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The Camassa–Holm–KP equations with compact and noncompact travelling wave solutions
Applied Mathematics and Computation 170 (2005) 347–360 www.elsevier.com/locate/amc
The Camassa–Holm–KP equations with compact and noncompact travelling wave solutions Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA
Abstract In this paper, two variants of Camassa–Holm–KP equations are investigated. Compactons: solitons with the absence of infinite tails, solitons: nonlinear localized waves of infinite support, solitary patterns having infinite slopes or cusps, and plane periodic solutions are formally derived. The work highlights the qualitative change in the physical structures of the obtained solutions. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Camassa–Holm–KP equation; Compactons; Solitons; Solitary patterns solutions; Periodic solutions; Sine–Cosine method; The tanh method
1. Introduction Nonlinear partial differential equations with dispersion and dissipation effects, that arise in scientific applications, have been under huge size of investigations. Many powerful methods, such as Ba¨cklund transformation, inverse E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.12.002
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scattering method, Hirota bilinear forms, pseudo-spectral method, the tanh– sech method, the sine–cosine method, and many others were successfully used to investigate these types of equations. Practically, there is no unified method that can be used to handle all types of nonlinearity. The KdV equation ut þ auux þ uxxx ¼ 0
ð1Þ
is a typical soliton equation [1–3] where the balance between the nonlinear convection term uux and the dispersion effect term uxxx gives rise to solitons, while dissipation effects are small enough to be neglected in the lowest order approximation [4–6]. However, the best known two-dimensional generalizations of the KdV equations are the integrable Kadomtsov–Petviashivilli (KP) equation, and the nonintegrable Zakharov–Kuznetsov (ZK) equation, given by fut þ auux þ uxxx gx þ uyy ¼ 0;
ð2Þ
and ut þ auux þ ðr2 uÞx ¼ 0; 2
@ 2x
ð3Þ @ 2y
@ 2z
respectively, where r ¼ þ þ is the isotropic Laplacian [7–19]. The now well-known K(n, n) equation introduced in [13] ut þ aðun Þx þ ðun Þxxx ¼ 0;
n > 1;
ð4Þ
where the delicate interaction between nonlinear convection with the genuine nonlinear dispersion generates solitary waves with exact compact support that are termed compactons. Compactons are defined as solitons with finite wavelengths or solitons free of exponential wings. Camassa and Holm [14] derived a completely integrable wave equation (CH) ut þ 2kux uxxt þ auux ¼ 2ux uxx þ uuxxx
ð5Þ
by retaining two terms that are usually neglected in the small amplitude, shallow water limit [15]. The constant k is related to the critical shallow water wave speed. Eq. (5) can be derived as an asymptotic model for long gravity waves at the surface of shallow water [14–16]. The CH equation, being a model equation for water waves, has its integrable bi-Hamiltonian structure. Constantin and Kolev [17] showed that the CH equation arises in the context of differential geometry. For k = 0, a = 3, it has been shown in [20] that the CH equation (5) has peaked solitary wave solutions of the form uðx; tÞ ¼ ceðjxctjÞ ;
ð6Þ
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349
where c is the wave speed. Solitary wave solutions (6) have discontinuous first derivative at the wave peak in contrast to the smoothness of most previously known species of solitary waves and thus are called peakons [20,21]. The name ‘‘peakons’’, that is, solitary waves with slope discontinuities, was used to single them from general solitary wave solutions since they have a corner at the peak of height c. However, for k 5 0, a 5 0, Qian and Tang [22] investigated the CH equation and obtained two peakons of the form rffiffiffi 6k a 6kt exp ; a 6¼ 3; ð7Þ uðx; tÞ ¼ x 3a 3 3 a and rffiffiffi 2k a 2kt uðx; tÞ ¼ 3 exp 2 : x 1þa 3 1 þ a
ð8Þ
The last peaked solitary wave (8) works for every a, a > 0. Boyd [15] noted that Camassa–Holm equation, which is exactly integrable, is of great interest for two reasons: it is a model for small amplitude shallow water waves such as the KdV equation, and it gives peaked periodic waves which have discontinuous first derivative at each peak and thus are called coshoidal waves or periodic cusp waves. Tian and Song [23] investigated a modified Camassa–Holm (mCH) equation ut þ 2kux uxxt þ aun ux ¼ 2ux uxx þ uuxxx ;
ð9Þ
where a > 0, k 2 R and n is called the strength of the nonlinearity. New peaked solitary wave solutions for (9) were obtained in [23]. Boyd [15] formally investigated that if the solitary wave varies slowly with n = x ct, then the two extra terms on the right-hand side will be small and the soliton is given to lowest order by the solutions of ut þ 2kux uxxt þ auux ¼ 0:
ð10Þ
In view of (10), Wazwaz [24,25] examined two modified forms of Camassa– Holm equation given by ut þ 2kux uxxt þ aun ux ¼ 0;
ð11Þ
ut þ 2kux uxxt þ aun ðun Þx ¼ 0;
ð12Þ
and where a > 0, k 2 R and n is called the strength of the nonlinearity. It is the objective of this work to further complement our previous study, by neglecting the two extra terms on the right-hand side of (9), exactly as used by Boyd [15]. Our first interest in the present work being in implementing the sine– cosine method and the tanh method to stress its power in handling nonlinear equations so that one can apply it to models of various types of nonlinearity.
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The next interest is in the determination of exact travelling wave solutions with distinct physical structures to variants in the (2 + 1)-dimensional, two spatial and one temporal variables, of the Camassa–Holm–KP equations given by Variant I: ðut þ 2kux uxxt aun ux Þx þ uyy ¼ 0;
a > 0;
and Variant II: ut þ 2kux uxxt þ aun ðun Þx x þ uyy ¼ 0:
ð13Þ
ð14Þ
The aforementioned variants (13) and (14) are developed similarly to the KP equation (2), derived from the modified (CH) equations (11) and (12). It is thus normal to call Eqs. (13) and (14) the Camassa–Holm–KP equations. As will be analyzed in the coming sections, our approach depends mainly on the sine–cosine method and the tanh method [18,19]. We stress here that the two algorithms have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be solved by using Mathematica or Maple. The two method provide systematic frameworks for many nonlinear dispersive and dissipative equations. Compactons, solitary patterns, plane periodic and solitary traveling waves solutions will be established. In what follows, we highlight the main steps of the sine–cosine algorithm. 2. The two methods The sine–cosine and the tanh method have been extensively studied and widely applied for a wide variety of nonlinear problems. The main features of the two methods will be reviewed briefly because details can be found in [18,19] and in [26–34]. 2.1. The sine–cosine method A PDE in two independent variables P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0
ð15Þ
can be carried into an ODE Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0
ð16Þ
upon using a wave variable n = (x ct). Eq. (16) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. The method admits the use of the solutions in the forms p ; fkcosb ðlnÞg; jnj 6 2l uðx; tÞ ¼ ð17Þ 0; otherwise;
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and
( uðx; tÞ ¼
fksinb ðlnÞg; jnj 6 pl ; 0; otherwise;
351
ð18Þ
where k, l, and b are parameters that will be determined, l and c are the wave number and the wave speed respectively. Eqs. (17) and (18) give ðun Þ00 ¼ n2 l2 b2 kn cosnb ðlnÞ þ nl2 kn bðnb 1Þcosnb2 ðlnÞ;
ð19Þ
and 00
ðun Þ ¼ n2 l2 b2 kn sinnb ðlnÞ þ nl2 kn bðnb 1Þsinnb2 ðlnÞ:
ð20Þ R
Substituting (19) or (20) into (16) gives a trigonometric equation of cos (ln) or sinR(ln) terms. The parameters are then determined by first balancing the exponents of each pair of cosine or sine to determine R. We next collect all coefficients of the same power in cosk(ln) or sink(ln), where these coefficients have to vanish. This gives a system of algebraic equations among the unknowns b, k and l that will be determined. The solutions proposed in (17) and (18) follow immediately. 2.2. The tanh method The tanh method is developed by Malfliet [18,19] 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;
ð21Þ
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
ð22Þ
We then apply the following series expansion uðlnÞ ¼ SðY Þ ¼
M X
ak Y k ;
ð23Þ
k¼0
where M is a positive integer, in most cases, that will be determined. However, if M is not an integer, a transformation formula is usually used. Substituting (22) and (23) into the simplified ODE 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
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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 (23) we obtain an analytic solution u(x, t) in a closed form. The algorithms described above certainly works well for a large class of very interesting nonlinear equations. The main advantage of the methods is that the great capability of reducing the size of computational work compared to existing techniques such as the pseudo-spectral method, the inverse scattering method, HirotaÕs bilinear method, and the truncated Painleve´ expansion. 3. The Camassa–Holm–KP equation: Variant I 3.1. Using the sine–cosine method The Camassa–Holm–KP equation reads ðut þ 2kux uxxt aun ux Þx þ uyy ¼ 0;
a > 0; n > 1:
ð24Þ
Using the wave variable n = x + y ct, integrating the resulting ODE twice and setting the constants of integration to zero we find a unþ1 þ cu00 ¼ 0: ð25Þ ð2k þ 1 cÞu nþ1 Substituting (17) into (25) gives a ð2k þ 1 cÞkcosb ðlnÞ knþ1 cosðnþ1Þb ðlnÞ cl2 b2 kcosb ðlnÞ nþ1 þ cl2 kbðb 1Þcosb2 ðlnÞ:
ð26Þ
Eq. (26) is satisfied only if the following system of algebraic equations holds: b 1 6¼ 0; ðn þ 1Þb ¼ b 2; cl2 b2 k ¼ ð2k þ 1 cÞk; a knþ1 ¼ cl2 bðb 1Þk; nþ1
ð27Þ
which leads to 2 b¼ ; n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r n 2k þ 1 c l¼ ; 2k þ 1 > c; 2 c 1 ð2k þ 1 cÞðn þ 1Þðn þ 2Þ n k¼ : 2a
ð28Þ
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353
The results (28) can be easily obtained if we also use the sine method (18). Combining (28) with (17) and (18), the following periodic solutions, for 2k + 1 > c, (
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n ð2k þ 1 cÞðn þ 1Þðn þ 2Þ 2 n 2k þ 1 c csc ðx þ y ctÞ uðx; y; tÞ ¼ ; 2a 2 c p ð29Þ 0 < ln < ; 2
and ( " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n ð2k þ 1 cÞðn þ 1Þðn þ 2Þ 2 n 2k þ 1 c uðx; y; tÞ ¼ ; sec ðx þ y ctÞ 2a 2 c ð30Þ
follow immediately. However, for 2k + 1 < c, we obtain the following solitons solutions ( uðx; y; tÞ ¼
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n ð2k þ 1 cÞðn þ 1Þðn þ 2Þ n ðc 2k 1Þ csch2 ðx þ y ctÞ ; 2a 2 c ð31Þ
and the bell shaped soliton ( uðx; y; tÞ ¼
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n ð2k þ 1 cÞðn þ 1Þðn þ 2Þ ðc 2k 1Þ 2 n sech : ðx þ y ctÞ 2a 2 c ð32Þ
3.2. For negative exponents We next consider the first variant with exponent n: ðut þ 2kux uxxt aun ux Þx þ uyy ¼ 0;
a > 0; n > 2:
ð33Þ
Substituting n by n in the results (29)–(32), we obtain the following compactons solutions for 2k + 1 > c: 8n h qffiffiffiffiffiffiffiffiffiffiffi io1n < 2 n 2a 2kþ1c ðx þ y ctÞ sin ; jx ctj 6 pl ; ð2kþ1cÞðn1Þðn2Þ 2 c uðx; y; tÞ ¼ : 0; otherwise; ð34Þ and
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uðx; ytÞ ¼
8n < :
h qffiffiffiffiffiffiffiffiffiffiffi io1n 2kþ1c p ðx þ y ctÞ ; jx ctj 6 2l ; c
2a cos2 n2 ð2kþ1cÞðn1Þðn2Þ
0;
otherwise; ð35Þ
and for 2k + 1 < c, we obtain the following solitary patterns solutions ( uðx; y; tÞ ¼
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n 2a c 2k 1 2 n ; sinh ðx þ y ctÞ ð2k þ 1 cÞðn 1Þðn 2Þ 2 c ð36Þ
and ( uðx; y; tÞ ¼
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n 2a c 2k 1 2 n cosh ðx þ y ctÞ : ð2k þ 1 cÞðn 1Þðn 2Þ 2 c ð37Þ
3.3. Using the tanh method The first variant of the Camassa–Holm–KP equation ðut þ 2kux uxxt aun ux Þx þ uyy ¼ 0;
a > 0; n > 1;
ð38Þ
will be handled by using the tanh method. Using the wave variable n = x + y ct, and proceeding as before we find a unþ1 þ cu00 ¼ 0: ð2k þ 1 cÞu ð39Þ nþ1 Balancing u00 with un + 1 we find 4 þ M 2 ¼ ðn þ 1ÞM;
ð40Þ
so that 2 M¼ : n
ð41Þ
To get a closed form analytic solution, then M should be an integer. A transformation formula 1
u ¼ vn
ð42Þ
should be used to achieve our goal. This in turn transforms (39) to n2 ðn þ 1Þð2k þ 1 cÞv2 an2 v3 þ cnðn þ 1Þvv00 þ cðn þ 1Þð2 nÞðv0 Þ2 ¼ 0: ð43Þ
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355
Balancing vv00 and v3 gives M = 2. The tanh method allows us to use the substitution vðx; y; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 :
ð44Þ
Substituting (44) into (43), collecting the coefficients of each power of Y, and using Mathematica to solve the resulting system of algebraic equations we obtain ð2k þ 1 cÞðn þ 2Þðn þ 1Þ ; 2a a1 ¼ 0;
a0 ¼
ð45Þ
ð2k þ 1 cÞðn þ 2Þðn þ 1Þ ; a2 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a n c 2k 1 M¼ : 2 c
This in turn gives the same results obtained before in (29)–(32) for positive exponent, and the results (34)–(37) for negative exponents that were obtained by using the sine–cosine method.
4. Variant II 4.1. Using the sine–cosine method The second variant of the Camassa–Holm–KP equation reads ðut þ 2kux uxxt þ aun ðun ÞÞx þ uyy ¼ 0; or equivalently a ut þ 2kux uxxt þ ðu2n Þx þ uyy ¼ 0; 2 x
a > 0; n > 1
a > 0; n > 1:
ð46Þ
ð47Þ
The wave variable n = x + y ct carries (47) into the ODE a ð48Þ ð2k þ 1 cÞu00 þ ðu2n Þ00 þ cuðivÞ ¼ 0: 2 Integrating (48) twice and setting the constant of integration to zero we find a ð2k þ 1 cÞu þ u2n þ cu00 ¼ 0: ð49Þ 2 Substituting (17) into (49) gives a ð2k þ 1 cÞkcosb ðlnÞ þ k2n cos2nb ðlnÞ cl2 b2 kcosb ðlnÞ 2 þ cl2 bðb 1Þkcosb2 ðlnÞ:
ð50Þ
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As a result we obtain the following system of algebraic equations holds: b 1 6¼ 0; 2nb ¼ b 2; cl2 b2 k ¼ ð2k þ 1 cÞk;
ð51Þ
a 2n k ¼ cl2 bðb 1Þk; 2 from which we find 2 ; n > 1; 2n 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n 1 2k þ 1 c ; l¼ 2 c 1 ðc 2k 1Þð2n þ 1Þ 2n1 k¼ ; a b¼
ð52Þ
which can be easily obtained by using the sine method (18). Combining (52) with (17) and (18), gives the following periodic solutions for 2k + 1 > c 1 ( " #)2n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2k 1Þð2n þ 1Þ 2 2n 1 2k þ 1 c uðx; y; tÞ ¼ ; csc ðx þ y ctÞ a 2 c p 0 < ln < ; 2
ð53Þ
and ( uðx; y; tÞ ¼
1 " #)2n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2k 1Þð2n þ 1Þ 2 2n 1 2k þ 1 c sec : ðx ctÞ a 2 c
ð54Þ However, for 2k + 1 < c, we obtain the following solitons solutions ( uðx; y; tÞ ¼
1 " #)2n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2k 1Þð2n þ 1Þ c 2k 1 2 2n 1 ; csch ðx þ y ctÞ a 2 c
ð55Þ
and ( uðx; y; tÞ ¼
1 " #)2n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2k 1Þð2n þ 1Þ c 2k 1 2 2n 1 sech ðx þ y ctÞ : a 2 c
ð56Þ
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357
4.2. For negative exponents We next consider the second variant with exponent n: ut þ 2kux uxxt þ aun ðun Þx x þ uyy ¼ 0; a > 0; n > 1 or equivalently a ut þ 2kux uxxt þ ðu2n Þx þ uyy ¼ 0; 2 x
ð57Þ
ð58Þ
a > 0; n > 1:
Replacing every n by n in the previous results, the following compactons solutions uðx; y; tÞ ¼
8n < :
h
a sin2 2nþ1 ðc2k1Þð12nÞ 2
qffiffiffiffiffiffiffiffiffiffiffi
2kþ1c ðx c
1 io2nþ1 þ y ctÞ ; jx þ y ctj 6 pl ;
0;
otherwise; ð59Þ
and uðx; y; tÞ ¼
8n < :
h
a cos2 2nþ1 ðc2k1Þð12nÞ 2
qffiffiffiffiffiffiffiffiffiffiffi
2kþ1c ðx c
þ y ctÞ
1 io2nþ1
0;
;
p jx þ y ctj 6 2l ;
otherwise; ð60Þ
for 2k + 1 > c. However, for 2k + 1 < c, we obtain the following solitary patterns solutions ( uðx; y; tÞ ¼
1 " #)2nþ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c 2k 1 2 2n þ 1 ; sinh ðx þ y ctÞ ðc 2k 1Þð1 2nÞ 2 c
ð61Þ
and ( uðx; y; tÞ ¼
1 " #)2nþ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c 2k 1 2 2n þ 1 cosh ðx þ y ctÞ : ðc 2k 1Þð1 2nÞ 2 c
ð62Þ
5. Using the Tanh method We now study the second variant of the Camassa–Holm–KP equation ðut þ 2kux uxxt þ aun ðun Þx Þ þ uyy ¼ 0; or equivalently
a > 0; n > 1
ð63Þ
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a ut þ 2kux uxxt þ ðu2n Þx þ uyy ¼ 0; 2 x
a > 0; n > 1;
ð64Þ
by using the tanh method. As stated before, using the wave variable n = x + y ct in (64) and integrating the resulting ODE twice we get a ð2k þ 1 cÞu þ u2n þ cu00 ¼ 0: 2
ð65Þ
Balancing u00 with u2n gives M¼
2 : 2n 1
ð66Þ
To obtain a closed form analytic solution, M should be an integer, hence we use the transformation 1
uðx; y; tÞ ¼ v2n1 ðx; y; tÞ;
ð67Þ
to carry out Eq. (65) into a 2 2 ð2k þ 1 cÞð2n 1Þ v2 þ ð2n 1Þ v3 þ cð2n 1Þvv00 2 þ cð3 2nÞvðv0 Þ2 ¼ 0:
ð68Þ
3
Balancing vv00 with v gives M = 2. The tanh methods allows us to substitute vðx; y; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2
ð69Þ
into (68). Collecting the coefficients of powers of Y, where the sum of coefficients of each power of Y should vanish, and solving the resulting system of algebraic equations we obtain a0 ¼
ð2k þ 1 cÞð2n þ 1Þ ; a
a1 ¼ 0; ð2k þ 1 cÞð2n þ 1Þ ; a2 ¼ a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ð2n 1Þ c 2k 1 M¼ ; 2 c
ð70Þ 2k þ 1 < c:
Using these results we obtain the same solutions obtained before in (53)–(56) for positive exponent, and the solutions in (59)–(62) for negative exponent.
6. Discussion In this work two variants of the Camassa–Holm equations constructed in the sense of the KP equation were investigated. It was formally derived that
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359
the solutions may come as compactons, solitary patterns, periodic solutions, or solitons as a result of the positive and the negative exponents. The analysis rests mainly on the sine–cosine method and on the tanh method. The goals of obtaining new travelling wave equations of compact and noncompact structures, and of carrying a comparative study between the two implemented schemes were achieved. Although the tanh method was not useful to derive compactons solutions in other works, it worked successfully in this work to obtain compact and noncompact physical structures for nonlinear evolution equations. The obtained results complement the useful works in [24,25] to enrich the properties of the qualitative change made in the genuinely nonlinear phenomenon.
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[19] W. Malfliet, The tanh method. II. Perturbation technique for conservative systems, Phys. Scr. 54 (1996) 569–575. [20] Z. Liu, R. Wang, Z. Jing, Peaked wave solutions of Camassa–Holm equation, Chaos Solitons Fractals 19 (2004) 77–92. [21] Z. Liu, T. Qian, Peakons of the Camassa–Holm equation, Appl. Math. Modell. 26 (2002) 473– 480. [22] T. Qian, M. Tang, Peakons and periodic cusp waves in a generalized Camassa–Holm equation, Chaos Solitons Fractals 12 (2001) 1347–1360. [23] L. Tian, X. Song, New peaked solitary wave solutions of the generalized Camassa–Holm equation, Chaos Solitons Fractals 19 (2004) 621–637. [24] A.M. Wazwaz, New compact and noncompact solutions for variants of a modified Camassa– Holm equation, Appl. Math. Comput., in press. [25] A.M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa–Holm equation with compact and noncompat solutions, Appl. Math. Comput., in press. [26] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [27] A.M. Wazwaz, New solitary-wave special solutions with compact support for the nonlinear dispersive K(m, n) equations, Chaos Solitons Fractals 13 (2) (2002) 321–330. [28] A.M. Wazwaz, Exact specific solutions with solitary patterns for the nonlinear dispersive K(m, n) equations, Chaos Solitons Fractals 13 (1) (2001) 161–170. [29] A.M. Wazwaz, General compactons solutions for the focusing branch of the nonlinear dispersive K(n, n) equations in higher dimensional spaces, Appl. Math. Comput. 133 (2/3) (2002) 213–227. [30] 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. [31] A.M. Wazwaz, An analytic study of compactons structures in a class of nonlinear dispersive equations, Math. Comput. Simul. 63 (1) (2003) 35–44. [32] A.M. Wazwaz, A study on nonlinear dispersive partial differential equations of compact and noncompact solutions, Appl. Math. Comput. 135 (2–3) (2003) 399–409. [33] A.M. Wazwaz, A construction of compact and noncompact solutions of nonlinear dispersive equations of even order, Appl. Math. Comput. 135 (2–3) (2003) 324–411. [34] A.M. Wazwaz, Compactons in a class of nonlinear dispersive equations, Math. Comput. Modell. 37 (3/4) (2003) 333–341.
The Camassa–Holm–KP equations with compact and noncompact travelling wave solutions Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA
Abstract In this paper, two variants of Camassa–Holm–KP equations are investigated. Compactons: solitons with the absence of infinite tails, solitons: nonlinear localized waves of infinite support, solitary patterns having infinite slopes or cusps, and plane periodic solutions are formally derived. The work highlights the qualitative change in the physical structures of the obtained solutions. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Camassa–Holm–KP equation; Compactons; Solitons; Solitary patterns solutions; Periodic solutions; Sine–Cosine method; The tanh method
1. Introduction Nonlinear partial differential equations with dispersion and dissipation effects, that arise in scientific applications, have been under huge size of investigations. Many powerful methods, such as Ba¨cklund transformation, inverse E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.12.002
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scattering method, Hirota bilinear forms, pseudo-spectral method, the tanh– sech method, the sine–cosine method, and many others were successfully used to investigate these types of equations. Practically, there is no unified method that can be used to handle all types of nonlinearity. The KdV equation ut þ auux þ uxxx ¼ 0
ð1Þ
is a typical soliton equation [1–3] where the balance between the nonlinear convection term uux and the dispersion effect term uxxx gives rise to solitons, while dissipation effects are small enough to be neglected in the lowest order approximation [4–6]. However, the best known two-dimensional generalizations of the KdV equations are the integrable Kadomtsov–Petviashivilli (KP) equation, and the nonintegrable Zakharov–Kuznetsov (ZK) equation, given by fut þ auux þ uxxx gx þ uyy ¼ 0;
ð2Þ
and ut þ auux þ ðr2 uÞx ¼ 0; 2
@ 2x
ð3Þ @ 2y
@ 2z
respectively, where r ¼ þ þ is the isotropic Laplacian [7–19]. The now well-known K(n, n) equation introduced in [13] ut þ aðun Þx þ ðun Þxxx ¼ 0;
n > 1;
ð4Þ
where the delicate interaction between nonlinear convection with the genuine nonlinear dispersion generates solitary waves with exact compact support that are termed compactons. Compactons are defined as solitons with finite wavelengths or solitons free of exponential wings. Camassa and Holm [14] derived a completely integrable wave equation (CH) ut þ 2kux uxxt þ auux ¼ 2ux uxx þ uuxxx
ð5Þ
by retaining two terms that are usually neglected in the small amplitude, shallow water limit [15]. The constant k is related to the critical shallow water wave speed. Eq. (5) can be derived as an asymptotic model for long gravity waves at the surface of shallow water [14–16]. The CH equation, being a model equation for water waves, has its integrable bi-Hamiltonian structure. Constantin and Kolev [17] showed that the CH equation arises in the context of differential geometry. For k = 0, a = 3, it has been shown in [20] that the CH equation (5) has peaked solitary wave solutions of the form uðx; tÞ ¼ ceðjxctjÞ ;
ð6Þ
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349
where c is the wave speed. Solitary wave solutions (6) have discontinuous first derivative at the wave peak in contrast to the smoothness of most previously known species of solitary waves and thus are called peakons [20,21]. The name ‘‘peakons’’, that is, solitary waves with slope discontinuities, was used to single them from general solitary wave solutions since they have a corner at the peak of height c. However, for k 5 0, a 5 0, Qian and Tang [22] investigated the CH equation and obtained two peakons of the form rffiffiffi 6k a 6kt exp ; a 6¼ 3; ð7Þ uðx; tÞ ¼ x 3a 3 3 a and rffiffiffi 2k a 2kt uðx; tÞ ¼ 3 exp 2 : x 1þa 3 1 þ a
ð8Þ
The last peaked solitary wave (8) works for every a, a > 0. Boyd [15] noted that Camassa–Holm equation, which is exactly integrable, is of great interest for two reasons: it is a model for small amplitude shallow water waves such as the KdV equation, and it gives peaked periodic waves which have discontinuous first derivative at each peak and thus are called coshoidal waves or periodic cusp waves. Tian and Song [23] investigated a modified Camassa–Holm (mCH) equation ut þ 2kux uxxt þ aun ux ¼ 2ux uxx þ uuxxx ;
ð9Þ
where a > 0, k 2 R and n is called the strength of the nonlinearity. New peaked solitary wave solutions for (9) were obtained in [23]. Boyd [15] formally investigated that if the solitary wave varies slowly with n = x ct, then the two extra terms on the right-hand side will be small and the soliton is given to lowest order by the solutions of ut þ 2kux uxxt þ auux ¼ 0:
ð10Þ
In view of (10), Wazwaz [24,25] examined two modified forms of Camassa– Holm equation given by ut þ 2kux uxxt þ aun ux ¼ 0;
ð11Þ
ut þ 2kux uxxt þ aun ðun Þx ¼ 0;
ð12Þ
and where a > 0, k 2 R and n is called the strength of the nonlinearity. It is the objective of this work to further complement our previous study, by neglecting the two extra terms on the right-hand side of (9), exactly as used by Boyd [15]. Our first interest in the present work being in implementing the sine– cosine method and the tanh method to stress its power in handling nonlinear equations so that one can apply it to models of various types of nonlinearity.
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The next interest is in the determination of exact travelling wave solutions with distinct physical structures to variants in the (2 + 1)-dimensional, two spatial and one temporal variables, of the Camassa–Holm–KP equations given by Variant I: ðut þ 2kux uxxt aun ux Þx þ uyy ¼ 0;
a > 0;
and Variant II: ut þ 2kux uxxt þ aun ðun Þx x þ uyy ¼ 0:
ð13Þ
ð14Þ
The aforementioned variants (13) and (14) are developed similarly to the KP equation (2), derived from the modified (CH) equations (11) and (12). It is thus normal to call Eqs. (13) and (14) the Camassa–Holm–KP equations. As will be analyzed in the coming sections, our approach depends mainly on the sine–cosine method and the tanh method [18,19]. We stress here that the two algorithms have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be solved by using Mathematica or Maple. The two method provide systematic frameworks for many nonlinear dispersive and dissipative equations. Compactons, solitary patterns, plane periodic and solitary traveling waves solutions will be established. In what follows, we highlight the main steps of the sine–cosine algorithm. 2. The two methods The sine–cosine and the tanh method have been extensively studied and widely applied for a wide variety of nonlinear problems. The main features of the two methods will be reviewed briefly because details can be found in [18,19] and in [26–34]. 2.1. The sine–cosine method A PDE in two independent variables P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0
ð15Þ
can be carried into an ODE Qðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0
ð16Þ
upon using a wave variable n = (x ct). Eq. (16) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. The method admits the use of the solutions in the forms p ; fkcosb ðlnÞg; jnj 6 2l uðx; tÞ ¼ ð17Þ 0; otherwise;
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and
( uðx; tÞ ¼
fksinb ðlnÞg; jnj 6 pl ; 0; otherwise;
351
ð18Þ
where k, l, and b are parameters that will be determined, l and c are the wave number and the wave speed respectively. Eqs. (17) and (18) give ðun Þ00 ¼ n2 l2 b2 kn cosnb ðlnÞ þ nl2 kn bðnb 1Þcosnb2 ðlnÞ;
ð19Þ
and 00
ðun Þ ¼ n2 l2 b2 kn sinnb ðlnÞ þ nl2 kn bðnb 1Þsinnb2 ðlnÞ:
ð20Þ R
Substituting (19) or (20) into (16) gives a trigonometric equation of cos (ln) or sinR(ln) terms. The parameters are then determined by first balancing the exponents of each pair of cosine or sine to determine R. We next collect all coefficients of the same power in cosk(ln) or sink(ln), where these coefficients have to vanish. This gives a system of algebraic equations among the unknowns b, k and l that will be determined. The solutions proposed in (17) and (18) follow immediately. 2.2. The tanh method The tanh method is developed by Malfliet [18,19] 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;
ð21Þ
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
ð22Þ
We then apply the following series expansion uðlnÞ ¼ SðY Þ ¼
M X
ak Y k ;
ð23Þ
k¼0
where M is a positive integer, in most cases, that will be determined. However, if M is not an integer, a transformation formula is usually used. Substituting (22) and (23) into the simplified ODE 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
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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 (23) we obtain an analytic solution u(x, t) in a closed form. The algorithms described above certainly works well for a large class of very interesting nonlinear equations. The main advantage of the methods is that the great capability of reducing the size of computational work compared to existing techniques such as the pseudo-spectral method, the inverse scattering method, HirotaÕs bilinear method, and the truncated Painleve´ expansion. 3. The Camassa–Holm–KP equation: Variant I 3.1. Using the sine–cosine method The Camassa–Holm–KP equation reads ðut þ 2kux uxxt aun ux Þx þ uyy ¼ 0;
a > 0; n > 1:
ð24Þ
Using the wave variable n = x + y ct, integrating the resulting ODE twice and setting the constants of integration to zero we find a unþ1 þ cu00 ¼ 0: ð25Þ ð2k þ 1 cÞu nþ1 Substituting (17) into (25) gives a ð2k þ 1 cÞkcosb ðlnÞ knþ1 cosðnþ1Þb ðlnÞ cl2 b2 kcosb ðlnÞ nþ1 þ cl2 kbðb 1Þcosb2 ðlnÞ:
ð26Þ
Eq. (26) is satisfied only if the following system of algebraic equations holds: b 1 6¼ 0; ðn þ 1Þb ¼ b 2; cl2 b2 k ¼ ð2k þ 1 cÞk; a knþ1 ¼ cl2 bðb 1Þk; nþ1
ð27Þ
which leads to 2 b¼ ; n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r n 2k þ 1 c l¼ ; 2k þ 1 > c; 2 c 1 ð2k þ 1 cÞðn þ 1Þðn þ 2Þ n k¼ : 2a
ð28Þ
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353
The results (28) can be easily obtained if we also use the sine method (18). Combining (28) with (17) and (18), the following periodic solutions, for 2k + 1 > c, (
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n ð2k þ 1 cÞðn þ 1Þðn þ 2Þ 2 n 2k þ 1 c csc ðx þ y ctÞ uðx; y; tÞ ¼ ; 2a 2 c p ð29Þ 0 < ln < ; 2
and ( " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n ð2k þ 1 cÞðn þ 1Þðn þ 2Þ 2 n 2k þ 1 c uðx; y; tÞ ¼ ; sec ðx þ y ctÞ 2a 2 c ð30Þ
follow immediately. However, for 2k + 1 < c, we obtain the following solitons solutions ( uðx; y; tÞ ¼
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n ð2k þ 1 cÞðn þ 1Þðn þ 2Þ n ðc 2k 1Þ csch2 ðx þ y ctÞ ; 2a 2 c ð31Þ
and the bell shaped soliton ( uðx; y; tÞ ¼
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n ð2k þ 1 cÞðn þ 1Þðn þ 2Þ ðc 2k 1Þ 2 n sech : ðx þ y ctÞ 2a 2 c ð32Þ
3.2. For negative exponents We next consider the first variant with exponent n: ðut þ 2kux uxxt aun ux Þx þ uyy ¼ 0;
a > 0; n > 2:
ð33Þ
Substituting n by n in the results (29)–(32), we obtain the following compactons solutions for 2k + 1 > c: 8n h qffiffiffiffiffiffiffiffiffiffiffi io1n < 2 n 2a 2kþ1c ðx þ y ctÞ sin ; jx ctj 6 pl ; ð2kþ1cÞðn1Þðn2Þ 2 c uðx; y; tÞ ¼ : 0; otherwise; ð34Þ and
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uðx; ytÞ ¼
8n < :
h qffiffiffiffiffiffiffiffiffiffiffi io1n 2kþ1c p ðx þ y ctÞ ; jx ctj 6 2l ; c
2a cos2 n2 ð2kþ1cÞðn1Þðn2Þ
0;
otherwise; ð35Þ
and for 2k + 1 < c, we obtain the following solitary patterns solutions ( uðx; y; tÞ ¼
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n 2a c 2k 1 2 n ; sinh ðx þ y ctÞ ð2k þ 1 cÞðn 1Þðn 2Þ 2 c ð36Þ
and ( uðx; y; tÞ ¼
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1n 2a c 2k 1 2 n cosh ðx þ y ctÞ : ð2k þ 1 cÞðn 1Þðn 2Þ 2 c ð37Þ
3.3. Using the tanh method The first variant of the Camassa–Holm–KP equation ðut þ 2kux uxxt aun ux Þx þ uyy ¼ 0;
a > 0; n > 1;
ð38Þ
will be handled by using the tanh method. Using the wave variable n = x + y ct, and proceeding as before we find a unþ1 þ cu00 ¼ 0: ð2k þ 1 cÞu ð39Þ nþ1 Balancing u00 with un + 1 we find 4 þ M 2 ¼ ðn þ 1ÞM;
ð40Þ
so that 2 M¼ : n
ð41Þ
To get a closed form analytic solution, then M should be an integer. A transformation formula 1
u ¼ vn
ð42Þ
should be used to achieve our goal. This in turn transforms (39) to n2 ðn þ 1Þð2k þ 1 cÞv2 an2 v3 þ cnðn þ 1Þvv00 þ cðn þ 1Þð2 nÞðv0 Þ2 ¼ 0: ð43Þ
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355
Balancing vv00 and v3 gives M = 2. The tanh method allows us to use the substitution vðx; y; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2 :
ð44Þ
Substituting (44) into (43), collecting the coefficients of each power of Y, and using Mathematica to solve the resulting system of algebraic equations we obtain ð2k þ 1 cÞðn þ 2Þðn þ 1Þ ; 2a a1 ¼ 0;
a0 ¼
ð45Þ
ð2k þ 1 cÞðn þ 2Þðn þ 1Þ ; a2 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a n c 2k 1 M¼ : 2 c
This in turn gives the same results obtained before in (29)–(32) for positive exponent, and the results (34)–(37) for negative exponents that were obtained by using the sine–cosine method.
4. Variant II 4.1. Using the sine–cosine method The second variant of the Camassa–Holm–KP equation reads ðut þ 2kux uxxt þ aun ðun ÞÞx þ uyy ¼ 0; or equivalently a ut þ 2kux uxxt þ ðu2n Þx þ uyy ¼ 0; 2 x
a > 0; n > 1
a > 0; n > 1:
ð46Þ
ð47Þ
The wave variable n = x + y ct carries (47) into the ODE a ð48Þ ð2k þ 1 cÞu00 þ ðu2n Þ00 þ cuðivÞ ¼ 0: 2 Integrating (48) twice and setting the constant of integration to zero we find a ð2k þ 1 cÞu þ u2n þ cu00 ¼ 0: ð49Þ 2 Substituting (17) into (49) gives a ð2k þ 1 cÞkcosb ðlnÞ þ k2n cos2nb ðlnÞ cl2 b2 kcosb ðlnÞ 2 þ cl2 bðb 1Þkcosb2 ðlnÞ:
ð50Þ
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As a result we obtain the following system of algebraic equations holds: b 1 6¼ 0; 2nb ¼ b 2; cl2 b2 k ¼ ð2k þ 1 cÞk;
ð51Þ
a 2n k ¼ cl2 bðb 1Þk; 2 from which we find 2 ; n > 1; 2n 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n 1 2k þ 1 c ; l¼ 2 c 1 ðc 2k 1Þð2n þ 1Þ 2n1 k¼ ; a b¼
ð52Þ
which can be easily obtained by using the sine method (18). Combining (52) with (17) and (18), gives the following periodic solutions for 2k + 1 > c 1 ( " #)2n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2k 1Þð2n þ 1Þ 2 2n 1 2k þ 1 c uðx; y; tÞ ¼ ; csc ðx þ y ctÞ a 2 c p 0 < ln < ; 2
ð53Þ
and ( uðx; y; tÞ ¼
1 " #)2n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2k 1Þð2n þ 1Þ 2 2n 1 2k þ 1 c sec : ðx ctÞ a 2 c
ð54Þ However, for 2k + 1 < c, we obtain the following solitons solutions ( uðx; y; tÞ ¼
1 " #)2n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2k 1Þð2n þ 1Þ c 2k 1 2 2n 1 ; csch ðx þ y ctÞ a 2 c
ð55Þ
and ( uðx; y; tÞ ¼
1 " #)2n1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðc 2k 1Þð2n þ 1Þ c 2k 1 2 2n 1 sech ðx þ y ctÞ : a 2 c
ð56Þ
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357
4.2. For negative exponents We next consider the second variant with exponent n: ut þ 2kux uxxt þ aun ðun Þx x þ uyy ¼ 0; a > 0; n > 1 or equivalently a ut þ 2kux uxxt þ ðu2n Þx þ uyy ¼ 0; 2 x
ð57Þ
ð58Þ
a > 0; n > 1:
Replacing every n by n in the previous results, the following compactons solutions uðx; y; tÞ ¼
8n < :
h
a sin2 2nþ1 ðc2k1Þð12nÞ 2
qffiffiffiffiffiffiffiffiffiffiffi
2kþ1c ðx c
1 io2nþ1 þ y ctÞ ; jx þ y ctj 6 pl ;
0;
otherwise; ð59Þ
and uðx; y; tÞ ¼
8n < :
h
a cos2 2nþ1 ðc2k1Þð12nÞ 2
qffiffiffiffiffiffiffiffiffiffiffi
2kþ1c ðx c
þ y ctÞ
1 io2nþ1
0;
;
p jx þ y ctj 6 2l ;
otherwise; ð60Þ
for 2k + 1 > c. However, for 2k + 1 < c, we obtain the following solitary patterns solutions ( uðx; y; tÞ ¼
1 " #)2nþ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c 2k 1 2 2n þ 1 ; sinh ðx þ y ctÞ ðc 2k 1Þð1 2nÞ 2 c
ð61Þ
and ( uðx; y; tÞ ¼
1 " #)2nþ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c 2k 1 2 2n þ 1 cosh ðx þ y ctÞ : ðc 2k 1Þð1 2nÞ 2 c
ð62Þ
5. Using the Tanh method We now study the second variant of the Camassa–Holm–KP equation ðut þ 2kux uxxt þ aun ðun Þx Þ þ uyy ¼ 0; or equivalently
a > 0; n > 1
ð63Þ
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A.-M. Wazwaz / Appl. Math. Comput. 170 (2005) 347–360
a ut þ 2kux uxxt þ ðu2n Þx þ uyy ¼ 0; 2 x
a > 0; n > 1;
ð64Þ
by using the tanh method. As stated before, using the wave variable n = x + y ct in (64) and integrating the resulting ODE twice we get a ð2k þ 1 cÞu þ u2n þ cu00 ¼ 0: 2
ð65Þ
Balancing u00 with u2n gives M¼
2 : 2n 1
ð66Þ
To obtain a closed form analytic solution, M should be an integer, hence we use the transformation 1
uðx; y; tÞ ¼ v2n1 ðx; y; tÞ;
ð67Þ
to carry out Eq. (65) into a 2 2 ð2k þ 1 cÞð2n 1Þ v2 þ ð2n 1Þ v3 þ cð2n 1Þvv00 2 þ cð3 2nÞvðv0 Þ2 ¼ 0:
ð68Þ
3
Balancing vv00 with v gives M = 2. The tanh methods allows us to substitute vðx; y; tÞ ¼ SðY Þ ¼ a0 þ a1 Y þ a2 Y 2
ð69Þ
into (68). Collecting the coefficients of powers of Y, where the sum of coefficients of each power of Y should vanish, and solving the resulting system of algebraic equations we obtain a0 ¼
ð2k þ 1 cÞð2n þ 1Þ ; a
a1 ¼ 0; ð2k þ 1 cÞð2n þ 1Þ ; a2 ¼ a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ð2n 1Þ c 2k 1 M¼ ; 2 c
ð70Þ 2k þ 1 < c:
Using these results we obtain the same solutions obtained before in (53)–(56) for positive exponent, and the solutions in (59)–(62) for negative exponent.
6. Discussion In this work two variants of the Camassa–Holm equations constructed in the sense of the KP equation were investigated. It was formally derived that
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the solutions may come as compactons, solitary patterns, periodic solutions, or solitons as a result of the positive and the negative exponents. The analysis rests mainly on the sine–cosine method and on the tanh method. The goals of obtaining new travelling wave equations of compact and noncompact structures, and of carrying a comparative study between the two implemented schemes were achieved. Although the tanh method was not useful to derive compactons solutions in other works, it worked successfully in this work to obtain compact and noncompact physical structures for nonlinear evolution equations. The obtained results complement the useful works in [24,25] to enrich the properties of the qualitative change made in the genuinely nonlinear phenomenon.
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