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Two classes of variants of the generalized KdV equations with compact and noncompact solutions
Applied Mathematics and Computation 154 (2004) 835–846 www.elsevier.com/locate/amc
Two classes of variants of the generalized KdV equations with compact and noncompact solutions Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, 3700 West 103rd Street, Chicago, IL 60655, USA
Abstract In this paper we study compact and noncompact solutions for two classes of new variants of the generalized KdV equations. The study is conducted on the focusing and the defocusing structures of genuinely nonlinear dispersive equations. The results emphasize the different physical structures of the compact and the noncompact solutions. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Compactons; Solitons; Generalized KdV equation
1. Introduction It is well-known that the nonlinear term uux in the standard KdV equation ut þ auux þ uxxx ¼ 0;
ð1Þ
causes the steepening of the wave form. However, the dispersion effect term uxxx in Eq. (1) makes the wave form spread [1]. Due to the balance between this weak nonlinearity and dispersion, solitons exist. Wadati [1–3] defined soliton as a nonlinear wave characterized by the following properties: (1) A localized wave propagates without change of its properties (shape, velocity, etc.).
E-mail address: [email protected] (A.-M. Wazwaz). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00752-5
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(2) Localized waves are stable against mutual collisions and retain their identities. This in turn means that soliton has the property of a particle [4]. Soliton is a localized wave that has an infinite support or a localized wave with exponential tails. A genuinely nonlinear dispersive equation Kðn; nÞ, a special type of the KdV equation defined by ut þ aðun Þx þ ðun Þxxx ¼ 0;
n > 1;
ð2Þ
introduced by Rosenau and Hyman [5] and was the first to draw the attention to a new discovery. It is clear that the convection term in the Kðn; nÞ equation (2) is nonlinear, whereas the dispersion effect term in Eq. (2) is genuinely nonlinear. It was emphasized in [5–10] that the delicate interaction between nonlinear convection with genuine nonlinear dispersion generates solitary waves with exact compact support that are called compactons. Compactons are solitons with finite wavelength, waves with a compact support or solitons free of exponential tails. Unlike soliton that narrows as the amplitude increases, the compactonÕs width is independent of the amplitude. The models (2) are particularly interesting due to the local nature of their solutions, which serves as a caricature of a wide range of phenomena in nature [11]. The nonsmooth interfaces of the compactons and the strong nonlinearity of the equation present significant theoretical and numerical challenges [11]. Compactons were proved to collide elastically and vanish outside a finite core region. Two important features of compactons structures are observed: (1) The compacton is a soliton characterized by the absence of exponential wings or infinite tails. (2) The width of the compacton is independent of the amplitude. Compactons such as drops do not possess infinite wings, hence they interact among themselves only across short distances. In modern physics, a suffix-on is used to indicate the particle property [1], for example phonon, photon, and soliton have particle property. For this reason, the solitary wave with compact support is called compacton to indicate that it has the property of a particle. The compactons are nonanalytic solutions [4–20], whereas classical solitons are analytic solutions. The points of nonanalyticity at the edge of the compacton correspond to points of genuine nonlinearity for the differential equation. Several analytical and numerical methods have been employed for obtaining compactons solutions in one spatial dimension: the inverse scattering method, the B€ acklund transformation, the Darboux transformation, and the Painleve analysis were used to study the soliton concept. On the other hand, the compacton concept has been studied by many analytical and numerical methods, such as the pseudo spectral method [5–8], the tri-Hamiltonian operators [9], the
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837
finite difference method [10], particle method [11], and Adomian decomposition method [21–24]. Some of these methods may be extended for higher spatial dimensions. The study of compactons may give insight into many scientific processes such as the super deformed nuclei, preformation of cluster in hydrodynamic models, the fission of liquid drops (nuclear physics), inertial fusion. For more details about the role of nonlinear dispersion on patterns formation as well as the formation of nonlinear structures like liquid drops, and for more insight through the physical properties of compacton structures, see the works in [4–20]. The compactons discovery motivated a considerable work to make compactons be practically realized in scientific applications, such as the super deformed nuclei, preformation of cluster in hydrodynamic models, the fission of liquid drops (nuclear physics), inertial fusion and others, as indicated in [2,8–10]. Recently, Rosenau [8] investigated the model ut þ aðunþ1 Þx þ ½uðun Þxx x ¼ 0;
a > 0; n P 1;
ð3Þ
that emerges in nonlinear lattices and was used to describe the dispersion of dilute suspensions for n ¼ 1. The aim of Rosenau work in [8] was to explore a number of formal mathematical extensions of soliton supporting equations with the aim of producing compact dispersive structures. Rosenau [8] examined the focusing branch of (3), where (a > 0), and developed general formulas in terms of the cosine function only. Recently, Wazwaz [18] studied the model (3) for both types, the focusing branch where a > 0 and the defocusing branch where a is replaced by a. For the compactons supporting equations, where a > 0, the solutions were defined in terms of sine and cosine profiles, whereas for the solitary patterns supporting models, the solutions were formally derived in terms of the hyperbolic functions sinh and cosh. For a > 0, the following compactons solutions 8 pffiffiffi 1=n > a p < 2c sin2 ðx ctÞ ; jx ctj 6 ; uðx; tÞ ¼ ð4Þ 2 a l > : 0; otherwise; 8 pffiffiffi 1=n > a p < 2c cos2 ðx ctÞ ; ; jx ctj 6 2 a 2l uðx; tÞ ¼ ð5Þ > : 0; otherwise; pffiffiffi where l ¼ a=2. In [18], the defocusing branches of (3), where a is replaced by a, the following solitary patterns solutions
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pffiffiffi 1=n a 2c ðx ctÞ sinh2 ; 2 a pffiffiffi 1=n a 2c 2 ðx ctÞ cosh uðx; tÞ ¼ ; 2 a
uðx; tÞ ¼
ð6Þ ð7Þ
were formally obtained. However, Wazwaz [19] studied the variants of the Kðn; nÞ equations given by ut þ auðun Þx þ ½uðun Þxx x ¼ 0;
ð8Þ
a > 0; n > 1:
For a > 0, the following compactons solutions 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n > an < 2ðn þ 1Þc sin2 ; ðx ctÞ uðx; tÞ ¼ an 4ðn þ 1Þ > : 0; 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n > an < 2ðn þ 1Þc cos2 ; ðx ctÞ uðx; tÞ ¼ an 4ðn þ 1Þ > : 0;
jx ctj 6
p ; a
otherwise; ð9Þ jx ctj 6
p ; 2a
otherwise; ð10Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ an=ð4ðn þ 1ÞÞ. In [19], the defocusing branch of (8), where a is replaced by a, the following solitary patterns solutions
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n 2ðn þ 1Þc an 2 uðx; tÞ ¼ sinh ; ðx ctÞ an 4ðn þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n 2ðn þ 1Þc an cosh2 ; uðx; tÞ ¼ ðx ctÞ an 4ðn þ 1Þ
ð11Þ ð12Þ
were formally obtained. The main motivation for this work is to extend the results of the work in [8] and our works in [18,19] to more dispersive models. In particular, our main concern will be examining four variants of the generalized KdV (gKdV) equation, where wave dispersion is purely nonlinear, given by Variant I ut þ aðunþ1 Þx þ bux ðun Þxx ¼ 0; ut aðu
nþ1
n
Þx þ bux ðu Þxx ¼ 0;
a; b > 0; n > 1; a; b > 0; n > 1:
ð13Þ
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Variant II ut þ aðunþ1 Þx þ b½un ðuÞxx x ¼ 0; ut aðu
nþ1
n
Þx þ b½u ðuÞxx x ¼ 0;
a; b > 0; n > 2; a; b > 0; n > 2:
ð14Þ
Variant III ut þ auðun Þx þ bux ðun Þxx ¼ 0; n
a; b > 0; n > 1;
n
ut auðu Þx þ bux ðu Þxx ¼ 0;
a; b > 0; n > 1:
ð15Þ
and Variant IV ut þ aðun Þux þ bux ðun Þxx ¼ 0; ut aðun Þux þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1; a; b > 0; n > 1:
ð16Þ
By careful examining the four variants, we can easily observe that variants can be classified into two classes. In the first class, variants I and II have the same convection term but differ only in nonlinear dispersive term. However, in the second class, variants III and IV have the same nonlinear dispersive term and differ only in the convection term. Our work relies significantly on the results obtained in [8,18,19] where focusing and defocusing branches will be approached independently.
2. Variant I 2.1. The focusing branch We first consider the one dimensional equation ut þ aðunþ1 Þx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1:
ð17Þ
The results obtained in [8,18,19] allows us to set the compactons solutions of Eq. (17) in the form uðx; tÞ ¼ fa sin2 ½bðx ctÞg
1=n
;
ð18Þ
1=n
ð19Þ
or in the form uðx; tÞ ¼ fa cos2 ½bðx ctÞg
;
where a and b are constants that will be determined. Substituting (18) or (19) into (17) and solving the resulting equations for a and b we obtain
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rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 aðn þ 1Þ ; b¼ 2 b 2c a¼ ; n > 1: aðn þ 1Þ
ð20Þ
Substituting (20) into (18) and (19) gives the following set of general compactons solutions 8( " #)1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2c 1 aðn þ 1Þ p < 2 sin ðx ctÞ ; jx ctj6 ; n > 1; uðx;tÞ ¼ aðn þ 1Þ 2 b b > : 0; otherwise; ð21Þ and
uðx;tÞ ¼
8( > > < > > :
" #)1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c 1 aðn þ 1Þ p cos2 ðx ctÞ ; jx ctj6 ; n > 1; aðn þ 1Þ 2 b 2b
0;
otherwise: ð22Þ
2.2. The defocusing branch We next consider the defocusing branch ut aðunþ1 Þx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1;
ð23Þ
where a in the model discussed above (17) is replaced by a. It is formally derived in [18,19] that the defocusing branch gives the solutions in terms of the hyperbolic functions. Consequently, it is normal to set the general solution of (23) in the form ~ ctÞg1=n ; uðx; tÞ ¼ f~ a sinh2 ½bðx
ð24Þ
or of the form ~ ctÞg1=n ; uðx; tÞ ¼ f~ a cosh2 ½bðx
ð25Þ
where a~ and b~ are constants that will be determined. Substituting (24) or (25) into (23) we immediately obtain rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 aðn þ 1Þ 2c b~ ¼ ; a~ ¼ : ð26Þ 2 b aðn þ 1Þ For a~ to be positive, then c < 0. Substituting (26) into (24) and (25) gives the following set of general solutions
A.-M. Wazwaz / Appl. Math. Comput. 154 (2004) 835–846
(
2c sinh2 uðx; tÞ ¼ aðn þ 1Þ
"
1 2
#)1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðn þ 1Þ ðx ctÞ ; b
841
ð27Þ
and ( uðx; tÞ ¼
2c cosh2 aðn þ 1Þ
"
1 2
#)1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðn þ 1Þ ðx ctÞ : b
ð28Þ
3. Variant II 3.1. The focusing branch We consider the nonlinear dispersive equation ut þ aðunþ1 Þx þ b½un ðuÞxx x ¼ 0;
ð29Þ
a; b > 0; n > 2:
It is normal to set the compactons solutions in the form uðx; tÞ ¼ fA sin2 ½Bðx ctÞg1=n ;
ð30Þ
or in the form uðx; tÞ ¼ fA cos2 ½Bðx ctÞg
1=n
ð31Þ
;
where A and B are constants. Substituting (30) or (31) into (29) and solving we find rffiffiffi n a 2c ; n > 2: ð32Þ B¼ ; A¼ 2 b aðn 2Þ The compactons solutions 8 rffiffiffi 1=n > 2c n a < 2 sin ; ðx ctÞ uðx; tÞ ¼ aðn 2Þ 2 b > : 0;
p ; B otherwise;
jx ctj 6
n > 2;
ð33Þ and
8 rffiffiffi 1=n > 2c n a p < 2 cos ; n > 2; ; jx ctj 6 ðx ctÞ uðx; tÞ ¼ aðn 2Þ 2 b 2B > : 0; otherwise; ð34Þ
follow immediately.
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3.2. The defocusing branch We next consider the defocusing branch ut aðunþ1 Þx þ b½un ðuÞxx x ¼ 0;
a; b > 0; n > 2:
ð35Þ
Following our approach presented before, it is convenient to set the general solution of (35) in the form e ctÞg1=n ; uðx; tÞ ¼ fA~sinh2 ½ Bðx
ð36Þ
or of the form e ctÞg1=n ; uðx; tÞ ¼ fA~cosh2 ½ Bðx
ð37Þ
e are constants. Substituting (36) or (37) into (35) and solving the where A~ and B resulting equations we obtain rffiffiffi 2c e ¼ n a; A~ ¼ : ð38Þ B 2 b aðn 2Þ It then follows that rffiffiffi 1=n 2c n a 2 uðx; tÞ ¼ sinh ; ðx ctÞ aðn 2Þ 2 b
ð39Þ
and uðx; tÞ ¼
rffiffiffi 1=n 2c n a 2 cosh : ðx ctÞ aðn 2Þ 2 b
ð40Þ
4. Variant III 4.1. The focusing branch We now consider the nonlinear dispersive equation ut þ auðun Þx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1:
ð41Þ
Proceeding as before, we set the compactons solutions of Eq. (41) in the form uðx; tÞ ¼ fk sin2 ½lðx ctÞg1=n ;
ð42Þ
or of the form uðx; tÞ ¼ fk cos2 ½lðx ctÞg1=n ;
ð43Þ
A.-M. Wazwaz / Appl. Math. Comput. 154 (2004) 835–846
843
where k and l are constants. Substituting (42) or (43) into (41) and solving the resulting equations we obtain rffiffiffiffiffi 1 an 2c ð44Þ l¼ ; k¼ : 2 b an Substituting (44) into (42) and (43) gives the following set of general compactons solutions 8 rffiffiffiffiffi 1=n > < 2c 1 an p 2 sin ; jx ctj 6 ; ðx ctÞ uðx; tÞ ¼ ð45Þ an 2 b l > : 0; otherwise; and
8 rffiffiffiffiffi 1=n > < 2c 1 an 2 cos ; ðx ctÞ uðx; tÞ ¼ an 2 b > : 0;
p ; 2l otherwise:
jx ctj 6
ð46Þ
4.2. The defocusing branch We next consider the defocusing branch ut auðun Þx þ bux ðun Þxx ¼ 0; a; b > 0; n > 1:
ð47Þ
It is convenient to set the general solution of (47) in the form lðx ctÞg1=n ; uðx; tÞ ¼ fk~ sinh2 ½~
ð48Þ
or in the form lðx ctÞg1=n ; uðx; tÞ ¼ fk~ cosh2 ½~
ð49Þ
where k~ and l~ are constants. Substituting (48) or (49) into (47) and solving the resulting equations we obtain rffiffiffiffiffi 1 an 2c : ð50Þ l~ ¼ ; k~ ¼ 2 b an As a consequence of (50), the general solutions rffiffiffiffiffi 1=n 2c 1 an uðx; tÞ ¼ sinh2 ; ðx ctÞ an 2 b
ð51Þ
and uðx; tÞ ¼
rffiffiffiffiffi 1=n 2c 1 an cosh2 ; ðx ctÞ an 2 b
follow immediately.
ð52Þ
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5. Variant IV 5.1. The focusing branch We finally consider the nonlinear dispersive equation ut þ aun ðuÞx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1:
ð53Þ
It is normal to set the compactons solutions of Eq. (53) in the form uðx; tÞ ¼ fk sin2 ½lðx ctÞg1=n ;
ð54Þ
or of the form uðx; tÞ ¼ fk cos2 ½lðx ctÞg
1=n
ð55Þ
;
where k and l are constants. Substituting (54) or (55) into (53) and solving the resulting equations we obtain rffiffiffi 1 a 2c l¼ ð56Þ ; k¼ : 2 b a We can easily observe that l and k do not depend on n. Substituting (56) into (54) and (55) gives the following set of general compactons solutions 8 rffiffiffi 1=n > < 2c 1 a p 2 sin ; jx ctj 6 ; ðx ctÞ uðx; tÞ ¼ ð57Þ a 2 b l > : 0; otherwise; and 8 rffiffiffi 1=n > < 2c 1 a 2 cos ; ðx ctÞ uðx; tÞ ¼ a 2 b > : 0;
p ; 2l otherwise:
jx ctj 6
ð58Þ
5.2. The defocusing branch We next consider the defocusing branch ut auðun Þx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1:
ð59Þ
It is convenient to set the general solution of (59) in the form uðx; tÞ ¼ fk~ sinh2 ½~ lðx ctÞg
1=n
;
ð60Þ
or in the form uðx; tÞ ¼ fk~ cosh2 ½~ lðx ctÞg1=n ;
ð61Þ
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845
where k~ and l~ are constants. Substituting (60) or (61) into (59) and solving the resulting equations we obtain 1 l~ ¼ 2
rffiffiffi a ; b
2c : k~ ¼ a
ð62Þ
It can be easily observed that l~ and k~ do not depend on n. As a consequence of (62), the general solutions rffiffiffi 1=n 2c 1 a 2 uðx; tÞ ¼ sinh ; ðx ctÞ a 2 b
ð63Þ
and uðx; tÞ ¼
rffiffiffi 1=n 2c 1 a cosh2 ; ðx ctÞ a 2 b
ð64Þ
follow immediately. This completes the derivation for the compactons solutions and for the solitary patterns solutions for the two classes of variants.
6. Concluding remarks The basic goal of this work has been to extend the works in [8,18,19]. The focusing branch and the defocusing branch of the four variants of the gKdV equation were thoroughly studied. It is easily noticed that the coefficients of the arguments of the compactons solutions and the patterns solutions are the same for each variant. However, the coefficients of the trigonometric functions and the hyperbolic functions for each variant are the same with opposite signs. Moreover, it is easily observed that the focusing branch of each variant provides compactons solutions, whereas the defocusing branch gives solitary patterns solutions having infinite slopes or cusps. This confirms the reality that the focusing branch and the defocusing branch represent two different sets of models, each leading to a different physical structure. The obtained results in this work clearly demonstrate the effect of the purely nonlinear dispersion and the qualitative change made in the genuinely nonlinear phenomenon. The variants introduced in this work are more general compared to those investigated in [8,18,19]. The significance of the presented approach is that it can be used to search for exact solutions of other nonlinear dispersive equations.
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References [1] M. Wadati, Introduction to solitons, Pramana: J. Phys. 57 (5–6) (2001) 841–847. [2] M. Wadati, The exact solution of the modified Kortweg–de Vries equation, J. Phys. Soc. Jpn. 32 (1972) 1681–1687. [3] M. Wadati, The modified Kortweg–de Vries equation, J. Phys. Soc. Jpn. 34 (1973) 1289–1296. [4] F. Cooper, J. Hyman, A. Khare, Compacton solution in a class of generalized fifth-order Korteweg–de Vries equations, Phys. Rev. E 64 (2001) 1–5. [5] P. Rosenau, J.M. Hyman, Compactons: solitons with finite wavelengths, Phys. Rev. Lett. 70 (5) (1993) 564–567. [6] P. Rosenau, Nonlinear dispersion and compact structures, Phys. Rev. Lett. 73 (13) (1994) 1737–1741. [7] P. Rosenau, On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A 230 (5/6) (1997) 305–318. [8] P. Rosenau, Compact and noncompact dispersive structures, Phys. Lett. A 275 (3) (2000) 193– 203. [9] P.J. Olver, P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (2) (1996) 1900–1906. [10] M.S. Ismail, T. Taha, A numerical study of compactons, Math. Comput. Simul. 47 (1998) 519– 530. [11] A. Chertock, D. Levy, Particle methods for dispersive equations, J. Comput. Phys. 171 (2001) 708–730. [12] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [13] A.M. Wazwaz, New solitary-wave special solutions with compact support for the nonlinear dispersive Kðm; nÞ equations, Chaos, Solitons Fract. 13 (2) (2002) 321–330. [14] A.M. Wazwaz, Exact specific solutions with solitary patterns for the nonlinear dispersive Kðm; nÞ equations, Chaos, Solitons Fract. 13 (1) (2001) 161–170. [15] A.M. Wazwaz, General compactons solutions for the focusing branch of the nonlinear dispersive Kðn; nÞ equations in higher dimensional spaces, Appl. Math. Comput. 133 (2/3) (2002) 213–227. [16] A.M. Wazwaz, General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive Kðn; nÞ equations in higher dimensional spaces, Appl. Math. Comput. 133 (2/3) (2002) 229–244. [17] A.M. Wazwaz, A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simul. 56 (2001) 269–276. [18] A.M. Wazwaz, Compactons dispersive structures for variants of the Kðn; nÞ and the KP equations, Chaos, Solitons Fract. 13 (5) (2002) 1053–1062. [19] 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. [20] A.M. Wazwaz, Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos, Solitons Fract. 12 (8) (2001) 1549– 1556. [21] A.M. Wazwaz, A computational approach to soliton solutions of the Kadomtsev–Petviashili equation, Appl. Math. Comput. 123 (2) (2001) 205–217. [22] A.M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, 1997. [23] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, MA, 1994. [24] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1998) 501–544.
Two classes of variants of the generalized KdV equations with compact and noncompact solutions Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, 3700 West 103rd Street, Chicago, IL 60655, USA
Abstract In this paper we study compact and noncompact solutions for two classes of new variants of the generalized KdV equations. The study is conducted on the focusing and the defocusing structures of genuinely nonlinear dispersive equations. The results emphasize the different physical structures of the compact and the noncompact solutions. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Compactons; Solitons; Generalized KdV equation
1. Introduction It is well-known that the nonlinear term uux in the standard KdV equation ut þ auux þ uxxx ¼ 0;
ð1Þ
causes the steepening of the wave form. However, the dispersion effect term uxxx in Eq. (1) makes the wave form spread [1]. Due to the balance between this weak nonlinearity and dispersion, solitons exist. Wadati [1–3] defined soliton as a nonlinear wave characterized by the following properties: (1) A localized wave propagates without change of its properties (shape, velocity, etc.).
E-mail address: [email protected] (A.-M. Wazwaz). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00752-5
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(2) Localized waves are stable against mutual collisions and retain their identities. This in turn means that soliton has the property of a particle [4]. Soliton is a localized wave that has an infinite support or a localized wave with exponential tails. A genuinely nonlinear dispersive equation Kðn; nÞ, a special type of the KdV equation defined by ut þ aðun Þx þ ðun Þxxx ¼ 0;
n > 1;
ð2Þ
introduced by Rosenau and Hyman [5] and was the first to draw the attention to a new discovery. It is clear that the convection term in the Kðn; nÞ equation (2) is nonlinear, whereas the dispersion effect term in Eq. (2) is genuinely nonlinear. It was emphasized in [5–10] that the delicate interaction between nonlinear convection with genuine nonlinear dispersion generates solitary waves with exact compact support that are called compactons. Compactons are solitons with finite wavelength, waves with a compact support or solitons free of exponential tails. Unlike soliton that narrows as the amplitude increases, the compactonÕs width is independent of the amplitude. The models (2) are particularly interesting due to the local nature of their solutions, which serves as a caricature of a wide range of phenomena in nature [11]. The nonsmooth interfaces of the compactons and the strong nonlinearity of the equation present significant theoretical and numerical challenges [11]. Compactons were proved to collide elastically and vanish outside a finite core region. Two important features of compactons structures are observed: (1) The compacton is a soliton characterized by the absence of exponential wings or infinite tails. (2) The width of the compacton is independent of the amplitude. Compactons such as drops do not possess infinite wings, hence they interact among themselves only across short distances. In modern physics, a suffix-on is used to indicate the particle property [1], for example phonon, photon, and soliton have particle property. For this reason, the solitary wave with compact support is called compacton to indicate that it has the property of a particle. The compactons are nonanalytic solutions [4–20], whereas classical solitons are analytic solutions. The points of nonanalyticity at the edge of the compacton correspond to points of genuine nonlinearity for the differential equation. Several analytical and numerical methods have been employed for obtaining compactons solutions in one spatial dimension: the inverse scattering method, the B€ acklund transformation, the Darboux transformation, and the Painleve analysis were used to study the soliton concept. On the other hand, the compacton concept has been studied by many analytical and numerical methods, such as the pseudo spectral method [5–8], the tri-Hamiltonian operators [9], the
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finite difference method [10], particle method [11], and Adomian decomposition method [21–24]. Some of these methods may be extended for higher spatial dimensions. The study of compactons may give insight into many scientific processes such as the super deformed nuclei, preformation of cluster in hydrodynamic models, the fission of liquid drops (nuclear physics), inertial fusion. For more details about the role of nonlinear dispersion on patterns formation as well as the formation of nonlinear structures like liquid drops, and for more insight through the physical properties of compacton structures, see the works in [4–20]. The compactons discovery motivated a considerable work to make compactons be practically realized in scientific applications, such as the super deformed nuclei, preformation of cluster in hydrodynamic models, the fission of liquid drops (nuclear physics), inertial fusion and others, as indicated in [2,8–10]. Recently, Rosenau [8] investigated the model ut þ aðunþ1 Þx þ ½uðun Þxx x ¼ 0;
a > 0; n P 1;
ð3Þ
that emerges in nonlinear lattices and was used to describe the dispersion of dilute suspensions for n ¼ 1. The aim of Rosenau work in [8] was to explore a number of formal mathematical extensions of soliton supporting equations with the aim of producing compact dispersive structures. Rosenau [8] examined the focusing branch of (3), where (a > 0), and developed general formulas in terms of the cosine function only. Recently, Wazwaz [18] studied the model (3) for both types, the focusing branch where a > 0 and the defocusing branch where a is replaced by a. For the compactons supporting equations, where a > 0, the solutions were defined in terms of sine and cosine profiles, whereas for the solitary patterns supporting models, the solutions were formally derived in terms of the hyperbolic functions sinh and cosh. For a > 0, the following compactons solutions 8 pffiffiffi 1=n > a p < 2c sin2 ðx ctÞ ; jx ctj 6 ; uðx; tÞ ¼ ð4Þ 2 a l > : 0; otherwise; 8 pffiffiffi 1=n > a p < 2c cos2 ðx ctÞ ; ; jx ctj 6 2 a 2l uðx; tÞ ¼ ð5Þ > : 0; otherwise; pffiffiffi where l ¼ a=2. In [18], the defocusing branches of (3), where a is replaced by a, the following solitary patterns solutions
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pffiffiffi 1=n a 2c ðx ctÞ sinh2 ; 2 a pffiffiffi 1=n a 2c 2 ðx ctÞ cosh uðx; tÞ ¼ ; 2 a
uðx; tÞ ¼
ð6Þ ð7Þ
were formally obtained. However, Wazwaz [19] studied the variants of the Kðn; nÞ equations given by ut þ auðun Þx þ ½uðun Þxx x ¼ 0;
ð8Þ
a > 0; n > 1:
For a > 0, the following compactons solutions 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n > an < 2ðn þ 1Þc sin2 ; ðx ctÞ uðx; tÞ ¼ an 4ðn þ 1Þ > : 0; 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n > an < 2ðn þ 1Þc cos2 ; ðx ctÞ uðx; tÞ ¼ an 4ðn þ 1Þ > : 0;
jx ctj 6
p ; a
otherwise; ð9Þ jx ctj 6
p ; 2a
otherwise; ð10Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ an=ð4ðn þ 1ÞÞ. In [19], the defocusing branch of (8), where a is replaced by a, the following solitary patterns solutions
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n 2ðn þ 1Þc an 2 uðx; tÞ ¼ sinh ; ðx ctÞ an 4ðn þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=n 2ðn þ 1Þc an cosh2 ; uðx; tÞ ¼ ðx ctÞ an 4ðn þ 1Þ
ð11Þ ð12Þ
were formally obtained. The main motivation for this work is to extend the results of the work in [8] and our works in [18,19] to more dispersive models. In particular, our main concern will be examining four variants of the generalized KdV (gKdV) equation, where wave dispersion is purely nonlinear, given by Variant I ut þ aðunþ1 Þx þ bux ðun Þxx ¼ 0; ut aðu
nþ1
n
Þx þ bux ðu Þxx ¼ 0;
a; b > 0; n > 1; a; b > 0; n > 1:
ð13Þ
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Variant II ut þ aðunþ1 Þx þ b½un ðuÞxx x ¼ 0; ut aðu
nþ1
n
Þx þ b½u ðuÞxx x ¼ 0;
a; b > 0; n > 2; a; b > 0; n > 2:
ð14Þ
Variant III ut þ auðun Þx þ bux ðun Þxx ¼ 0; n
a; b > 0; n > 1;
n
ut auðu Þx þ bux ðu Þxx ¼ 0;
a; b > 0; n > 1:
ð15Þ
and Variant IV ut þ aðun Þux þ bux ðun Þxx ¼ 0; ut aðun Þux þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1; a; b > 0; n > 1:
ð16Þ
By careful examining the four variants, we can easily observe that variants can be classified into two classes. In the first class, variants I and II have the same convection term but differ only in nonlinear dispersive term. However, in the second class, variants III and IV have the same nonlinear dispersive term and differ only in the convection term. Our work relies significantly on the results obtained in [8,18,19] where focusing and defocusing branches will be approached independently.
2. Variant I 2.1. The focusing branch We first consider the one dimensional equation ut þ aðunþ1 Þx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1:
ð17Þ
The results obtained in [8,18,19] allows us to set the compactons solutions of Eq. (17) in the form uðx; tÞ ¼ fa sin2 ½bðx ctÞg
1=n
;
ð18Þ
1=n
ð19Þ
or in the form uðx; tÞ ¼ fa cos2 ½bðx ctÞg
;
where a and b are constants that will be determined. Substituting (18) or (19) into (17) and solving the resulting equations for a and b we obtain
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rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 aðn þ 1Þ ; b¼ 2 b 2c a¼ ; n > 1: aðn þ 1Þ
ð20Þ
Substituting (20) into (18) and (19) gives the following set of general compactons solutions 8( " #)1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2c 1 aðn þ 1Þ p < 2 sin ðx ctÞ ; jx ctj6 ; n > 1; uðx;tÞ ¼ aðn þ 1Þ 2 b b > : 0; otherwise; ð21Þ and
uðx;tÞ ¼
8( > > < > > :
" #)1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c 1 aðn þ 1Þ p cos2 ðx ctÞ ; jx ctj6 ; n > 1; aðn þ 1Þ 2 b 2b
0;
otherwise: ð22Þ
2.2. The defocusing branch We next consider the defocusing branch ut aðunþ1 Þx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1;
ð23Þ
where a in the model discussed above (17) is replaced by a. It is formally derived in [18,19] that the defocusing branch gives the solutions in terms of the hyperbolic functions. Consequently, it is normal to set the general solution of (23) in the form ~ ctÞg1=n ; uðx; tÞ ¼ f~ a sinh2 ½bðx
ð24Þ
or of the form ~ ctÞg1=n ; uðx; tÞ ¼ f~ a cosh2 ½bðx
ð25Þ
where a~ and b~ are constants that will be determined. Substituting (24) or (25) into (23) we immediately obtain rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 aðn þ 1Þ 2c b~ ¼ ; a~ ¼ : ð26Þ 2 b aðn þ 1Þ For a~ to be positive, then c < 0. Substituting (26) into (24) and (25) gives the following set of general solutions
A.-M. Wazwaz / Appl. Math. Comput. 154 (2004) 835–846
(
2c sinh2 uðx; tÞ ¼ aðn þ 1Þ
"
1 2
#)1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðn þ 1Þ ðx ctÞ ; b
841
ð27Þ
and ( uðx; tÞ ¼
2c cosh2 aðn þ 1Þ
"
1 2
#)1=n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðn þ 1Þ ðx ctÞ : b
ð28Þ
3. Variant II 3.1. The focusing branch We consider the nonlinear dispersive equation ut þ aðunþ1 Þx þ b½un ðuÞxx x ¼ 0;
ð29Þ
a; b > 0; n > 2:
It is normal to set the compactons solutions in the form uðx; tÞ ¼ fA sin2 ½Bðx ctÞg1=n ;
ð30Þ
or in the form uðx; tÞ ¼ fA cos2 ½Bðx ctÞg
1=n
ð31Þ
;
where A and B are constants. Substituting (30) or (31) into (29) and solving we find rffiffiffi n a 2c ; n > 2: ð32Þ B¼ ; A¼ 2 b aðn 2Þ The compactons solutions 8 rffiffiffi 1=n > 2c n a < 2 sin ; ðx ctÞ uðx; tÞ ¼ aðn 2Þ 2 b > : 0;
p ; B otherwise;
jx ctj 6
n > 2;
ð33Þ and
8 rffiffiffi 1=n > 2c n a p < 2 cos ; n > 2; ; jx ctj 6 ðx ctÞ uðx; tÞ ¼ aðn 2Þ 2 b 2B > : 0; otherwise; ð34Þ
follow immediately.
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3.2. The defocusing branch We next consider the defocusing branch ut aðunþ1 Þx þ b½un ðuÞxx x ¼ 0;
a; b > 0; n > 2:
ð35Þ
Following our approach presented before, it is convenient to set the general solution of (35) in the form e ctÞg1=n ; uðx; tÞ ¼ fA~sinh2 ½ Bðx
ð36Þ
or of the form e ctÞg1=n ; uðx; tÞ ¼ fA~cosh2 ½ Bðx
ð37Þ
e are constants. Substituting (36) or (37) into (35) and solving the where A~ and B resulting equations we obtain rffiffiffi 2c e ¼ n a; A~ ¼ : ð38Þ B 2 b aðn 2Þ It then follows that rffiffiffi 1=n 2c n a 2 uðx; tÞ ¼ sinh ; ðx ctÞ aðn 2Þ 2 b
ð39Þ
and uðx; tÞ ¼
rffiffiffi 1=n 2c n a 2 cosh : ðx ctÞ aðn 2Þ 2 b
ð40Þ
4. Variant III 4.1. The focusing branch We now consider the nonlinear dispersive equation ut þ auðun Þx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1:
ð41Þ
Proceeding as before, we set the compactons solutions of Eq. (41) in the form uðx; tÞ ¼ fk sin2 ½lðx ctÞg1=n ;
ð42Þ
or of the form uðx; tÞ ¼ fk cos2 ½lðx ctÞg1=n ;
ð43Þ
A.-M. Wazwaz / Appl. Math. Comput. 154 (2004) 835–846
843
where k and l are constants. Substituting (42) or (43) into (41) and solving the resulting equations we obtain rffiffiffiffiffi 1 an 2c ð44Þ l¼ ; k¼ : 2 b an Substituting (44) into (42) and (43) gives the following set of general compactons solutions 8 rffiffiffiffiffi 1=n > < 2c 1 an p 2 sin ; jx ctj 6 ; ðx ctÞ uðx; tÞ ¼ ð45Þ an 2 b l > : 0; otherwise; and
8 rffiffiffiffiffi 1=n > < 2c 1 an 2 cos ; ðx ctÞ uðx; tÞ ¼ an 2 b > : 0;
p ; 2l otherwise:
jx ctj 6
ð46Þ
4.2. The defocusing branch We next consider the defocusing branch ut auðun Þx þ bux ðun Þxx ¼ 0; a; b > 0; n > 1:
ð47Þ
It is convenient to set the general solution of (47) in the form lðx ctÞg1=n ; uðx; tÞ ¼ fk~ sinh2 ½~
ð48Þ
or in the form lðx ctÞg1=n ; uðx; tÞ ¼ fk~ cosh2 ½~
ð49Þ
where k~ and l~ are constants. Substituting (48) or (49) into (47) and solving the resulting equations we obtain rffiffiffiffiffi 1 an 2c : ð50Þ l~ ¼ ; k~ ¼ 2 b an As a consequence of (50), the general solutions rffiffiffiffiffi 1=n 2c 1 an uðx; tÞ ¼ sinh2 ; ðx ctÞ an 2 b
ð51Þ
and uðx; tÞ ¼
rffiffiffiffiffi 1=n 2c 1 an cosh2 ; ðx ctÞ an 2 b
follow immediately.
ð52Þ
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5. Variant IV 5.1. The focusing branch We finally consider the nonlinear dispersive equation ut þ aun ðuÞx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1:
ð53Þ
It is normal to set the compactons solutions of Eq. (53) in the form uðx; tÞ ¼ fk sin2 ½lðx ctÞg1=n ;
ð54Þ
or of the form uðx; tÞ ¼ fk cos2 ½lðx ctÞg
1=n
ð55Þ
;
where k and l are constants. Substituting (54) or (55) into (53) and solving the resulting equations we obtain rffiffiffi 1 a 2c l¼ ð56Þ ; k¼ : 2 b a We can easily observe that l and k do not depend on n. Substituting (56) into (54) and (55) gives the following set of general compactons solutions 8 rffiffiffi 1=n > < 2c 1 a p 2 sin ; jx ctj 6 ; ðx ctÞ uðx; tÞ ¼ ð57Þ a 2 b l > : 0; otherwise; and 8 rffiffiffi 1=n > < 2c 1 a 2 cos ; ðx ctÞ uðx; tÞ ¼ a 2 b > : 0;
p ; 2l otherwise:
jx ctj 6
ð58Þ
5.2. The defocusing branch We next consider the defocusing branch ut auðun Þx þ bux ðun Þxx ¼ 0;
a; b > 0; n > 1:
ð59Þ
It is convenient to set the general solution of (59) in the form uðx; tÞ ¼ fk~ sinh2 ½~ lðx ctÞg
1=n
;
ð60Þ
or in the form uðx; tÞ ¼ fk~ cosh2 ½~ lðx ctÞg1=n ;
ð61Þ
A.-M. Wazwaz / Appl. Math. Comput. 154 (2004) 835–846
845
where k~ and l~ are constants. Substituting (60) or (61) into (59) and solving the resulting equations we obtain 1 l~ ¼ 2
rffiffiffi a ; b
2c : k~ ¼ a
ð62Þ
It can be easily observed that l~ and k~ do not depend on n. As a consequence of (62), the general solutions rffiffiffi 1=n 2c 1 a 2 uðx; tÞ ¼ sinh ; ðx ctÞ a 2 b
ð63Þ
and uðx; tÞ ¼
rffiffiffi 1=n 2c 1 a cosh2 ; ðx ctÞ a 2 b
ð64Þ
follow immediately. This completes the derivation for the compactons solutions and for the solitary patterns solutions for the two classes of variants.
6. Concluding remarks The basic goal of this work has been to extend the works in [8,18,19]. The focusing branch and the defocusing branch of the four variants of the gKdV equation were thoroughly studied. It is easily noticed that the coefficients of the arguments of the compactons solutions and the patterns solutions are the same for each variant. However, the coefficients of the trigonometric functions and the hyperbolic functions for each variant are the same with opposite signs. Moreover, it is easily observed that the focusing branch of each variant provides compactons solutions, whereas the defocusing branch gives solitary patterns solutions having infinite slopes or cusps. This confirms the reality that the focusing branch and the defocusing branch represent two different sets of models, each leading to a different physical structure. The obtained results in this work clearly demonstrate the effect of the purely nonlinear dispersion and the qualitative change made in the genuinely nonlinear phenomenon. The variants introduced in this work are more general compared to those investigated in [8,18,19]. The significance of the presented approach is that it can be used to search for exact solutions of other nonlinear dispersive equations.
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References [1] M. Wadati, Introduction to solitons, Pramana: J. Phys. 57 (5–6) (2001) 841–847. [2] M. Wadati, The exact solution of the modified Kortweg–de Vries equation, J. Phys. Soc. Jpn. 32 (1972) 1681–1687. [3] M. Wadati, The modified Kortweg–de Vries equation, J. Phys. Soc. Jpn. 34 (1973) 1289–1296. [4] F. Cooper, J. Hyman, A. Khare, Compacton solution in a class of generalized fifth-order Korteweg–de Vries equations, Phys. Rev. E 64 (2001) 1–5. [5] P. Rosenau, J.M. Hyman, Compactons: solitons with finite wavelengths, Phys. Rev. Lett. 70 (5) (1993) 564–567. [6] P. Rosenau, Nonlinear dispersion and compact structures, Phys. Rev. Lett. 73 (13) (1994) 1737–1741. [7] P. Rosenau, On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A 230 (5/6) (1997) 305–318. [8] P. Rosenau, Compact and noncompact dispersive structures, Phys. Lett. A 275 (3) (2000) 193– 203. [9] P.J. Olver, P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E 53 (2) (1996) 1900–1906. [10] M.S. Ismail, T. Taha, A numerical study of compactons, Math. Comput. Simul. 47 (1998) 519– 530. [11] A. Chertock, D. Levy, Particle methods for dispersive equations, J. Comput. Phys. 171 (2001) 708–730. [12] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [13] A.M. Wazwaz, New solitary-wave special solutions with compact support for the nonlinear dispersive Kðm; nÞ equations, Chaos, Solitons Fract. 13 (2) (2002) 321–330. [14] A.M. Wazwaz, Exact specific solutions with solitary patterns for the nonlinear dispersive Kðm; nÞ equations, Chaos, Solitons Fract. 13 (1) (2001) 161–170. [15] A.M. Wazwaz, General compactons solutions for the focusing branch of the nonlinear dispersive Kðn; nÞ equations in higher dimensional spaces, Appl. Math. Comput. 133 (2/3) (2002) 213–227. [16] A.M. Wazwaz, General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive Kðn; nÞ equations in higher dimensional spaces, Appl. Math. Comput. 133 (2/3) (2002) 229–244. [17] A.M. Wazwaz, A study of nonlinear dispersive equations with solitary-wave solutions having compact support, Math. Comput. Simul. 56 (2001) 269–276. [18] A.M. Wazwaz, Compactons dispersive structures for variants of the Kðn; nÞ and the KP equations, Chaos, Solitons Fract. 13 (5) (2002) 1053–1062. [19] 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. [20] A.M. Wazwaz, Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos, Solitons Fract. 12 (8) (2001) 1549– 1556. [21] A.M. Wazwaz, A computational approach to soliton solutions of the Kadomtsev–Petviashili equation, Appl. Math. Comput. 123 (2) (2001) 205–217. [22] A.M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, 1997. [23] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, MA, 1994. [24] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1998) 501–544.