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Compactons, solitons and periodic solutions for variants of the KdV and the KP equations
Applied Mathematics and Computation 161 (2005) 561–575 www.elsevier.com/locate/amc
Compactons, solitons and periodic solutions for variants of the KdV and the KP equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA
Abstract Studying solitons and solitons with compact support is of important significance in nonlinear phenomena. In this paper we study nonlinear variants of the KdV and the KP equations with positive and negative exponents. We employ the sine–cosine algorithm to back up our analysis. Exact solutions with compactons, solitons, solitary patterns, and periodic structures are obtained for these variants. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Compactons; Solitons; Periodic solutions; KP equation; KdV equation; Sine–cosine method
1. Introduction The KdV equation is a model that governs the one dimensional propagation of small-amplitude, weakly dispersive waves [1]. The nonlinear term uux in the KdV equation ut þ auux þ uxxx ¼ 0
ð1Þ
causes the steepening of wave form as described by Wadati [2]. However, the dispersion effect term uxxx in the same equation makes the wave form spread [2]. The balance between this weak nonlinearity and dispersion gives rise to
E-mail address: [email protected] (A.-M. Wazwaz). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.049
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solitons. Soliton is a localized wave that has an infinite support or a localized wave with exponential wings. Wadati [2–4] defined soliton as a nonlinear wave that has the following properties: (1) A localized wave propagates without change of its properties (shape, velocity, etc.). (2) Localized waves are stable against mutual collisions and retain their identities. This means that soliton has the property of a particle. However, the convection term in the Kðn; nÞ equation, introduced in [5] ut þ aðun Þx þ ðun Þxxx ¼ 0;
n > 1;
ð2Þ
is nonlinear, whereas the dispersion effect term ðun Þxxx in this equation is genuinely nonlinear. It is now formally proved by many researchers that the delicate interaction between nonlinear convection with genuine nonlinear dispersion generates solitary waves with exact compact support that are termed compactons. Partial differential equations that have the feature of nonlinear dispersion generate compactons that are solitons with finite wavelength, solitons with compact support or solitons free of exponential tails. Compactons are compact solutions that are usually expressed by powers of trigonometric functions sine and cosine. Unlike soliton that narrows as the amplitude increases, the compactonÕs width is independent of the amplitude. In modern physics, a suffix-on is used to indicate the particle property as described by Wadati [2], for example phonon, photon, and soliton. For this reason, the solitary wave with compact support is called compacton to indicate that it has the property of a particle. The classical solitons are analytic solutions, whereas compactons are nonanalytic [6] solutions. 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. The soliton concept has been examined by many mathematical methods. There are many approaches such as the inverse scattering method, the B€acklund transformation, the Darboux transformation, and the Painleve analysis. However, the compacton concept has been studied by using many analytical and numerical methods in [5–31]. There are many algorithms such as the pseudospectral method [5], the tri-Hamiltonian operators [10], the finite difference method [11], and Adomian decomposition method [32–34]. The compactons discovery motivated a considerable work to make compactons be practically realized in scientific applications, such as the super
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563
deformed nuclei, preformation of cluster in hydrodynamic models, the fission of liquid drops (nuclear physics), inertial fusion and others, as indicated in [4– 19]. For the compactons supporting equations, where a > 0, the solutions were defined in terms of the powers of the trigonometric functions sine and cosine profiles, whereas for the solitary patterns supporting models, where a < 0, the solutions were formally derived in terms of powers of the hyperbolic functions sinh and cosh. Recently, Wazwaz [22] examined two variants of the KdV and the KP equations, where wave dispersion is purely nonlinear, given by ut auðun Þx þ b½uðun Þxx x ¼ 0;
ð3Þ
a; b > 0; n > 1;
and fut aðun Þx þ b½uðun Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0;
ð4Þ
a; b > 0; n > 1;
where k ¼ 2; 3 so that k gives the dimensions of the two and three dimensional spaces. The study examined both types of physical structures, the compact and the noncompact behavior. For a > 0, the compactons solutions 8 qffiffiffiffiffiffiffiffiffiffiffiffi 1n < 2ðnþ1Þc 2 an sin ; jx ctj 6 pl ; ðx ctÞ 4bðnþ1Þ an uðx; tÞ ¼ ð5Þ : 0 otherwise; and uðx; tÞ ¼
8 < 2ðnþ1Þc :
an
qffiffiffiffiffiffiffiffiffiffiffiffi 1n an cos ; ðx ctÞ 4bðnþ1Þ 2
0
jx ctj 6
p ; 2l
ð6Þ
otherwise;
and uðx; r; tÞ ¼
8 < :
1 a ðtþbÞ2
2ðnþ1Þc an
2
sin
qffiffiffiffiffiffiffiffiffiffiffiffi
an 4bðnþ1Þ
xþ
r2 4ðtþbÞ
cs
1n
;
0
jvj 6 pl ; otherwise; ð7Þ
and uðx; r; tÞ ¼
8 < :
1 a ðtþbÞ2
2ðnþ1Þc an
cos2
qffiffiffiffiffiffiffiffiffiffiffiffi
an 4bðnþ1Þ
2
r
cs x þ 4ðtþbÞ
0
where a ¼ k 1, r2 ¼ y 2 þ ðk 2Þz2 , k ¼ 2; 3, and
1n
;
jvj 6
p ; 2l
otherwise; ð8Þ
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A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
v¼
r2
cs xþ 4ðt þ bÞ
ð9Þ
were formally derived in [22] for the solutions of (3) and (4) respectively. However, for a < 0, the following solitary patterns solutions rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1n 2ðn þ 1Þc an 2 sinh uðx; tÞ ¼ ; ð10Þ ðx ctÞ an 4bðn þ 1Þ and uðx; tÞ ¼
2ðn þ 1Þc cosh2 an
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1n an ; ðx ctÞ 4bðn þ 1Þ
ð11Þ
and uðx; tÞ ¼
1 a
ðt þ bÞ2
2ðn þ 1Þc sinh2 an
1n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi an r2
cs ; xþ 4bðn þ 1Þ 4ðt þ bÞ ð12Þ
and uðx; tÞ ¼
1 a
ðt þ bÞ2
2ðn þ 1Þc cosh2 an
1n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi an r2
cs xþ 4bðn þ 1Þ 4ðt þ bÞ ð13Þ
were formally obtained for the defocusing branch of (3) and (4). The present paper is motivated by the desire to extend the works made in the literature to make further progress. More importantly, it is the objective of this work to show that compactons, solitary patterns, periodic solutions, and solitons arise from variants of the KdV and the KP equations. This variation in physical structures of the resulting solutions, as will be shown, depends mainly on the exponents, positive or negative, of the unknown function uðx; tÞ, and on the coefficient of the convection term a as well. To achieve our goal, we selected two nonlinear variants of the KdV and the KP equations, given by ut aðu3n Þx þ b½un ðu2n Þxx x ¼ 0;
ð14Þ
a; b > 0; n > 1;
and fut aðu3n Þx þ b½un ðu2n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0; > 1;
a; b > 0; n ð15Þ
where k ¼ 2; 3 to mindicate the two and three dimensional spaces respectively. In this work, the sine–cosine method will be used to back up our analysis. The two physical structures where a > 0 and a < 0 will be examined for po-
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565
sitive n and for negative n as well. It is reasonable to highlight the main steps of this algorithm.
2. The sine–cosine method 1. We introduce the wave variable n ¼ ðx ctÞ into the nonlinear PDE, say in two variables: P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0;
ð16Þ
where uðx; tÞ is the traveling-type wave solution. This enables us to use the following changes o d ¼ c ; ot dn
o2 d2 ¼ c2 2 ; 2 ot dn
o d ¼ ; ox dn
o2 d2 ¼ 2: 2 ox dn
ð17Þ
One can immediately reduce the nonlinear PDE (16) into a nonlinear ODE Qðu; un ; unn ; unnn ; . . .Þ ¼ 0;
ð18Þ
where un denotes du . dn 2. The ordinary differential equation (18) is then integrated as long as all terms contain derivatives, where the associated integration constants can be set zeros. 3. The sine–cosine algorithm admits the use of the solutions of the resulting equation in the form uðx; tÞ ¼ fk cosb ðlnÞg;
jnj 6
p ; 2l
ð19Þ
jnj 6
p ; l
ð20Þ
or in the form uðx; tÞ ¼ fk sinb ðlnÞg;
where k, l, and b are parameters that will be determined. 4. In view of (19) we use uðnÞ ¼ k cosb ðlnÞ; un ðnÞ ¼ kn cosnb ðlnÞ; ðun Þn ¼ nlbkn sinðlnÞ cosnb 1 ðnÞ; ðun Þnn ¼ n2 l2 b2 kn cosnb þnl2 kn bðnb 1Þ cosnb 2 ðlnÞ; and for (20) we use
ð21Þ
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uðnÞ ¼ k sinb ðlnÞ; un ðnÞ ¼ kn sinnb ðlnÞ; ðun Þn ¼ nlbkn cosðlnÞ sinnb 1 ðnÞ;
ð22Þ
ðun Þnn ¼ n2 l2 b2 kn sinnb þnl2 kn bðnb 1Þ sinnb 2 ðlnÞ: 5. Substituting (21) or (22) into the reduced ODE obtained above after integrating (18) gives a trigonometric equation of cosine or sine terms. 6. The main task is to balance the exponents of the trigonometric functions cosine or sine. Collect all terms with same power in cosk ðlnÞ or sink ðlnÞ and set to zero their coefficients to get a system of algebraic equations among the unknowns b, k and l. The problem is now completely reduced to an algebraic one. Having determined k, l, and b by using computerized symbolic calculations, the solutions proposed in (19) and in (20) follow immediately. The algorithm described above certainly works well for a large class of very interesting nonlinear equations. The main advantage of the method is that it is capable of greatly reducing the size of computational work compared to existing techniques such as the pseudospectral method, the inverse scattering method, HirotaÕs bilinear method, and the truncated Painleve expansion.
3. Nonlinear variant of the KdV equation 3.1. The positive exponents 3.1.1. The case where n > 1 and þa We first rewrite a variant of the KdV equation given in (14) in the form ut þ aðu3n Þx þ b½un ðu2n Þxx x ¼ 0;
a; b > 0; n > 1;
ð23Þ
where uðx; tÞ is an unknown function. With the analysis presented above, Eq. (23) will be transformed to
cun þ aðu3n Þn þ b½un ðu2n Þnn n ¼ 0:
ð24Þ
Integrating (24), using the constant of integration to be zero we obtain
cu þ au3n þ bun ðu2n Þnn ¼ 0:
ð25Þ
Substituting (21) into (25) gives
ck cosb ðlnÞ þ ak3n cos3nb ðlnÞ 4bn2 l2 b2 k3n cos3nb ðlnÞ þ 2bnk3n l2 bð2nb 1Þ cos3nb 2 ðlnÞ ¼ 0:
ð26Þ
A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
567
Eq. (26) is satisfied only if the following system of algebraic equations holds: 2nb 1 6¼ 0; 3nb 2 ¼ b;
ð27Þ
4bn2 l2 b2 ¼ a; 2bnk3n l2 bð2nb 1Þ ¼ ck:
The system (27) is straightforwardly calculated by hand or easily computed with any symbolic manipulation package to give 2 ; n > 1; b¼ 3n ffi 1 rffiffi
a 3n 1 l¼ ; ð28Þ b 4n 1
3n 1 4nc k¼ : aðn þ 1Þ Using the sine method (22) leads to the same derivations (28). Consequently, the following compactons solutions (n 1 hpffiffi io3n 1 4nc a 3n 1 sin2 ðx ctÞ ; jx ctj 6 pl ; aðnþ1Þ b 4n uðx; tÞ ¼ ð29Þ 0 otherwise; and
(n uðx; tÞ ¼
4nc aðnþ1Þ
cos2
hpffiffi a 3n 1 b 4n
ðx ctÞ
1 io3n 1
0
;
p jx ctj 6 2l ; otherwise
ð30Þ
are readily obtained. 3.1.2. The case where n > 1 and a In this case, the variant of the KdV becomes ut aðu3n Þx þ b½un ðu2n Þxx x ¼ 0;
a; b > 0;
ð31Þ
well known as the defocusing branch. Replacing a by a into (29) and (30) we obtain the solitary patterns solutions 1 rffiffiffi 3n 1 4nc a 3n 1 2 sinh ðx ctÞ uðx; tÞ ¼ ; ð32Þ aðn þ 1Þ b 4n and uðx; tÞ ¼
4nc cosh2 aðn þ 1Þ
1 rffiffiffi 3n 1 a 3n 1 ðx ctÞ ; b 4n
which are in the form obtained in [22].
ð33Þ
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3.2. The case of negative exponents 3.2.1. The case where n > 1 and þa For negative exponents, we consider a variant of the KdV equation in the form ut þ aðu 3n Þx þ b½u n ðu 2n Þxx x ¼ 0;
ð34Þ
where a; b > 0 are constants, and uðx; tÞ is an unknown function. Following the analysis presented above, Eq. (34) will be transformed to
cun þ aðu 3n Þn þ b½u n ðu 2n Þnn n ¼ 0:
ð35Þ
Integrating (35), using the constant of integration to be zero, and dividing by u we obtain
cu þ au 3n þ bu n ðu 2n Þnn ¼ 0:
ð36Þ
Substituting (21) into (36) yields
ck cosb ðlnÞ þ ak 3n cos 3nb ðlnÞ 4bn2 l2 b2 k 3n cos 3nb ðlnÞ þ 2bnk 3n l2 bð2nb þ 1Þ cos 3nb 2 ðlnÞ ¼ 0:
ð37Þ
Eq. (37) is satisfied only if the following system of algebraic equations holds: 2nb þ 1 6¼ 0;
3nb 2 ¼ b; 4bn2 l2 b2 ¼ a;
ð38Þ
2nbk 3n l2 bð2nb þ 1Þ ¼ ck: This in turn gives 2 ; b¼
3n þ 1 rffiffiffi a 3n þ 1 l¼ ; b 4n
1 aðn 1Þ 3nþ1 k¼ : 4nc
ð39Þ
The results (39) can be easily obtained if we also use the sine method (22). In view of (39), we obtain the periodic solutions 1 rffiffiffi 3nþ1 aðn 1Þ 2 a 3n þ 1 csc ðx ctÞ uðx; tÞ ¼ ; ð40Þ 4nc b 4n and
A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
uðx; tÞ ¼
aðn 1Þ sec2 4nc
1 rffiffiffi 3nþ1 a 3n þ 1 ðx ctÞ : b 4n
569
ð41Þ
We can easily observe that the effect of the negative exponents is the existence of periodic solutions. Recall that compactons and solitary patterns solutions were obtained before for the positive exponents and þa, a > 0. 3.2.2. The case where n > 1 and a In this case, replacing a by a into (34) gives ut aðu 3n Þx þ b½u n ðu 2n Þxx x ¼ 0;
ð42Þ
a; b > 0; n > 1;
where a; b > 0 are constants, and uðx; tÞ is an unknown function. Replacing a by a into the periodic solutions obtained above in (40) and (41), the following solitons solutions 1 rffiffiffi 3nþ1 aðn 1Þ a 3n þ 1 2 csch ðx ctÞ uðx; tÞ ¼ ; ð43Þ 4nc b 4n and uðx; tÞ ¼
aðn 1Þ sech2
4nc
1 rffiffiffi 3nþ1 a 3n þ 1 ðx ctÞ b 4n
ð44Þ
are readily obtained.
4. Nonlinear variant of the KP equation 4.1. The case of positive exponents 4.1.1. The case where n > 1 and þa We now consider a variant of the KP equation expressed in the form fut þ aðu3n Þx þ b½un ðu2n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0;
a; b > 0; n ð45Þ
> 1;
where k ¼ 2; 3 indicates the dimensions of the two and three dimensional spaces. Proceeding as before, Eq. (45) will be transformed to 3
f cu~n þ aðu3n Þ~n þ b½un ðu2n Þ~n~n ~n g~n þ u~n~n þ ðk 2Þ u~n~n ¼ 0;
ð46Þ
where ~ n ¼ x þ y þ ðk 2Þz ct:
ð47Þ
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A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
Integrating (46) twice, setting the constants of integration to be zero we find
ðc aÞu þ au3n þ bun ðu2n Þ~n~n ¼ 0;
ð48Þ
where a ¼ 1 þ ðk 2Þ3 ;
for k ¼ 2; 3:
ð49Þ
In other words, a ¼ 1; 2 for k ¼ 2; 3 respectively. Substituting the cosine assumptions made before in (21) into (48) yields
ðc aÞk cosðl~ nÞ þ ak3n cos3nb ðl~ nÞ 4bn2 l2 b2 k3n cos3nb ðl~nÞ nÞ ¼ 0: þ 2bnk3n l2 bð2nb 1Þ cos3nb 2 ðl~
ð50Þ
Eq. (50) is justified provided that the following system of algebraic equations holds: 2nb 1 6¼ 0; 3nb 2 ¼ b;
ð51Þ
4bn2 l2 b2 ¼ a; 2nbk3n l2 bð3nb 1Þ ¼ ðc aÞk;
where a ¼ 1; 2 for k ¼ 2; 3 as stated before. A straightforward computation yields 2 ; 3n ffi 1 rffiffi
a 3n 1 l¼ ; b 4n
1 4nðc aÞ 3n 1 k¼ : aðn þ 1Þ b¼
ð52Þ
The results (52) can be easily obtained if we also use the sine method (22). Consequently, we obtain the compactons solutions (n 1 h io3n 1 4nðc aÞ 2 pffiffia 3n 1 ~ n sin ; j~nj 6 pl ; b 4n aðnþ1Þ uðx; y; z; tÞ ¼ ð53Þ 0 otherwise; and (n uðx; y; z; tÞ ¼ 0
4nðc aÞ aðnþ1Þ
cos2
hpffiffi
a 3n 1 ~ n b 4n
1 io3n 1
;
p j~nj 6 2l ; otherwise;
where ~ n ¼ x þ y þ zðk 2Þ ct, a ¼ 1; 2 for k ¼ 2; 3 respectively.
ð54Þ
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571
4.1.2. The case where n > 1 and a In this case, the variant of the KP equation becomes fut aðu3n Þx þ b½un ðu2n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0; a; b > 0; n > 1; k ¼ 2; 3:
ð55Þ
Substituting a by a into (53) and (54), the solitary patterns solutions rffiffiffi 1 4nðc aÞ a 3n 1 ~ 3n 1 2 uðx; y; z; tÞ ¼ sinh n ; aðn þ 1Þ b 4n
ð56Þ
and uðx; y; z; tÞ ¼
4nðc aÞ cosh2
aðn þ 1Þ
rffiffiffi 1 a 3n 1 ~ 3n 1 n b 4n
ð57Þ
follow immediately, where k ¼ 2; 3. 4.2. The case of negative exponents 4.2.1. The case where n > 1 and þa In this case, the variant of the KP equation becomes fut þ aðu 3n Þx þ b½u n ðu 2n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0; b > 0; n > 1;
ð58Þ
where k ¼ 2; 3. Proceeding as before, Eq. (58) will be transformed to f cu~n þ aðu 3n Þ~n þ b½u n ðu 2n Þ~n~n ~n g~n þ u~n~n þ ðk 2Þ3 u~n~n ¼ 0:
ð59Þ
Integrating (59) twice and following the approach presented above we find
ðc aÞu þ au 3n þ bu n ðu 2n Þ~n~n ¼ 0:
ð60Þ
Substituting (21) into (60) yields
ðc aÞk cosb ðlnÞ þ ak 3n cos 3nb ðlnÞ 4bn2 l2 b2 k 3n cos 3nb ðlnÞ þ 2bnk 3n l2 bð2nb þ 1Þ cos 3nb 2 ðlnÞ ¼ 0:
ð61Þ
Applying the balance process gives the following system of algebraic equations 2nb þ 1 6¼ 0;
3nb 2 ¼ b; 4bn2 l2 b2 ¼ a; nbk 3n l2 bð2nb þ 1Þ ¼ ðc aÞk:
ð62Þ
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A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
Solving the system (62) gives 2 ; b¼
3n þ 1 rffiffiffi a 3n þ 1 l¼ ; b 2n
1 aðn 1Þ 3nþ1 k¼ : 4nðc aÞ
ð63Þ
The results (63) can be easily obtained if we also use the sine method (22). The results (63) lead to the following periodic solutions uðx; y; z; tÞ ¼
aðn 1Þ csc2 4nðc aÞ
rffiffiffi 1 a 3n þ 1 ~ 3nþ1 n ; b 2n
ð64Þ
and uðx; y; z; tÞ ¼
aðn 1Þ sec2 4nðc aÞ
rffiffiffi 1 a 3n þ 1 ~ 3nþ1 n ; b 2n
ð65Þ
where a ¼ 1; 2 for k ¼ 2; 3 respectively. 4.2.2. The case where n > 1 and a In this case, we consider the equation fut aðu 3n Þx þ b½u n ðu n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0; b > 0; n > 1:
ð66Þ
Substituting a by a into the periodic solutions obtained above we obtain the following solitons solutions uðx; y; z; tÞ ¼
aðn 1Þ csch2 4nðc aÞ
rffiffiffi 1 a 3n þ 1 ~ 3nþ1 n ; b 2n
ð67Þ
and uðx; y; z; tÞ ¼
aðn 1Þ sech2
4nðc aÞ
rffiffiffi 1 a 3n þ 1 ~ 3nþ1 n : b 2n
ð68Þ
This completes the analysis for the nonlinear variants of the KdV and the KP equations for positive and negative exponents, and for þa and for a as well.
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573
5. Discussion The basic goals of this work has been to extend the works in [7,21,22] on new variants of the KdV and the KP equations. It is formally derived in this work that the solutions may come as compactons, solitary patterns, periodic solutions, or solitons as a result of the positive and the negative exponents, and as a result of using the coefficient þa or a, a > 0 as well. The effect of exponents, positive or negative, and the change of the convection coefficient, þa or a, cause a qualitative difference in the physical structures of the resulting solutions. In Table 1 below, we summarize the physical structures of solutions for variants of KdV and KP equations that were investigated in this work. Table 2 below shows a comparison of the resulting values of the parameters k, and l for positive and negative exponents, and for þa and a. In closing, the introduced variants of the KdV and the KP equations represent appropriate candidates for studying the concepts of compact and noncompact dispersive structures. The variants introduced in this paper are more general compared to that investigated in [7,21,22]. 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.
Table 1 The physical structures of solutions for KdV and KP variants n
a
Solution structure
Positive Positive Negative Negative
þa
a þa
a
Compacton solutions Solitary patterns solutions Periodic solutions Solitons solutions
Table 2 The coefficients k and l for KdV and KP variants n
a
Positive
þa
kKdV 1
3n 1
kKP
a
1
3n 1 4nc aðnþ1Þ
1
3n 1 4nðc aÞ aðnþ1Þ
Positive Negative Negative
þa
a
4nc aðnþ1Þ
1 3nþ1
aðn 1Þ 4nc
1
3nþ1 aðn 1Þ 4nc
1 3n 1
4nðc aÞ aðnþ1Þ
1 3nþ1
aðn 1Þ 4nðc aÞ
1 aðn 1Þ 3nþ1
4nðc aÞ
lKdV ¼ lKP pffiffia 3n 1 b 4n
pffiffia 3n 1 b 4n
pffiffia 3nþ1 b 4n
pffiffia 3nþ1 b 4n
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[25] A.M. Wazwaz, A computational approach to soliton solutions of the Kadomtsev–Petviashili equation, Appl. Math. Comput. 123 (2) (2001) 205–217. [26] A.M. Wazwaz, An analytic study of compactons structures in a class of nonlinear dispersive equations, Math. Comput. Simul. 63 (1) (2003) 35–44. [27] A.M. Wazwaz, T. Taha, Compact and noncompact structures in a class of nonlinearly dispersive equations, Math. Comput. Simul. 62 (1–2) (2003) 171–189. [28] A.M. Wazwaz, Existence and construction of compacton solutions, Chaos, Solitons and Fractals 19 (3) (2004) 463–470. [29] 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. [30] 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. [31] A.M. Wazwaz, Compactons in a class of nonlinear dispersive equations, Math. Comput. Modell. 37 (3–4) (2003) 333–341. [32] A.M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, 1997. [33] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, MA, 1994. [34] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1998) 501–544.
Compactons, solitons and periodic solutions for variants of the KdV and the KP equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA
Abstract Studying solitons and solitons with compact support is of important significance in nonlinear phenomena. In this paper we study nonlinear variants of the KdV and the KP equations with positive and negative exponents. We employ the sine–cosine algorithm to back up our analysis. Exact solutions with compactons, solitons, solitary patterns, and periodic structures are obtained for these variants. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Compactons; Solitons; Periodic solutions; KP equation; KdV equation; Sine–cosine method
1. Introduction The KdV equation is a model that governs the one dimensional propagation of small-amplitude, weakly dispersive waves [1]. The nonlinear term uux in the KdV equation ut þ auux þ uxxx ¼ 0
ð1Þ
causes the steepening of wave form as described by Wadati [2]. However, the dispersion effect term uxxx in the same equation makes the wave form spread [2]. The balance between this weak nonlinearity and dispersion gives rise to
E-mail address: [email protected] (A.-M. Wazwaz). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.049
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solitons. Soliton is a localized wave that has an infinite support or a localized wave with exponential wings. Wadati [2–4] defined soliton as a nonlinear wave that has the following properties: (1) A localized wave propagates without change of its properties (shape, velocity, etc.). (2) Localized waves are stable against mutual collisions and retain their identities. This means that soliton has the property of a particle. However, the convection term in the Kðn; nÞ equation, introduced in [5] ut þ aðun Þx þ ðun Þxxx ¼ 0;
n > 1;
ð2Þ
is nonlinear, whereas the dispersion effect term ðun Þxxx in this equation is genuinely nonlinear. It is now formally proved by many researchers that the delicate interaction between nonlinear convection with genuine nonlinear dispersion generates solitary waves with exact compact support that are termed compactons. Partial differential equations that have the feature of nonlinear dispersion generate compactons that are solitons with finite wavelength, solitons with compact support or solitons free of exponential tails. Compactons are compact solutions that are usually expressed by powers of trigonometric functions sine and cosine. Unlike soliton that narrows as the amplitude increases, the compactonÕs width is independent of the amplitude. In modern physics, a suffix-on is used to indicate the particle property as described by Wadati [2], for example phonon, photon, and soliton. For this reason, the solitary wave with compact support is called compacton to indicate that it has the property of a particle. The classical solitons are analytic solutions, whereas compactons are nonanalytic [6] solutions. 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. The soliton concept has been examined by many mathematical methods. There are many approaches such as the inverse scattering method, the B€acklund transformation, the Darboux transformation, and the Painleve analysis. However, the compacton concept has been studied by using many analytical and numerical methods in [5–31]. There are many algorithms such as the pseudospectral method [5], the tri-Hamiltonian operators [10], the finite difference method [11], and Adomian decomposition method [32–34]. The compactons discovery motivated a considerable work to make compactons be practically realized in scientific applications, such as the super
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563
deformed nuclei, preformation of cluster in hydrodynamic models, the fission of liquid drops (nuclear physics), inertial fusion and others, as indicated in [4– 19]. For the compactons supporting equations, where a > 0, the solutions were defined in terms of the powers of the trigonometric functions sine and cosine profiles, whereas for the solitary patterns supporting models, where a < 0, the solutions were formally derived in terms of powers of the hyperbolic functions sinh and cosh. Recently, Wazwaz [22] examined two variants of the KdV and the KP equations, where wave dispersion is purely nonlinear, given by ut auðun Þx þ b½uðun Þxx x ¼ 0;
ð3Þ
a; b > 0; n > 1;
and fut aðun Þx þ b½uðun Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0;
ð4Þ
a; b > 0; n > 1;
where k ¼ 2; 3 so that k gives the dimensions of the two and three dimensional spaces. The study examined both types of physical structures, the compact and the noncompact behavior. For a > 0, the compactons solutions 8 qffiffiffiffiffiffiffiffiffiffiffiffi 1n < 2ðnþ1Þc 2 an sin ; jx ctj 6 pl ; ðx ctÞ 4bðnþ1Þ an uðx; tÞ ¼ ð5Þ : 0 otherwise; and uðx; tÞ ¼
8 < 2ðnþ1Þc :
an
qffiffiffiffiffiffiffiffiffiffiffiffi 1n an cos ; ðx ctÞ 4bðnþ1Þ 2
0
jx ctj 6
p ; 2l
ð6Þ
otherwise;
and uðx; r; tÞ ¼
8 < :
1 a ðtþbÞ2
2ðnþ1Þc an
2
sin
qffiffiffiffiffiffiffiffiffiffiffiffi
an 4bðnþ1Þ
xþ
r2 4ðtþbÞ
cs
1n
;
0
jvj 6 pl ; otherwise; ð7Þ
and uðx; r; tÞ ¼
8 < :
1 a ðtþbÞ2
2ðnþ1Þc an
cos2
qffiffiffiffiffiffiffiffiffiffiffiffi
an 4bðnþ1Þ
2
r
cs x þ 4ðtþbÞ
0
where a ¼ k 1, r2 ¼ y 2 þ ðk 2Þz2 , k ¼ 2; 3, and
1n
;
jvj 6
p ; 2l
otherwise; ð8Þ
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A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
v¼
r2
cs xþ 4ðt þ bÞ
ð9Þ
were formally derived in [22] for the solutions of (3) and (4) respectively. However, for a < 0, the following solitary patterns solutions rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1n 2ðn þ 1Þc an 2 sinh uðx; tÞ ¼ ; ð10Þ ðx ctÞ an 4bðn þ 1Þ and uðx; tÞ ¼
2ðn þ 1Þc cosh2 an
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1n an ; ðx ctÞ 4bðn þ 1Þ
ð11Þ
and uðx; tÞ ¼
1 a
ðt þ bÞ2
2ðn þ 1Þc sinh2 an
1n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi an r2
cs ; xþ 4bðn þ 1Þ 4ðt þ bÞ ð12Þ
and uðx; tÞ ¼
1 a
ðt þ bÞ2
2ðn þ 1Þc cosh2 an
1n rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi an r2
cs xþ 4bðn þ 1Þ 4ðt þ bÞ ð13Þ
were formally obtained for the defocusing branch of (3) and (4). The present paper is motivated by the desire to extend the works made in the literature to make further progress. More importantly, it is the objective of this work to show that compactons, solitary patterns, periodic solutions, and solitons arise from variants of the KdV and the KP equations. This variation in physical structures of the resulting solutions, as will be shown, depends mainly on the exponents, positive or negative, of the unknown function uðx; tÞ, and on the coefficient of the convection term a as well. To achieve our goal, we selected two nonlinear variants of the KdV and the KP equations, given by ut aðu3n Þx þ b½un ðu2n Þxx x ¼ 0;
ð14Þ
a; b > 0; n > 1;
and fut aðu3n Þx þ b½un ðu2n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0; > 1;
a; b > 0; n ð15Þ
where k ¼ 2; 3 to mindicate the two and three dimensional spaces respectively. In this work, the sine–cosine method will be used to back up our analysis. The two physical structures where a > 0 and a < 0 will be examined for po-
A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
565
sitive n and for negative n as well. It is reasonable to highlight the main steps of this algorithm.
2. The sine–cosine method 1. We introduce the wave variable n ¼ ðx ctÞ into the nonlinear PDE, say in two variables: P ðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0;
ð16Þ
where uðx; tÞ is the traveling-type wave solution. This enables us to use the following changes o d ¼ c ; ot dn
o2 d2 ¼ c2 2 ; 2 ot dn
o d ¼ ; ox dn
o2 d2 ¼ 2: 2 ox dn
ð17Þ
One can immediately reduce the nonlinear PDE (16) into a nonlinear ODE Qðu; un ; unn ; unnn ; . . .Þ ¼ 0;
ð18Þ
where un denotes du . dn 2. The ordinary differential equation (18) is then integrated as long as all terms contain derivatives, where the associated integration constants can be set zeros. 3. The sine–cosine algorithm admits the use of the solutions of the resulting equation in the form uðx; tÞ ¼ fk cosb ðlnÞg;
jnj 6
p ; 2l
ð19Þ
jnj 6
p ; l
ð20Þ
or in the form uðx; tÞ ¼ fk sinb ðlnÞg;
where k, l, and b are parameters that will be determined. 4. In view of (19) we use uðnÞ ¼ k cosb ðlnÞ; un ðnÞ ¼ kn cosnb ðlnÞ; ðun Þn ¼ nlbkn sinðlnÞ cosnb 1 ðnÞ; ðun Þnn ¼ n2 l2 b2 kn cosnb þnl2 kn bðnb 1Þ cosnb 2 ðlnÞ; and for (20) we use
ð21Þ
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uðnÞ ¼ k sinb ðlnÞ; un ðnÞ ¼ kn sinnb ðlnÞ; ðun Þn ¼ nlbkn cosðlnÞ sinnb 1 ðnÞ;
ð22Þ
ðun Þnn ¼ n2 l2 b2 kn sinnb þnl2 kn bðnb 1Þ sinnb 2 ðlnÞ: 5. Substituting (21) or (22) into the reduced ODE obtained above after integrating (18) gives a trigonometric equation of cosine or sine terms. 6. The main task is to balance the exponents of the trigonometric functions cosine or sine. Collect all terms with same power in cosk ðlnÞ or sink ðlnÞ and set to zero their coefficients to get a system of algebraic equations among the unknowns b, k and l. The problem is now completely reduced to an algebraic one. Having determined k, l, and b by using computerized symbolic calculations, the solutions proposed in (19) and in (20) follow immediately. The algorithm described above certainly works well for a large class of very interesting nonlinear equations. The main advantage of the method is that it is capable of greatly reducing the size of computational work compared to existing techniques such as the pseudospectral method, the inverse scattering method, HirotaÕs bilinear method, and the truncated Painleve expansion.
3. Nonlinear variant of the KdV equation 3.1. The positive exponents 3.1.1. The case where n > 1 and þa We first rewrite a variant of the KdV equation given in (14) in the form ut þ aðu3n Þx þ b½un ðu2n Þxx x ¼ 0;
a; b > 0; n > 1;
ð23Þ
where uðx; tÞ is an unknown function. With the analysis presented above, Eq. (23) will be transformed to
cun þ aðu3n Þn þ b½un ðu2n Þnn n ¼ 0:
ð24Þ
Integrating (24), using the constant of integration to be zero we obtain
cu þ au3n þ bun ðu2n Þnn ¼ 0:
ð25Þ
Substituting (21) into (25) gives
ck cosb ðlnÞ þ ak3n cos3nb ðlnÞ 4bn2 l2 b2 k3n cos3nb ðlnÞ þ 2bnk3n l2 bð2nb 1Þ cos3nb 2 ðlnÞ ¼ 0:
ð26Þ
A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
567
Eq. (26) is satisfied only if the following system of algebraic equations holds: 2nb 1 6¼ 0; 3nb 2 ¼ b;
ð27Þ
4bn2 l2 b2 ¼ a; 2bnk3n l2 bð2nb 1Þ ¼ ck:
The system (27) is straightforwardly calculated by hand or easily computed with any symbolic manipulation package to give 2 ; n > 1; b¼ 3n ffi 1 rffiffi
a 3n 1 l¼ ; ð28Þ b 4n 1
3n 1 4nc k¼ : aðn þ 1Þ Using the sine method (22) leads to the same derivations (28). Consequently, the following compactons solutions (n 1 hpffiffi io3n 1 4nc a 3n 1 sin2 ðx ctÞ ; jx ctj 6 pl ; aðnþ1Þ b 4n uðx; tÞ ¼ ð29Þ 0 otherwise; and
(n uðx; tÞ ¼
4nc aðnþ1Þ
cos2
hpffiffi a 3n 1 b 4n
ðx ctÞ
1 io3n 1
0
;
p jx ctj 6 2l ; otherwise
ð30Þ
are readily obtained. 3.1.2. The case where n > 1 and a In this case, the variant of the KdV becomes ut aðu3n Þx þ b½un ðu2n Þxx x ¼ 0;
a; b > 0;
ð31Þ
well known as the defocusing branch. Replacing a by a into (29) and (30) we obtain the solitary patterns solutions 1 rffiffiffi 3n 1 4nc a 3n 1 2 sinh ðx ctÞ uðx; tÞ ¼ ; ð32Þ aðn þ 1Þ b 4n and uðx; tÞ ¼
4nc cosh2 aðn þ 1Þ
1 rffiffiffi 3n 1 a 3n 1 ðx ctÞ ; b 4n
which are in the form obtained in [22].
ð33Þ
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3.2. The case of negative exponents 3.2.1. The case where n > 1 and þa For negative exponents, we consider a variant of the KdV equation in the form ut þ aðu 3n Þx þ b½u n ðu 2n Þxx x ¼ 0;
ð34Þ
where a; b > 0 are constants, and uðx; tÞ is an unknown function. Following the analysis presented above, Eq. (34) will be transformed to
cun þ aðu 3n Þn þ b½u n ðu 2n Þnn n ¼ 0:
ð35Þ
Integrating (35), using the constant of integration to be zero, and dividing by u we obtain
cu þ au 3n þ bu n ðu 2n Þnn ¼ 0:
ð36Þ
Substituting (21) into (36) yields
ck cosb ðlnÞ þ ak 3n cos 3nb ðlnÞ 4bn2 l2 b2 k 3n cos 3nb ðlnÞ þ 2bnk 3n l2 bð2nb þ 1Þ cos 3nb 2 ðlnÞ ¼ 0:
ð37Þ
Eq. (37) is satisfied only if the following system of algebraic equations holds: 2nb þ 1 6¼ 0;
3nb 2 ¼ b; 4bn2 l2 b2 ¼ a;
ð38Þ
2nbk 3n l2 bð2nb þ 1Þ ¼ ck: This in turn gives 2 ; b¼
3n þ 1 rffiffiffi a 3n þ 1 l¼ ; b 4n
1 aðn 1Þ 3nþ1 k¼ : 4nc
ð39Þ
The results (39) can be easily obtained if we also use the sine method (22). In view of (39), we obtain the periodic solutions 1 rffiffiffi 3nþ1 aðn 1Þ 2 a 3n þ 1 csc ðx ctÞ uðx; tÞ ¼ ; ð40Þ 4nc b 4n and
A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
uðx; tÞ ¼
aðn 1Þ sec2 4nc
1 rffiffiffi 3nþ1 a 3n þ 1 ðx ctÞ : b 4n
569
ð41Þ
We can easily observe that the effect of the negative exponents is the existence of periodic solutions. Recall that compactons and solitary patterns solutions were obtained before for the positive exponents and þa, a > 0. 3.2.2. The case where n > 1 and a In this case, replacing a by a into (34) gives ut aðu 3n Þx þ b½u n ðu 2n Þxx x ¼ 0;
ð42Þ
a; b > 0; n > 1;
where a; b > 0 are constants, and uðx; tÞ is an unknown function. Replacing a by a into the periodic solutions obtained above in (40) and (41), the following solitons solutions 1 rffiffiffi 3nþ1 aðn 1Þ a 3n þ 1 2 csch ðx ctÞ uðx; tÞ ¼ ; ð43Þ 4nc b 4n and uðx; tÞ ¼
aðn 1Þ sech2
4nc
1 rffiffiffi 3nþ1 a 3n þ 1 ðx ctÞ b 4n
ð44Þ
are readily obtained.
4. Nonlinear variant of the KP equation 4.1. The case of positive exponents 4.1.1. The case where n > 1 and þa We now consider a variant of the KP equation expressed in the form fut þ aðu3n Þx þ b½un ðu2n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0;
a; b > 0; n ð45Þ
> 1;
where k ¼ 2; 3 indicates the dimensions of the two and three dimensional spaces. Proceeding as before, Eq. (45) will be transformed to 3
f cu~n þ aðu3n Þ~n þ b½un ðu2n Þ~n~n ~n g~n þ u~n~n þ ðk 2Þ u~n~n ¼ 0;
ð46Þ
where ~ n ¼ x þ y þ ðk 2Þz ct:
ð47Þ
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A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
Integrating (46) twice, setting the constants of integration to be zero we find
ðc aÞu þ au3n þ bun ðu2n Þ~n~n ¼ 0;
ð48Þ
where a ¼ 1 þ ðk 2Þ3 ;
for k ¼ 2; 3:
ð49Þ
In other words, a ¼ 1; 2 for k ¼ 2; 3 respectively. Substituting the cosine assumptions made before in (21) into (48) yields
ðc aÞk cosðl~ nÞ þ ak3n cos3nb ðl~ nÞ 4bn2 l2 b2 k3n cos3nb ðl~nÞ nÞ ¼ 0: þ 2bnk3n l2 bð2nb 1Þ cos3nb 2 ðl~
ð50Þ
Eq. (50) is justified provided that the following system of algebraic equations holds: 2nb 1 6¼ 0; 3nb 2 ¼ b;
ð51Þ
4bn2 l2 b2 ¼ a; 2nbk3n l2 bð3nb 1Þ ¼ ðc aÞk;
where a ¼ 1; 2 for k ¼ 2; 3 as stated before. A straightforward computation yields 2 ; 3n ffi 1 rffiffi
a 3n 1 l¼ ; b 4n
1 4nðc aÞ 3n 1 k¼ : aðn þ 1Þ b¼
ð52Þ
The results (52) can be easily obtained if we also use the sine method (22). Consequently, we obtain the compactons solutions (n 1 h io3n 1 4nðc aÞ 2 pffiffia 3n 1 ~ n sin ; j~nj 6 pl ; b 4n aðnþ1Þ uðx; y; z; tÞ ¼ ð53Þ 0 otherwise; and (n uðx; y; z; tÞ ¼ 0
4nðc aÞ aðnþ1Þ
cos2
hpffiffi
a 3n 1 ~ n b 4n
1 io3n 1
;
p j~nj 6 2l ; otherwise;
where ~ n ¼ x þ y þ zðk 2Þ ct, a ¼ 1; 2 for k ¼ 2; 3 respectively.
ð54Þ
A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
571
4.1.2. The case where n > 1 and a In this case, the variant of the KP equation becomes fut aðu3n Þx þ b½un ðu2n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0; a; b > 0; n > 1; k ¼ 2; 3:
ð55Þ
Substituting a by a into (53) and (54), the solitary patterns solutions rffiffiffi 1 4nðc aÞ a 3n 1 ~ 3n 1 2 uðx; y; z; tÞ ¼ sinh n ; aðn þ 1Þ b 4n
ð56Þ
and uðx; y; z; tÞ ¼
4nðc aÞ cosh2
aðn þ 1Þ
rffiffiffi 1 a 3n 1 ~ 3n 1 n b 4n
ð57Þ
follow immediately, where k ¼ 2; 3. 4.2. The case of negative exponents 4.2.1. The case where n > 1 and þa In this case, the variant of the KP equation becomes fut þ aðu 3n Þx þ b½u n ðu 2n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0; b > 0; n > 1;
ð58Þ
where k ¼ 2; 3. Proceeding as before, Eq. (58) will be transformed to f cu~n þ aðu 3n Þ~n þ b½u n ðu 2n Þ~n~n ~n g~n þ u~n~n þ ðk 2Þ3 u~n~n ¼ 0:
ð59Þ
Integrating (59) twice and following the approach presented above we find
ðc aÞu þ au 3n þ bu n ðu 2n Þ~n~n ¼ 0:
ð60Þ
Substituting (21) into (60) yields
ðc aÞk cosb ðlnÞ þ ak 3n cos 3nb ðlnÞ 4bn2 l2 b2 k 3n cos 3nb ðlnÞ þ 2bnk 3n l2 bð2nb þ 1Þ cos 3nb 2 ðlnÞ ¼ 0:
ð61Þ
Applying the balance process gives the following system of algebraic equations 2nb þ 1 6¼ 0;
3nb 2 ¼ b; 4bn2 l2 b2 ¼ a; nbk 3n l2 bð2nb þ 1Þ ¼ ðc aÞk:
ð62Þ
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A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
Solving the system (62) gives 2 ; b¼
3n þ 1 rffiffiffi a 3n þ 1 l¼ ; b 2n
1 aðn 1Þ 3nþ1 k¼ : 4nðc aÞ
ð63Þ
The results (63) can be easily obtained if we also use the sine method (22). The results (63) lead to the following periodic solutions uðx; y; z; tÞ ¼
aðn 1Þ csc2 4nðc aÞ
rffiffiffi 1 a 3n þ 1 ~ 3nþ1 n ; b 2n
ð64Þ
and uðx; y; z; tÞ ¼
aðn 1Þ sec2 4nðc aÞ
rffiffiffi 1 a 3n þ 1 ~ 3nþ1 n ; b 2n
ð65Þ
where a ¼ 1; 2 for k ¼ 2; 3 respectively. 4.2.2. The case where n > 1 and a In this case, we consider the equation fut aðu 3n Þx þ b½u n ðu n Þxx x gx þ uyy þ ðk 2Þuzz ¼ 0; b > 0; n > 1:
ð66Þ
Substituting a by a into the periodic solutions obtained above we obtain the following solitons solutions uðx; y; z; tÞ ¼
aðn 1Þ csch2 4nðc aÞ
rffiffiffi 1 a 3n þ 1 ~ 3nþ1 n ; b 2n
ð67Þ
and uðx; y; z; tÞ ¼
aðn 1Þ sech2
4nðc aÞ
rffiffiffi 1 a 3n þ 1 ~ 3nþ1 n : b 2n
ð68Þ
This completes the analysis for the nonlinear variants of the KdV and the KP equations for positive and negative exponents, and for þa and for a as well.
A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
573
5. Discussion The basic goals of this work has been to extend the works in [7,21,22] on new variants of the KdV and the KP equations. It is formally derived in this work that the solutions may come as compactons, solitary patterns, periodic solutions, or solitons as a result of the positive and the negative exponents, and as a result of using the coefficient þa or a, a > 0 as well. The effect of exponents, positive or negative, and the change of the convection coefficient, þa or a, cause a qualitative difference in the physical structures of the resulting solutions. In Table 1 below, we summarize the physical structures of solutions for variants of KdV and KP equations that were investigated in this work. Table 2 below shows a comparison of the resulting values of the parameters k, and l for positive and negative exponents, and for þa and a. In closing, the introduced variants of the KdV and the KP equations represent appropriate candidates for studying the concepts of compact and noncompact dispersive structures. The variants introduced in this paper are more general compared to that investigated in [7,21,22]. 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.
Table 1 The physical structures of solutions for KdV and KP variants n
a
Solution structure
Positive Positive Negative Negative
þa
a þa
a
Compacton solutions Solitary patterns solutions Periodic solutions Solitons solutions
Table 2 The coefficients k and l for KdV and KP variants n
a
Positive
þa
kKdV 1
3n 1
kKP
a
1
3n 1 4nc aðnþ1Þ
1
3n 1 4nðc aÞ aðnþ1Þ
Positive Negative Negative
þa
a
4nc aðnþ1Þ
1 3nþ1
aðn 1Þ 4nc
1
3nþ1 aðn 1Þ 4nc
1 3n 1
4nðc aÞ aðnþ1Þ
1 3nþ1
aðn 1Þ 4nðc aÞ
1 aðn 1Þ 3nþ1
4nðc aÞ
lKdV ¼ lKP pffiffia 3n 1 b 4n
pffiffia 3n 1 b 4n
pffiffia 3nþ1 b 4n
pffiffia 3nþ1 b 4n
574
A.-M. Wazwaz / Appl. Math. Comput. 161 (2005) 561–575
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