(2, 2) and B(3, 3) equations by a new extended sn–cn method. Some conclusions of Wazwaz [34,35] are special cases of ours. In addition, the presented solutions of (21) and (36) were not found by any of the methods [23–35]. These solutions may be useful to explain some physical phenomena. The present method is direct and efficient to obtain new compacton, Jacobi elliptic function and solitary wave solutions of Eq. (3). Our method is very easily applied to both this type of nonlinear dispersive and the modified nonlinear dispersive equations in higher dimensional spaces.
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Chaos, Solitons and Fractals 37 (2008) 792–798 www.elsevier.com/locate/chaos
New solitary wave solutions with compact support and Jacobi elliptic function solutions for the nonlinearly dispersive Boussinesq equations Mustafa Inc Department of Mathematics, Fırat University, 23119 Elazıg˘, Turkey Accepted 19 September 2006
Communicated by Prof. El Naschie
Abstract In this paper, we establish exact special solutions for the nonlinearly dispersive Boussinesq equations. A new extended sn–cn method is used for obtaining compacton, solitary patterns, solitary wave, periodic wave and Jacobi elliptic function solutions for this equation with minimal calculations. As a result, abundant new compactons:solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave, periodic wave solutions and Jacobi elliptic function solutions are obtained. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Nonlinear wave phenomena appear in many fields, such as fluid mechanics, plasma physics, biology, hydrodynamics, solid state physics and optical fibers. These nonlinear phenomena are often related to nonlinear wave equations. In order to better understand these phenomena as well as further apply them in practical situations, it is important to seek their more exact solutions. Moreover, new exact solutions may help find new phenomena. Meantime powerful methods had been developed such as Backlund transformation [1,2], Darboux transformation [3], the inverse scattering transformation [4], the bilinear method [5], the tanh method [6,7], the sine–cosine method [8,9], the homogeneous balance method [10], the Riccati method [11] and the Jacobi elliptic function method [12–14], etc. In the well-known Korteweg–de Vries (KdV) equation ut auux þ uxxx ¼ 0;
ð1Þ
the nonlinear term uux causes the steepening of the wave form. On the other hand, the dispersion term uxxx in this equation makes the wave form spread [15]. The soliton is defined [15–17] as a nonlinear wave that has the following properties:
E-mail address: minc@firat.edu.tr 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.064
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(a) Some certain waves propogates which does not change its special behaviour, (b) Reginal waves does not loose their properties and also they are stable towards to the collisions. Another words it means that the solution has important properties of particle [18]. Solution is replaced the part of the region which is localized wave and also it has an infinite support or exponential term of wave equation. Indeed nonlinear scattering or dispersive equation K(m, n) is a special case of KdV equation can be expressed as ut þ aðum Þx þ ðun Þxxx ¼ 0;
ð2Þ
m; n > 1
by Rosenau and Hyman [19] and also sketch the graph of it is called compacton presented firstly by these authors. The solution of Eq. (2) is also very interesting because of the local nature in which can serve the reflaction of a wide range of extraordinary in nature [20]. Compactons were shown that it strikes elastically and also vanish in finite nuclei of exterior region. The compactons is a solitons which classifies not contain the infinite tail or carrying no wings and the extensiveness of compactons does not depends on amplitude. The investigation on compactons presents different proceses. For instance, internal and external fusion, deformed bundle in hydrodynamic models, deformed particle, the fussion of fluid drip in nuclear physics. In further studies at physical properties of compacton structures and the effection on nonlinear dispersion on pattern formation can be seen in Refs. [18–27]. To understand the role of nonlinear dispersions in the formation of patterns in liquid drops, we consider the following nonlinearly dispersive Boussinesq equations (shortly B(m, n) equation): utt uxx aðun Þxx þ bðum Þxxxx ¼ 0;
m; n > 1:
ð3Þ
If we take m = 1 and n = 2 in Eq. (3), then we get the Boussinesq equation [28] which has been proposed as a model of a large number of LC-circuits and as a model to describe vibrations of a single one-dimensional dense lattice. Recently, Yan and Bluman [29] studied Eq. (3) by the sine–cosine method and they obtained the following compacton and solitary pattern solutions: rffiffiffiffiffiffiffi 1=ðm1Þ 2mðk2 1Þ am 1 2 uðx; tÞ ¼ cos ðx ktÞ ; ð4Þ aðm þ 1Þ b 2m rffiffiffiffiffiffiffi 1=ðm1Þ 2mðk2 1Þ am 1 ð5Þ uðx; tÞ ¼ sin2 ðx ktÞ aðm þ 1Þ b 2m pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi for j ab m1 ðx ktÞj 6 p2 and u(x, t) = 0 for j ab m1 ðx ktÞj > p2. 2m 2m ffiffi ffi r 1=ðm1Þ 2mðk2 1Þ am 1 uðx; tÞ ¼ ; ð6Þ cosh2 ðx ktÞ aðm þ 1Þ b 2m r ffiffi ffi 1=ðm1Þ 2mð1 k2 Þ am 1 sinh2 ðx ktÞ : ð7Þ uðx; tÞ ¼ aðm þ 1Þ b 2m This paper is organized as follows: Section 1 is introduction. In Section 2, we present a new extended sn–cn method. In Section 3, we give a new type compacton, solitary pattern and Jacobi elliptic function solutions of Eq. (3). Finally, some conclusions are given in Section 4.
2. The extended sn–cn method In this section, we present a new general method, namely, the extended sn–cn method, to obtain exact special solutions for the genuinely nonlinear dispersive equations based on the methods given in [30,31] by Shang and in [32,33] by Inc. Given nonlinear partial differential equation, as follows: F ðu; ux ; ut ; uxx ; uxt ; . . .Þ ¼ 0;
ð8Þ
where F is a nonlinear function and the subscripts denote the partial derivatives. Let u(x, t) = u(n), n = x + ct, then Eq. (8) reduces to a nonlinear ordinary differential equation (ODE): F ðu; u0 ; u00 ; . . .Þ ¼ 0:
ð9Þ
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M. Inc / Chaos, Solitons and Fractals 37 (2008) 792–798
We suppose that the solution of Eq. (9) is of the form: n X ai ðvðnÞÞi ; uðx; tÞ ¼ uðnÞ ¼
ð10Þ
i¼0
where ai (i = 0, 1, 2, . . . , n) are constants to be determined and v = v(n) satisfies a nonlinear ODE of first order pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dv ¼ e a þ bv2 ; a; b 2 R; v0 ¼ dn
ð11Þ
where e = ± 1. Substituting (10) into (9) and using (11), then we obtain a set of nonlinear algebraic equations on ai (i = 0, 1, 2, . . . n), a, b, c. With the aid of Mathematica, we can solve the set of algebraic equations and obtain all the constants. So Eq. (11) has the following Jacobi elliptic function solutions: rffiffiffi a pffiffiffiffiffiffiffi ð12Þ vðnÞ ¼ e sn½ bðn þ n0 Þ; m; a; b > 0; 0 < m < 1; e ¼ 1; b rffiffiffiffiffiffiffi a pffiffiffiffiffiffiffi ð13Þ vðnÞ ¼ cn½ bðn þ n0 Þ; m; a < 0; b > 0; 0 < m < 1; e ¼ 1; b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ð14Þ vðnÞ ¼ c1 sn½ bðn þ n0 Þ; m þ c2 cn½ bðn þ n0 Þ; m; a ¼ 0; b < 0; e ¼ 1: If we take m ! 1, then we obtain solitary wave solutions rffiffiffi pffiffiffiffiffiffiffi a tanh½ bðx þ ct þ þn0 Þ; a; b > 0; e ¼ 1; vðnÞ ¼ e b rffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a vðnÞ ¼ sech½ bðx þ ct þ þn0 Þ; a < 0; b > 0; e ¼ 1; b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðnÞ ¼ c1 tanh½ bðx þ ct þ þn0 Þ þ c2 sech½ bðx þ ct þ þn0 Þ;
ð15Þ ð16Þ a ¼ 0; b < 0:
If we take m ! 0, then we get the solitary wave solutions with compact support rffiffiffi pffiffiffiffiffiffiffi a sin½ bðx þ ct þ þn0 Þ; a; b > 0; e ¼ 1; vðnÞ ¼ e b rffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a vðnÞ ¼ cos½ bðx þ ct þ þn0 Þ; a < 0; b > 0; e ¼ 1; b pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi vðnÞ ¼ c1 sin½ bðx þ ctÞ þ c2 cos½ bðx þ ct þ þn0 Þ; a ¼ 0; b < 0; e ¼ 1;
ð17Þ
ð18Þ ð19Þ ð20Þ
where n0 is an arbitrary real parameter. Thus, many exact special solutions of nonlinear partial differential equation (8) are obtained by using (10), (12)–(20). Remark 1. Since when m ! 1, sn(n; m) ! tanh n and cn(n; m) ! sech n; while when m ! 0, sn(n; m) ! sinn and cn(n; m) ! cos n, thus it is easy to see that the present method is used to obtain both compacton solutions, solitary wave solutions and Jacobi elliptic function solutions. Therefore, it is easy to see that the solutions derived from the present method include both the results of the extend sine–cosine method [30,31], the sn–cn function method [32,33], the sine–cosine method [34], and new solutions.
3. Some illustrative examples In this section, we would like to choose two special equations, B(2, 2) and B(3, 3) to illustrate the concrete scheme of our method. Example 1. We first consider the B(2, 2) equation (a = b = 1 in Eq. (3)): utt uxx ðu2 Þxx þ ðu2 Þxxxx ¼ 0;
ð21Þ
where u(x, t) is an unknown function. Integrating twice (19) and setting the constants of integration to be zero, we have ðc2 1Þu u2 þ ðu2 Þ00 ¼ 0:
ð22Þ
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We suppose that (22) has solutions in the form of n X ai ðvÞi : uðx; tÞ ¼ uðnÞ ¼
795
ð23Þ
i¼0
We select the balance constant n = 1. Thus we can assume that uðx; tÞ ¼ uðnÞ ¼ a0 þ a1 v:
ð24Þ
Substituting (24) into (22) and using (12)–(20), one gets a set of nonlinear algebraic equations c2 a0 a0 a20 þ 2a21 ¼ 0; c2 a1 2a0 a1 2a0 a1 b ¼ 0;
ð25Þ
a21 þ 4a21 b ¼ 0: To obtain a nontrivial solution, we need to suppose a1 5 0. Solving Eq. (25), we have 2 a0 ¼ c2 ; 3
1 b¼ ; 4
a1 ;
c 6¼ 0 arbitrary:
Thus we get the following new Jacobi elliptic function solutions: pffiffiffi 1 2 uðx; tÞ ¼ c2 þ 2ei asn iðx þ ct þ n0 Þ; m ; a > 0; 0 < m < 1; e ¼ 1; 3 2 pffiffiffi 2 2 1 uðx; tÞ ¼ c þ 2 acn ðx þ ct þ n0 Þ; m ; a < 0; 0 < m < 1; e ¼ 1; 3 2 2 2 1 1 uðx; tÞ ¼ c þ c1 sn ðx þ ct þ n0 Þ; m þ c2 cn ðx þ ct þ n0 Þ; m ; a ¼ 0; b < 0: 3 2 2 By combining Eqs. (15)–(17), (24) and (26), we obtain the following solitary wave solutions for Eq. (21) 2 1 pffiffiffi 1 uðx; tÞ ¼ c2 þ ei a tanh ðx þ ct þ n0 Þ ; a > 0; e ¼ 1; 3 2 2 2 2 1 pffiffiffi 1 uðx; tÞ ¼ c þ a sech ðx þ ct þ n0 Þ ; a < 0; e ¼ 1; 3 2 2 1 1 uðx; tÞ ¼ c1 tanh ðx þ ct þ n0 Þ þ c2 sech ðx þ ct þ n0 Þ ; a ¼ 0; b < 0: 2 2
ð26Þ
ð27Þ ð28Þ ð29Þ
ð30Þ ð31Þ ð32Þ
By combining Eqs. (18)–(20), (24) and (26), we obtain solitary wave solutions with compact support of the B(2, 2) given as ( pffiffiffi 2 2 c þ 1 ei a sin 12 ðx þ ct þ n0 Þ ; 0 6 12 ðx þ ct þ n0 Þ 6 2p; uðx; tÞ ¼ 32 2 2 ð33Þ c; Otherwise; 3 ( pffiffiffi 2 2 c þ 12 a cos 12 ðx þ ct þ n0 Þ ; 12 ðx þ ct þ n0 Þ 6 p; 3 ð34Þ uðx; tÞ ¼ 2 2 c; Otherwise; 3 ( 2 2 c þ c1 sin 12 ðx þ ctÞ þ c2 cos 12 ðx þ ctÞ ; 2h0 6 12 ðx þ ct þ n0 Þ 6 2ðp h0 Þ; ð35Þ uðx; tÞ ¼ 32 2 c; Otherwise; 3 1j ffi where h0 ¼ arccos pjcffiffiffiffiffiffiffiffi . 2 2
c1 þc2
Example 2. We now consider the B(3, 3) equation utt uxx ðu3 Þxx þ ðu3 Þxxxx ¼ 0;
ð36Þ
where u(x, t) is an unknown function. Integrating twice (36) and setting the constants of integration to be zero, we have ðc2 1Þu u3 þ ðu3 Þ00 ¼ 0: We select the balance constant n = 1. Thus we can assume that
ð37Þ
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uðx; tÞ ¼ uðnÞ ¼ a0 þ a1 v:
ð38Þ
Substituting (38) into (37) and, using (10) and (11), one gets a set of nonlinear algebraic equations ðc2 1Þa0 a30 þ 6a0 a21 a þ 6a31 a ¼ 0; ðc2 1Þa1 þ 3a20 a1 þ 3a20 a1 b ¼ 0; 3a0 a21 þ 12a0 a21 b þ 6a31 b ¼ 0;
ð39Þ
a31 þ 9a31 b ¼ 0: To obtain a nontrivial solution, we need to suppose a1 5 0. Solving Eq. (39), we have ffi ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 ¼ 3ð1 c2 Þ; b ¼ ; a1 ¼ 3ð1 c2 Þ; c 6¼ 0 arbitrary: 2 9 4 Thus we get the following new Jacobi elliptic function solutions: ffi 15ei pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ 3ð1 c2 Þ þ 3að1 c2 Þsn ðx þ ct þ n0 Þ; m ; a > 0; 0 < m < 1; e ¼ 1; 2 4 3 ffi 15e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3ð1 c2 Þ þ 3að1 c2 Þcn ðx þ ct þ n0 Þ; m ; a < 0; 0 < m < 1; e ¼ 1; uðx; tÞ ¼ 2 4 3 ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 uðx; tÞ ¼ 3ð1 c2 Þ þ c1 sn ðx þ ct þ n0 Þ; m þ c2 cn ðx þ ct þ n0 Þ; m ; a ¼ 0; b < 0: 2 3 3 By combining Eqs. (15)–(17), (39) and (40), we obtain the following solitary wave solutions for Eq. (21): ffi 15ei pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 uðx; tÞ ¼ 3ð1 c2 Þ þ 3að1 c2 Þ tanh ðx þ ct þ n0 Þ ; a > 0; e ¼ 1; 2 4 3 ffi 15e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 uðx; tÞ ¼ 3ð1 c2 Þ þ 3að1 c2 Þ sech ðx þ ct þ n0 Þ ; a < 0; e ¼ 1; 2 4 3 1 1 uðx; tÞ ¼ c1 tanh2 ðx þ ct þ n0 Þ þ c2 sech2 ðx þ ct þ n0 Þ ; a ¼ 0; b < 0: 3 3
ð40Þ
ð41Þ ð42Þ ð43Þ
ð44Þ ð45Þ ð46Þ
By combining Eqs. (18)–(20), (39) and (40), we obtain solitary wave solutions with compact support of the B(3, 3) given as ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3ð1 c2 Þ þ 15ei 3að1 c2 Þ sin 13 ðx þ ct þ n0 Þ ; 0 6 13 ðx þ ct þ n0 Þ 6 2p; 4 uðx; tÞ ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð47Þ 1 Otherwise; 3ð1 c2 Þ; 2 ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ; 3ð1 c2 Þ þ 15e 3að1 c2 Þ cos 13 ðx þ ct þ n0 Þ ; 13 ðx þ ct þ n0 Þ 6 3p 2 4 2 ð48Þ uðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Otherwise; 3ð1 c Þ; 2 ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3ð1 c2 Þ þ c1 sin 12 ðx þ ctÞ þ c2 cos 12 ðx þ ctÞ ; 12 ðx þ ct þ n0 Þ 6 p2 ; ð49Þ uðx; tÞ ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Otherwise: 3ð1 c2 Þ; 2
4. General formulas of the compacton solutions of B(m, n) equations In this section, we investigate a general formula for the exact special solutions of the B(m, n) equations with m = n > 1. We have obtained many types of solitary wave, Jacobi elliptic function and compacton solutions of B(2, 2) and B(3, 3) equations. We consider B(n, n) equations utt uxx ðun Þxx þ ðun Þxxxx ¼ 0
ð50Þ
for all integers n > 1. Substituting travelling wave solution u(x, t) = u(x + ct) = u(n) into (50) and integrating the resultant equation two with respect to n, thus we get the following a nonlinear ODE: ðc 1Þu un þ ðun Þ00 ¼ 0:
ð51Þ
In order to solve Eq. (51) we introduce an auxiliary function u by the following transformation: 1
uðnÞ ¼ ½uðnÞn1 :
ð52Þ
M. Inc / Chaos, Solitons and Fractals 37 (2008) 792–798
797
Thus, we have ðc 1ÞuðnÞ u2 ðnÞ þ
nðc 1Þ ðn 1Þ2
ðu0 Þ2 ðnÞ þ
nðc 1Þ uðnÞu00 ðnÞ ¼ 0: n1
ð53Þ
We first suppose that Eq. (53) has solutions of the form uðnÞ ¼ a0 þ a1 vðnÞ
ð54Þ
and v(n) is the solution of Eq. (11). Substituting (54) into (53) yields a0 ¼
nðc 1Þ ; nþ1
b¼
ðn 1Þ2 ; n2 ðc 1Þ
a¼
ðn 1Þ2 ðc 1Þ ðn þ 1Þ2 a21
;
a1 6¼ 0;
c 6¼ 1:
So Eq. (11) has the following another new Jacobi elliptic function solutions: 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 ðn þ n0 Þ; m sn ; vðnÞ ¼ ie a1 ðn þ 1Þ n c1 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 cn ; ðn þ n0 Þ; m vðnÞ ¼ a1 ðn þ 1Þ n c1 1 ( " # " #)n1 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi n1 1 n1 1 : ðn þ n0 Þ; m þ c2 cn ðn þ n0 Þ; m vðnÞ ¼ c1 sn n c1 n c1 If we take m ! 1, then we obtain solitary wave solutions 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 vðnÞ ¼ ie tanh ; ½ðx þ ct þ n0 Þ a1 ðn þ 1Þ n c1 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 sech ; vðnÞ ¼ ðx þ ct þ n0 Þ a1 ðn þ 1Þ n c1 1 ( " # " #)n1 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi n1 1 n1 1 : ðx þ ct þ n0 Þ þ c2 sech ðx þ ct þ n0 Þ vðnÞ ¼ c1 tanh n c1 n c1
ð55Þ
ð56Þ
ð57Þ
ð58Þ
ð59Þ
ð60Þ
ð61Þ
If we take m ! 0, then we get that the B(n, n) equation with n being odd integer >1 admits the solitary wave solutions with compact support 8n 1 qffiffiffiffiffiffi h qffiffiffiffiffiffi ion1 > > 1 1 np < ie anðc1Þ ðx þ ct þ n0 Þ ðx þ ct þ n0 Þ 6 2ðn1Þ sin n1 ; n1 ; n c1 n c1 1 ðnþ1Þ vðnÞ ¼
ð62Þ 1 > > : ie nðc1Þ n1 ; Otherwise; a1 ðnþ1Þ 8n 1 qffiffiffiffiffiffi h qffiffiffiffiffiffi ion1 > n1 c1Þ > n1 1 1 np < anððnþ1Þ ðx þ ct þ n ðx þ ct þ n cos Þ ; Þ ; 6 2ðn1Þ 0 0 n c1 n c1 1 vðnÞ ¼
ð63Þ 1 > nðc1Þ n1 > : ; Otherwise; a1 ðnþ1Þ 1 ( " # " #)n1 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi n1 1 n1 1 ðx þ ct þ n0 Þ þ c2 cos ðx þ ct þ n0 Þ ð64Þ vðnÞ ¼ c1 sin n c1 n c1 qffiffiffiffiffiffi 1 np for j n1 and v(n) = 0 for otherwise. ðx þ ct þ n0 Þj 6 2ðn1Þ n c1 In addition, we obtain that the B(n, n) equation with n being even integer >1 admits the multiple exact special solutions with solitary patterns 1 ( " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 ðx þ ct þ n0 Þ vðnÞ ¼ ; ð65Þ sinh a1 ðn þ 1Þ n 1c
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(
1 " #)n1 rffiffiffiffiffiffiffiffiffiffiffi nðc 1Þ n1 1 cosh vðnÞ ¼ ; ðx þ ct þ n0 Þ a1 ðn þ 1Þ n 1c 1 ( " # " #)n1 rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi n1 1 n1 1 : ðx þ ct þ n0 Þ c2 cosh ðx þ ct þ n0 Þ vðnÞ ¼ c1 sinh n 1c n 1c
ð66Þ
ð67Þ
5. Conclusions In this paper we obtained several types of compacton, solitary patterns, Jacobi elliptic function and solitary wave solutions of the B(2, 2) and B(3, 3) equations by a new extended sn–cn method. Some conclusions of Wazwaz [34,35] are special cases of ours. In addition, the presented solutions of (21) and (36) were not found by any of the methods [23–35]. These solutions may be useful to explain some physical phenomena. The present method is direct and efficient to obtain new compacton, Jacobi elliptic function and solitary wave solutions of Eq. (3). Our method is very easily applied to both this type of nonlinear dispersive and the modified nonlinear dispersive equations in higher dimensional spaces.
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