Variants of the generalized KdV equation with compact and noncompact structures

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Computers and Mathematics with Applications 47 (2004) 583-591 www.elsevier.com/locate/camwa

Variants of the Generalized KdV Equation Compact and Noncompact Structures

with

A.-M. WAZWAZ Department of Mathematics and Computer Science Saint Xavier University, Chicago, IL 60655, U:S.A.

(Received February 2003; accepted March 2003)

A b s t r a c t - - In this paper, nonlinear dispersive variants of the generalized KdV equation are investigated. The compact and the noncompact physical structures are studied. Two general formulas for the focusing branch and the defocusing branch, that are of substantial interest, are formally derived. (~) 2004 Elsevier Ltd. All rights reserved.

Keywords-- Compactons, Solitons, Generalized KdV equation.

1. I N T R O D U C T I O N It is well known that, the standard KdV equation u t -~ OlUUx -I- Uxxx ~-- 0

(1)

is a model for m a n y wave related p h e n o m e n a and admits localized solutions called solitons. The nonlinear term uux and the linear dispersion term uxxx in equation (1) cause the steepening of wave form a n d the spread of the wave, respectively. Solitons appear as a result of the balance between the weak nonlinearity and dispersion and characterized as localized waves with exponential tails. Soliton has been defined by Wadati [1-3] and others as (1) a localized wave propagates without change of its properties (shape, velocity, etc.); (2) localized waves are stable against m u t u a l collisions and retain their identities. However, the genuinely nonlinear dispersive K ( n , n) equation

u, + a(un)x + (un)xx~= 0,

n > 1,

(2)

introduced by Rosenau and H y m a n [4], includes the nonlinear convection term (un)x and the genuinely nonlinear dispersion term (un)xxx. It is now formally derived by m a n y in [4-22] t h a t

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584

A.-M. WAZWAZ

the delicate interaction between nonlinear convection with genuine nonlinear dispersion generates solitary waves with exact compact support that are called compactons. It was revealed in [4-22] that equation (2) generates compactly supported solutions with nonsmooth fronts. Unlike solitons that are characterized as localized waves with exponential tails, compactons are solitons with compact support or solitons free of exponential wings. Moreover, soliton narrows as the amplitude increases whereas the compacton's width is independent of the amplitude. Two important features of compactons structures are observed. (1) The compacton is a soliton characterized by the absence of exponential wings. (2) The width of the compacton is independent of the amplitude. In modern physics, a suffix, -on, is used to indicate the particle property [1], 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 analysis of compactons: robust soliton-like solutions, has received a great deal of attention from both the mathematical theory and application viewpoints. Duseul et al. [9] showed that compactons can exist for specific velocities in physical systems modeled by a nonlinear KleinGordon equation with anharmonic coupling. A generalized ~-four model with nonlinear coupling was investigated in [9] to show that this model may exhibit compacton solutions. The existence and stability of these compact entities were addressed as well in [9]. A nonlinear hydrodynamic model describing new features of motion of the free surface of a liquid was introduced by Ludu et al. [10]. Compacton solutions have been found and new symmetries were realized. Dinda et al. [11] demonstrated the existence of a localized breathing mode with a compact support in a nonlinear Klein-Gordon system. It was revealed by many studies that the compactons are nonanalytic [4-12] solutions, whereas classical solitons are analytic solutions. The K(n, n) equation (2) cannot be derived from a first-order Lagrangian except for n = 1, and it does not possess the usual conservation laws of energy that KdV equation (1) possessed. The stability analysis has shown that compacton solutions are stable, where the stability condition is satisfied for arbitrary values of the nonlinearity parameter. The stability of the compactons solutions was investigated by means of both linear stability and by Lyapunov stability criteria as well. Compactons were proved to collide elastically and vanish outside a finite core region. The compacton phenomenon has been investigated by many analytical and numerical methods. There are many algorithms such as the pseudospectral method [4-7], tri-Hamiltonian operators [8], the finite-difference method [12], and Adomian decomposition method [23-25]. 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 [4-22]. Three variants of the standard KdV equation were recently studied. Rosenau [5] studied a variant of the KdV equation

(3) that emerges in nonlinear lattices and was used to describe the dispersion of dilute suspensions for n = 1. Moreover, Rosenau examined the focusing branch of (3), where (a > 0), and developed general formulas in terms of the cosine function only. Recently, Wazwaz [19] studied model (3) for both types, the focusing branch where a > 0 and the defocusing branch where a is replaced by - a . Two general distinct sets of formulas for each of the aforementioned types of problems were developed in [19]. For the compactons supporting equations, where a > 0, the solutions were defined in terms of the sine and cosine profiles, whereas for the solitary patterns supporting models, where a is replaced by - a , the solutions were formally derived in terms of the hyperbolic functions sinh and cosh.

GeneralizedKdVEquation

585

For a > 0, the following compactons solutions:

{{ u(x,t) =

2c sin= a

[_~_~

(x - ct)

]}l/n

7r

Ix - ctl < - , /* otherwise,

0,

{I

2c sin=

u(z,

t) =

(x - ct)

a

0,

11

(4)

Ix - ctl < '

-

(5)

2/,'

otherwise,

where p = v/-~/2 were derived for the focusing branch of equation (3). In [19], the defocusing branches of (3), where a is replaced by -a, the following solitary patterns solutions:

u(x,t) = { 2-~sinh2 [-~(x-ct)] } 1/~ , u(x,t)

= -

{

2c cosh 2

(6)

[~(x - ct) ]}~

(7)

were formally obtained. In addition, Wazwaz [20] studied a second variant of the KdV equation given by

us+au(u~)~+[u(u*~)zx]z=O,

a>0,

(8)

n>l.

For a > 0, the following compactons solutions: 2(n + 1)c sin2

an

[°~

(x - ct)

]}~,

7r

Ix-ctl S -,

(9)

otherwise,

O,

o/x,~/={{ - ~- cos~[~,~ ~,1}~

7F

an

2(n + 1)c

O,

I x - ctl < G '

(10)

otherwise,

were established, where a = v/an/4(n + 1). For the defocusing branch of (8), where a is replaced by - a , the following solitary patterns solutions:

/

(11)

an

u(x't) = - { 2(n + l)Cc°sh2 [i~(x-ct)]

}

(12)

were formally constructed. A third variant of the KdV equation, given by

ut÷au~(u)x+[un(u)~]x = 0 ,

a>0,

n_>3,

~ - a~(~)x

a > 0,

~ > 3,

+ b ~ (~)~x]~ = 0,

(13)

was studied in [20]. The compactons solutions were obtained as follows: 71"

u(x, t)

(14) O,

otherwise,

586

A.-M. WAZWAZ

and 71"

=

t aT

cos

,

0,

Ix - ctl <

2M'

(15)

otherwise,

where M = ( n / 2 ) x / a / ( n + 1). However, the solitary patterns solutions of the defocusing branch of (13) are given by

{2c(n + 1) [n~n___ ~ (x-ct) a ( n - 2 ) sinh2 -2

u(x,t)= -

]}l/n

(16)

and 2c(n + 1)

n

a

(17)

B

The present work is motivated by the desire to extend the works conducted so far to develop a fairly complete theoretical understanding of the compactons and the solitary patterns solutions. More precisely, our main concern will be examining three variants of the generalized KdV (gKdV) equation, where wave dispersion is purely nonlinear, given by the following. VARIANT I.

~+a(~

n+l

)x+b[~(~

n

)~&+k~(~")=~=O,

ut - a (u '~+1) x + b [u (un)xx]x + ku (un)xzz = 0, VARIANT

n>l,

a,b,k>0,

n>l.

a,b,k>0, a,b,k>0,

n>2, n>2.

(is)

II.

~t + a (~") ~x + b [(~") (~)x~]. + k~ ( ~ " ) ~ = 0, ~t - a (~") ~x + b [(~") (~)~]~ + k~(~")x~ = 0, VARIANT

a,b,k>0,

(19)

III.

~ +a~(~n)~ +b[~(

~n

)xx]x + k~(~n)xxx = 0,

a,b,k>0,

n>l,

~, - a~ (~")~ + b [~ (~")xx]~ + k u ( ~ " ) x ~ = O,

a,b,k>0,

n>l.

(20)

As we remarked before, these three distinct variants are models (3), (8), and (13) discussed before in [5,19,20] with a new term k u ( u ~ ) ~ added to each model. Our work will run parallel to the studies carried out in [5,19,20] where the focusing and the defocusing branches of these variants will be approached independently. 2. V A R I A N T

I

2.1. T h e Focusing B r a n c h

We first consider the one-dimensional equation

u t + a ( u n+l )x + b [ u ( u n )xx]x + ku(un)xxx = 0,

a, b,k > 0,

n > 1.

(21)

The significant derivations made in [5,19,20] defined the compactons solutions in terms of the sine and the cosine profiles raised to an exponent 2/n. This allows us to set the compactons solutions in the form u(x, t) = {a sin 2 [13(x - ct)] } 1/n (22) or in the form

~(~, t)

= { ~ cos ~ [Z (~ - ~t)] } v n ,

(2z)

Generalized KdV Equation

587

where a and fl are constants that will be determined. Substituting (22) or (23) into (21) and by solving the resulting equations, we find

1 ~/ a(n + I) fl=4-~,b+(b+k)n,

a=

2c[b +

(b + k)n] ab(n+l) '

n>l.

(24)

For a to be positive, then c > 0. It is interesting to note that, for the special case where b = k, then we have 1 / a(n + 1) 2c(2n+ 1) fl=±-2V~' o~= a(n-t-1) ' n>l. (25) Substituting (24) into (22) and (23) gives the following set of general compactons solutions:

u(x, t) =

ab(n + 1)

±

sin2

b + (b + k)n

'

-

0,

(26)

otherwise,

and

u(x,t)=

{

[f,

2c[b+(b+k)n] a ~ n :~ i?

[ 2 ~ a(n+l) ]}i/n 7r b+(b+k)n (x - ct) ' Ix - al -< V '

cos 2 4-

0,

(27)

otherwise.

2.2. T h e D e f o c u s i n g B r a n c h We next consider the defocusing branch ttt -- a (Un't-1)x Jr- b [~t (~n)xx] x -I- ]~u (un)xxx -~- O,

a, b, k > O,

n > 1,

(28)

where a in the model discussed above is replaced by - a , a > 0. As derived in [5,19,20], the defocusing branch usually gives solitary patterns solutions, with infinite slopes or cusps, that are expressed in terms of hyperbolic functions raised to an exponent 2/n. This allows us to set the general solutions of (28) in the form

u(x,t)

= {&sinh 2 [ f l ( x -

ct)] }l/n

(29)

or in the form

u(x,t):{&eosh2[~(x-ct)]} 1/n,

(30)

where 0 and fl are constants that will be determined. Substituting (29) and (30) into (28) and solving the resulting equations, we immediately obtain

1 i a(n + 1) = -t-3 "b-I- (b + k)n'

(~ =

2c[b + (b + k)n] ab(n + 1) '

(31)

and

1 ~/ a(n + 1) = +2vb

+ (b + k)n'

-2c[b + (b + k)n] a =

ab(~ + 1)

(32)

'

respectively. Substituting (31) and (32) into (29) and (30) gives the following set of general solutions:

ab(n + 1)

sinh2

±

b -t- (b + k)n

ab(n + 1)

c°sh2

+ Vb + (b + k)n kx -

and ct)

.

(34)

588

A.-M. WAZWAZ

3. V A R I A N T

II

3.1. The Focusing Branch We next consider the focusing branch of Variant II given by the nonlinear dispersive equation

ut+a(u~)u~+b[(u~)(u)x~]~+ku(

Un

a,b,k>O,

)~=0,

n>2.

(35)

Following the analysis presented above, it is normal to set the compactons solutions in the form

u(x,t) = {Asin 2 [B (x - ct)]} 1/~

(36)

u(x, t) = {Acos 2 [B (x - ct)]} 1/~ ,

(37)

or in the form where A and B are constants. equations for A and B, we find

B=

Substituting (36) or (37) into (35) and solving the resulting

+n i a -~ b + b n + k n 3'

A=

- 2 c ( b + b n + k n 3) ab(n-2) '

n>2.

(38)

n>2.

(39)

For b = k, we find n ~/ a B=4- 3 b(l+n+n3)

- 2 c (1 + n + n a) '

A=

a(n-2)

'

The compactons solutions

u(x, t) =

{{

-2c (b + bn + kn 3) ab-Cn: 2) sin 2 4-

a (x - ct) b ÷ bn--1- kn 3

0,

1}

,

7r

Ix- al < ~, (40) otherwise,

and 7r

Ix - ctl < 2-N'

0,

otherwise,

(41) are readily obtained upon substituting (38) into (36) and

(37),

respectively.

3.2. The Defocusing Branch We next consider the defocusing branch of Variant II given by ut - a (u n) us + b [(un) (u)~x] x + ku (u~)x,,~ = 0,

a, b, k > 0,

n > 2.

(42)

Following the analysis implemented above for handling the defocusing branch of Variant I, we set the general solutions of (42) in the form

}

(43)

or in the form (44)

Generalized KdV Equation

589

where i and /) are constants. Substituting (43) and (44) into (42) and by a straightforward computation, we obtain n ~/ a = ±-2 b + bn + kn 3'

~ = - 2 c (b + bn + kn 3) ab(n - 2)

/ ~ = + n -2 i b + bna+ kn 3'

fi = 2c (b + bn + kn 3) ab(~ - 2)

and

(45)

(46)

It then follows that the solitary patterns solutions

f -2c~+_b~ + k ~ ) ~ u(x,t) = [ a b ( n - 2) sinh 2 -t-~

[,/

and

U(X,t) : {2c(bq-b~q-]~n 3) ab(n - 2)

c°sh2

a (x - ~t) ))+br;-+ kn3

i}

(47)

[_1-2n / b + bna+ kn a (x-¢t)] }l/n

4. V A R I A N T

(4s)

III

4.1. T h e Focusing Branch We now consider the one-dimensional equation

ut+au(un)~+b[u(u~)x~]x+ku(un)xzx=O

,

a,b,k>O,

n>l.

(49)

The compactons solutions of the nonlinear dispersive equation (49) can be set in the form

u(x, t) = {A sin 2 [# (x - ct)]} 1/n

(50)

u(x, t) = {A cos ~ [~ (x - ct)]} 1/~ ,

(51)

or in the form where A and # are constants. Substituting (50) or (51) into (49) and solving directly for A and #, we find +1_I an 2c[b + (b + k)n] (52) #= 2 b+(b+k)n' A= abn For b = k, we obtain

1 ~/ an = ±~ b(1 + 2n)'

~-

2c(1 + 2n) a~

(53)

For A to be positive, then e > 0. The eompactons solutions

7r

(b + k)~] sin2 ±1 b+(b+k)n an (x - ~t) u(x,t)= { ~[ 2c[b + a;£

Ix - ctl _< -,

(54)

otherwise,

O, and 2c[b + (b + k)n] 1 abn c°s2 ± 2

u(x,t)= O,

are readily obtained.

an b+(b+k)n

- ct) (x

71"

'

Ix -

ctl

<

--

- 2#' otherwise,

(55)

590

A.-M. WAZWAZ

4.2. The Defocusing Branch We finally consider the defocusing branch

ut - au (un)x 4- b [u (un)x~]x + ku (u~)~ = O,

a,b,k>O,

n>l.

(56)

It is normal to set the general solutions of (56) in the form

u(x, t)

= {~ sinh2 [/i(x -

ct)]}

1/n

(57)

or in the form

~(X, t) = { ~ cosh2 [/~(x - ct)] } 1/n

(58)

,

where ~ and /1 are constants. Substituting (57) and (58) into (56) and solving the resulting equations, we obtain /1=41 i

an b + (b 4- k)n'

abn

'

(59)

and

/2=4_1/

= -2c[b + (b +

an

(60)

abn

respectively. As a result of (59), the general solutions an ])l/n u(x,t)= { 2c[b+(b+k)n] sinh2 [4-1 i abn b + (b 4- k)n(X -ct)] ~

and

U(X, t)

L

-2c[b + (b + k)n]

eosh2

[ 1 /

an

] 1/n

+ (b + k)n(X-et) }

(61)

(62)

are readily obtained. This completes the derivation of the general formulas that work for all three distinct variants of the generalized KdV equation: the focusing and the defocusing branches as well.

5. D I S C U S S I O N The phenomenon of compact and noncompact structures shows a rich variety of concepts and properties. The physical structures of the compactons solutions and the solitary patterns solutions give insight into many scientific processes other than fluid, such as the super deformed nuclei, preformation of cluster in hydrodynamic models, the fission of liquid drops (nuclear physics), and the inertiM fusion. The study of compactons is the first systematic research on the interaction between nonlinear convection and the genuinely nonlinear dispersion. It was shown that, while the compactons are the essence of the focusing branch a > 0, spikes, peaks, and cusps are the hallmark of the defocusing branch where a < 0 which also supports the motion of kinks. The basic goal of this work has been to extend the works in [5,19,20]. The focusing branch and the defocusing branch of each variant of the three variants of the generalized KdV equation were investigated. The study extended the previous works by adding an extra term for each of the models examined before. It is clearly observed that, the additional nonlinear term has changed the coefficients of the solutions, but the physical structures: compactons and solitary patterns solutions have been generated. 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 significance of the presented approach is that it can be used to search for exact solutions of models where nonlinear dispersion and nonlinear convection interact.

Generalized KdV Equation

591

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