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Transformations and multi-solitonic solutions for a generalized variable-coefficient Kadomtsev–Petviashvili equation
Computers and Mathematics with Applications 61 (2011) 3268–3277
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Transformations and multi-solitonic solutions for a generalized variable-coefficient Kadomtsev–Petviashvili equation Yueqian Liang ∗ , Guangmei Wei, Xiaonan Li LMIB and School of Mathematics and System Sciences, Beihang University, Beijing 100191, China
article
abstract
info
Article history: Received 17 June 2010 Received in revised form 7 April 2011 Accepted 7 April 2011
Kadomtsev–Petviashvili equations with variable coefficients can be used to characterize many nonlinear phenomena in fluid dynamics and plasma physics more realistically than the equations with constant coefficients. Hereby, a generalized variable-coefficient Kadomtsev–Petviashvili equation with nonlinearity, dispersion and perturbed terms is investigated. Transformations, of which the consistency conditions are exactly the Painlevé integrability conditions, to the Korteweg–de Vries equation, cylindrical Korteweg–de Vries equation, Kadomtsev–Petviashvili equation and cylindrical Kadomtsev–Petviashvili equation are presented by formal dependent variable transformation assumptions. Using the Hirota bilinear method, from the variable-coefficient bilinear equation, the multisolitonic solution, auto-Bäcklund transformation and Lax pair for the variable-coefficient Kadomtsev–Petviashvili equation are obtained. Moreover, the influence of inhomogeneity coefficients on solitonic structures and interaction properties is discussed for physical interest and possible applications. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Generalized variable-coefficient Kadomtsev–Petviashvili equation Dependent variable transformation Hirota bilinear method Multi-solitonic solution
1. Introduction The celebrated, historic Korteweg–de Vries (KdV) equation UT + 6UUX + UXXX = 0,
(1)
has been found to model many physical, mechanical and engineering phenomena, such as ion-acoustic waves, stratified internal waves, geophysical fluid dynamics, collision-free hydromagnetic waves, lattice dynamics, the jams in the congested traffic and so on (see Refs. [1–6] and references therein). As a two-spatial dimensional analog of the KdV equation (1), the Kadomtsev–Petviashvili (KP) equation
(UT + 6UUX + UXXX )X + σ UYY = 0,
(2)
where σ = ±1 has been discovered to describe the evolution of long water waves and small-amplitude surface waves with weak nonlinearity, weak dispersion, and weak perturbation in the y direction, weakly relativistic soliton interactions in the magnetized plasma, dust acoustic waves for hot dust plasmas and some other nonlinear models [1,7–12]. As a special case of the KdV equation with power-law nonlinearity and time-dependent coefficients [13], the cylindrical KdV equation UT + 6UUX + UXXX +
∗
1 2T
U = 0,
Corresponding author. E-mail addresses: [email protected] (Y. Liang), [email protected] (G. Wei).
0898-1221/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.04.007
(3)
Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
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which was first proposed by Maxon and Viecelli in 1974 when they studied propagation of radically ingoing acoustic waves in cylindrical geometry [14], has been given out in some different situations by using the stretched coordinates [14–18]. And its counterpart in (2 + 1)-dimensional, the cylindrical KP equation 1
σ
UYY = 0, (4) 2T T2 where σ = ±1, has also been constructed to describe the nearly straight wave propagation which varies in a very small angular region, the cylindrical shallow-water waves subject to certain kinds of the small transversal disturbances, electronacoustic solitary waves in dense quantum electron–ion plasmas, the dust ion-acoustic waves with azimuthal perturbation and so on [18–23]. Recently, when the inhomogeneous media and/or the nonuniform boundaries are considered, nonlinear evolution equations (NLEEs) with variable coefficients, which describe nonlinear phenomena of the complex world more realistically than their constant-coefficient forms, have been paid much attention [24–27]. Hereby, a new generalized variablecoefficient KP equation
(UT + 6UUX + UXXX )X +
UX +
(ut + χ1 (t )uux + χ2 (t )uxxx )x + χ3 (t )ux + χ4 (t )uyy + χ5 (t )uxy + χ6 (t )uxx = χ7 (t ),
(5)
where χ1 (t ) ̸= 0, χ2 (t ) ̸= 0, χ3 (t ), χ4 (t ) ̸= 0, χ5 (t ), χ6 (t ) and χ7 (t ) are all real analytic functions, is investigated. Of mathematical and physical interests, some examples of Eq. (5) are given below: 1. When χ5 (t ) = χ6 (t ) = 0, Eq. (5) becomes
(ut + χ1 (t )uux + χ2 (t )uxxx )x + χ3 (t )ux + χ4 (t )uyy = χ7 (t ),
(6)
where χ1 (t ) ̸= 0, χ2 (t ) ̸= 0, χ3 (t ), χ4 (t ) ̸= 0 and χ7 (t ) are all analytic functions and represent the nonlinearity, dispersion, perturbed effect, disturbed wave velocity along the y direction and external force, respectively. Eq. (6) can describe surface waves propagation in shallow seas and marine straits with varying depth and width [28,29], long waves in turbulent flow over a sloping bottom [30], long waves on the surface of a three-dimensional fluid domain bounded below by slowly varying topography when the initial data is modulated slowly in the direction parallel to the wavefronts [31]. Exact solutions for Eq. (6) have been obtained using computer algebra [32]. Painlevé property and analytic solutions for Eq. (6) without consideration of the external-force term χ7 (t ) have been discussed in Ref. [25], and pfaffianization procedure has been performed on it when χ3 (t ) = χ7 (t ) = 0 [33]. Besides, much attention has been paid to exact solutions for Eq. (6) when χ7 (t ) = 0, and both χ1 (t ) and χ2 (t ) are chosen to be constants [26,34–36]. 2. When χ3 (t ) = χ5 (t ) = χ7 (t ) = 0, Eq. (5) reduces to the following equation
(ut + χ1 (t )uux + χ2 (t )uxxx )x + χ4 (t )uyy + χ6 (t )uxx = 0,
(7)
which can be used to describe nonlinear waves with a weakly diffracted wave beam [8,37], internal waves propagating along the interface of two fluid layers [38], internal solitary waves in Alboran sea [39] and so on. 3. Integrability property and Jacobi elliptic function solutions for the following variable-coefficient KP equation
(ut + f1 (t )uux + f2 (t )uxxx )x + 6f (t )ux + g 2 (t )uyy + f3 (t ) = 0,
(8)
have been presented in Refs. [40,41]. Taking no account of external-force term in Eq. (5), we will mainly study the following generalized variable-coefficient KP equation
(ut + f (t )uux + g (t )uxxx )x + l(t )ux + m(t )uyy + n(t )uxy + q(t )uxx = 0,
(9)
in this paper. Grammian solutions have been presented in Ref. [27]. This paper is structured as follows. In Section 2, making use of computerized symbolic computation, four transformations from the generalized variable-coefficient KP equation (9) to KP equation (2), cylindrical KP equation (4), KdV equation (1) and cylindrical KdV equation (3) are given out. Analytic multi-solitonic solution, auto-Bäcklund transformation in bilinear form and Lax pair for Eq. (9) are derived from the variable-coefficient bilinear equation in Section 3. Conclusions and discussions of our results will be presented in the last section. 2. Transformations to Eqs. (1)–(4) Considering the complexity of the analysis of Eq. (9) due to the existence of six variable coefficients and abundant valuable results given for the completely integrable models (1)–(4), we aim at finding the transformations from Eq. (9) to Eqs. (1)–(4) and indirectly studying the variable-coefficient equation making use of the existing results of Eqs. (1)–(4). To seek transformations from Eq. (9) to Eqs. (2) and (4), we can assume u(x, y, t ) to be of the form u(x, y, t ) = A(x, y, t )U [X (x, y, t ), Y (x, y, t ), T (x, y, t )] + B(x, y, t ),
(10)
while to determine transformations from Eq. (9) to Eqs. (1) and (3), the following assumption can be taken u(x, y, t ) = A(x, y, t )U [X (x, y, t ), T (x, y, t )] + B(x, y, t ),
(11)
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with A(x, y, t ), B(x, y, t ), X (x, y, t ), Y (x, y, t ) and T (x, y, t ) being analytic functions with respect to {x, y, t } to be determined [24,42]. By substituting Eq. (10) into Eq. (9) and utilizing computerized symbolic computation, we can find that if 4fglm − 4ft gm + 3fgt m + fgmt = 0, 3gt2 m2
− 3g
2
m2t
(12)
− 2ggtt m + 2g mmtt = 0, 2
2
(13)
Eq. (9) can be transformed into Eq. (2) through the dependent variable transformation 6gF 2
u(x, y, t ) =
f
4gmFt + gt mF − gmt F
X (x, y, t ) = F (t )x −
Y (y, t ) = yF 2 T (t ) =
∫
H11 (t ) =
σg
8gm2
H13 (t ) =
fF
x + H11 (t )y2 + H12 (t )y + H13 (t ),
2
y −
nF 2m
Qt
+
2mF
(14)
σg
y + P (t ),
(15)
m
+ Q (t ),
m
(16)
gF 3 dt ,
1
l
2m
(17) Ft
−f
fF
Ft
2
+
fF
mnFt − mt nF + mnt F n2 F − 4mqF − 4mPt 4fmF
σg m
−
2fm2 F
Ft
,
fF
H12 (t ) =
Ft
U [X (x, y, t ), Y (y, t ), T (t )] −
(18)
t
(3gFt Qt + gt FQt − gFQtt ) 2σ fg 2 F 3
,
(19)
Qt2
−
(20)
4σ fgF 4
where F (t ) ̸= 0, P (t ), Q (t ) are all arbitrary functions of t, while through the transformation 6gF 2
u(x, y, t ) =
f
U [X (x, y, t ), Y (y, t ), T (t )] −
gF 4
X (x, y, t ) = F (t )x +
Y (y, t ) = y T (t ) =
∫
H21 (t ) =
F
2
T (t )
−
4mT (t )
σg
H23 (t ) =
x + H21 (t )y2 + H22 (t )y + H23 (t ),
4gmFt + gt mF − gmt F 8gm2
2
y −
nF 2m
+
T (t )Qt 2mF
(21)
y + P (t ),
σg
(23)
gF 3 dt , 1
2m
l
(22)
m
+ Q (t ),
m
(24) Ft
fF
−f
Ft
2
+
fF
H22 (t ) =
Ft fF
mnFt − mt nF + mnt F
−
2fm2 F n2 F − 4mqF − 4mPt 4fmF
−
Ft
,
fF σg m
(25)
t
(3gFt Qt + gt FQt − gFQtt )T (t ) 2σ
T 2 (t )Qt2 4σ fgF 4
fg 2 F 3
FQt
+
σf
σg m
,
,
(26)
(27)
where F (t ) ̸= 0, P (t ), Q (t ) are still all arbitrary functions of t, Eq. (9) can be converted to cylindrical KP equation (4). From Eqs. (12)–(13), we can derive that
g (t ) = C1 f (t )e
− l(t )dt
λ1 + λ2
∫
m(t ) = C2 f (t )e
− l(t )dt
λ1 + λ2
∫
f (t )e
− l(t )dt
dt ,
(28)
−3 f (t )e
− l(t )dt
dt
,
(29)
Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
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where C1 ̸= 0, C2 ̸= 0, λ1 and λ2 are all arbitrary constants with λ21 + λ22 ̸= 0, which coincide with the integrability conditions of Eq. (9). Following similar procedures, we can deduce the following transformation 6gF 2
u(x, y, t ) =
f
U [X (x, y, t ), T (t )] −
X (x, y, t ) = F (t )x − T (t ) =
∫
H31 (t ) =
Ft 2m
+
Ft fF
x + H31 (t )y2 + H32 (t )y + H33 (t ),
(30)
F 2fgm
(fgl − ft g + fgt ) y2 + P (t )y + Q (t ),
(31)
gF 3 dt ,
1
l
2m
H32 (t ) =
n f 2 gm
H33 (t ) = −
(32) Ft
−f
fF
2
+
fF
Ft
,
fF
(fgl − ft g + fgt ) +
f 2 gF
qF + nP + Qt
−
fF
mP 2 fF 2
(33)
t
2P
+
Ft
1 fmF 2
(nFFt + 2mFt P − mFPt ),
(34)
,
(35)
that can be used to transform Eq. (9) into KdV equation (1) with F (t ) ̸= 0, P (t ), Q (t ) being arbitrary functions of t and conditions (28)–(29) holding, and another dependent variable transformation 6gF 2
u(x, y, t ) =
f
U [X (x, y, t ), T (t )] −
X (x, y, t ) = F (t )x − T (t ) =
∫
H41 (t ) =
Ft 2m
+
F 2fgm
Ft fF
x + H41 (t )y2 + H42 (t )y + H43 (t ),
(fgl − ft g + fgt ) −
gF 4 4mT (t )
(36)
y2 + P (t )y + Q (t ),
(37)
gF 3 dt ,
1
l
2m
H42 (t ) = H43 (t ) = −
n f 2 gm
(38) Ft fF
+
−f
2
+
fF
Ft
fF
,
f 2 gF
qF + nP + Qt
(fgl − ft g + fgt ) + −
mP 2 fF 2
(39)
t
2P
fF
Ft
1 fmF 2
(nFFt + 2mFt P − mFPt ) −
gF 2 2fmT (t )
(nF + 2mP ),
,
(40)
(41)
through which Eq. (9) can be transformed into cylindrical KdV equation (3) with F (t ) ̸= 0, P (t ), Q (t ) being arbitrary functions of t under conditions (28)–(29). Making use of the afore-obtained transformations and abundant solutions of Eqs. (1)–(4) (See Refs. [15,17,18,21–23, 43–50] and related references therein), a large number of corresponding analytic solutions for Eq. (9), such as similarity solutions, positon-like solutions, breather-like solutions, complexiton solutions, travelling wave solutions and multisolitonic solutions, can be obtained. 3. Analytic multi-solitonic solutions It is straightforward to verify that Eq. (9) can be converted to the variable-coefficient bilinear equation
[Dx Dt + g (t )D4x + q(t )D2x + n(t )Dx Dy + m(t )D2y ]Φ · Φ = 0,
(42)
through the dependent variable transformation u(x, y, t ) =
12g (t ) ∂ 2 f ( t ) ∂ x2
log(Φ (x, y, t )),
(43)
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Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
under the condition g (t ) = C1 f (t )e− l(t )dt ,
(44)
with C1 a non-zero arbitrary constant. The Hirota bilinear operators in Eq. (42) are defined by [50,51] n l Dm x Dy Dt a(x, y, t ) · b(x, y, t ) =
∂m ∂n ∂l a(x + x′ , y + y′ , t + t ′ )b(x − x′ , y − y′ , t − t ′ )|x′ =0,y′ =0,t ′ =0 . ∂ x′ m ∂ y′ n ∂ t ′ l
(45)
Following the usual procedure of the formal parameter expansion method, analytic multi-solitonic solutions of Eq. (9) can be given by ∂2 uN (x, y, t ) = 12C1 e− l(t )dt 2 log(Φ (x, y, t )), ∂x
−
Φ (x, y, t ) =
exp
N −
µ=0,1
(46)
−
µi ηi +
µi µj Aij ,
(47)
1≤i
i =1
where
ηi = pi x + ri y − p3i aij = eAij =
∫
g (t )dt −
ri2
∫
pi
m(t )dt − ri
∫
n(t )dt − pi
3g (t )p2i p2j (pi − pj )2 − m(t )(pi rj − pj ri )2 3g (t )p2i p2j (pi + pj )2 − m(t )(pi rj − pj ri )2
∫
q(t )dt + ηi,0 ,
(48)
,
(49)
for i, j = 1, 2, . . . , N with N being the number of solitons. The notation µ=0,1 denotes the summation over all possible ∑ combinations of µi = 0, 1 (i = 1, 2, . . . , N ) and 1≤i
∑
m(t ) = C2 f (t )e− l(t )dt ,
(50)
where C2 is a non-zero arbitrary constant, has to be satisfied for N ≥ 2. In particular, we have the one-solitonic solution for Eq. (9) which reads
u1 ( x , y , t ) =
3C1 p21 e−
l(t )dt
2
sech
η1 2
= 3C1 p21 e−
l(t )dt
sech2
p1 x + r1 y −
p31 g (t ) + p1 q(t ) + r1 n(t ) +
r12 p1
m(t ) dt + η10
2
,
(51)
and the two-solitonic solution
u2 (x, y, t ) = 12C1 e
− l(t )dt
2e
η1 +η2 2
√
eA12 cosh
η1 + η2 + A12 2
+ cosh
η1 − η2 2
xx
2 η1 +η2 √ η −η + log ( p / p ) + A 1 2 1 2 12 + 2 p1 p2 cosh (p1 + p2 )e 2 2 − l(t )dt = 12C1 e − √ η +η + A η −η 1 2 12 1 2 A12 cosh 2 e + cosh 2 2 2 (p1 + p2 )2 e 2 +A12 + 2p1 p2 cosh η1 −η + log(p1 /p2 ) 2 + . √ η +η + A12 η −η 1 2 1 2 A 2 e 12 cosh + cosh 2 2 η1 +η2
(52)
Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
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Bäcklund transformation, which originated from differential geometry, is a transformation that relates pairs of solutions of NLEEs. Especially, auto-Bäcklund transformation in bilinear form performed on the bilinear equation derived by Hirota bilinear method, can provide analytic soliton solutions, Lax pair, an infinite number of conservation laws and the inverse scattering transformation for the original physical equation [51]. If Φ (x, y, t ) and Φ ′ (x, y, t ) are considered to be two different solutions of the bilinear equation (42), then the following equation P ≡ {[Dx Dt + g (t )D4x + q(t )D2x + n(t )Dx Dy + m(t )D2y ]Φ ′ · Φ ′ }Φ 2 2
− Φ ′ {[Dx Dt + g (t )D4x + q(t )D2x + n(t )Dx Dy + m(t )D2y ]Φ · Φ } = 0,
(53)
can be rewritten as P = 2Dx (Dt Φ ′ · Φ ) · ΦΦ ′ + 2g (t )Dx [(D3x Φ ′ · Φ ) · ΦΦ ′ + 3(D2x Φ ′ · Φ ) · (Dx Φ · Φ ′ )]
+ 2q(t )Dx (Dx Φ ′ · Φ ) · ΦΦ ′ + 2n(t )Dx (Dy Φ ′ · Φ ) · ΦΦ ′ + 2m(t )Dy (Dy Φ ′ · Φ ) · ΦΦ ′ = [3a(t )D2x − m(t )Dy − λ(t )]Φ ′ · Φ + 2Dx
Dt + g (t )D3x + q(t )Dx + n(t )Dy
λ(t )g (t ) ′ + 3a(t )Dx Dy + Dx − ρ(t ) Φ · Φ · ΦΦ ′ = 0, a(t )
(54)
where λ(t ), ρ(t ) are both arbitrary functions of t, and 3a(t )2 = g (t )m(t ). Therefore the auto-Bäcklund transformation in bilinear form for Eq. (9) can be expressed by
[3a(t )D2x − m(t )Dy ]Φ ′ · Φ = λ(t )ΦΦ ′ , [ ] λ(t )g (t ) Dt + g (t )D3x + q(t )Dx + n(t )Dy + 3a(t )Dx Dy + Dx Φ ′ · Φ = ρ(t )ΦΦ ′ . a( t )
(55) (56)
If we take ρ(t ) = 0 and Φ ′ (x, y, t ) = Ψ (x, y, t )Φ (x, y, t ), we can deduce from Eqs. (55)–(56) the Lax pair of Eq. (9) L = 3a(t )∂x2 − m(t )∂y +
a(t )f (t ) 2g (t )
[
u,
M = −[4a(t )∂x ∂y + n(t )∂y ] − q(t ) +
(57) 4λ(t )g (t ) 3a(t )
+
f (t ) 3
] u +
f (t ) 6g (t )
[
g (t )
∂u − 3a(t ) ∂x
∫
∂u dx , ∂y ]
(58)
where u = u(x, y, t ) is a solution of Eq. (9), Ψ = Ψ (x, y, t ) is an arbitrary function and the operators L and M satisfy LΨ = λ(t )Ψ , MΨ =
∂Ψ . ∂t
(59) (60)
4. Conclusions and discussions Variable-coefficient KP equation is an important model of various nonlinear real situations in hydrodynamics, plasma physics and some other nonlinear science when the inhomogeneities of media and nonuniformities of boundaries are taken into consideration. A generalized variable-coefficient KP equation (9) with perturbed effect terms has been investigated here. Relations between the generalized variable-coefficient KP equation and the KdV equation (1), cylindrical KdV equation (3), KP equation (2) and cylindrical KP equation (4) have been presented using the transformation assumptions (10) and (11). Analytic multi-solitonic solutions for Eq. (9) have been given out explicitly by Hirota bilinear method. What is more, autoBäcklund transformation in bilinear form and the Lax pair have been deduced from the variable-coefficient bilinear equation. Conclusions and discussions on our results are given below. 1. The four transformations, i.e., transformation (14)–(20), transformation (21)–(27), transformation (30)–(35) and transformation (36)–(41), which can be used to convert Eq. (9) to KP equation (2), cylindrical KP equation (4), KdV equation (1) and cylindrical KdV equation (3) respectively, share the same constraint conditions, which are consistent with the Painlevé integrability conditions of Eq. (9). And making use of the known solutions of Eqs. (1)–(4) [15,17,18, 21–23,43–50], many different kinds of analytic solutions can be derived from these transformations. 2. Under conditions (44) and (50), we have constructed multi-solitonic solutions for Eq. (9) from the bilinear equation (42) and expansion of the format parameter ε . We can find that the conditions for multi-solitonic solutions with N ≥ 2 are a special case of the Painlevé integrability conditions (28)–(29), i.e., the case when λ1 = 1, λ2 = 0 in Eqs. (28)–(29).
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Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
a
b -40
0.15 0.1 u
0 -40
0.2 0.1
-20 0 -50
u 0
x
-20
--40
0.3
-20
0.5
0
20
0
20
0
20
t
25
t
50
40 40
c u
x
-25
40
d
0.1 0.075 0.05 0.025
--20 -10 u
0
0
x
-4 -2
-20 -10 0 -5
10
0 t
0.3 0.2 0.1 0 -10
2
t 4
20
x
10
0 5
20 10
Fig. 1. The profile of the one-solitonic solution (51) with C1 = 0.3, n(t ) = 0.3, q(t ) = 0.1, p1 = 0.3, r1 = 0.5, η10 = 0, y = 0: (a) f (t ) = 0.3, m(t ) = 0.1, l(t ) = 0; (b) f (t ) = 1, m(t ) = 0.1 sin(0.2t ), l(t ) = 0.02; (c) f (t ) = 0.3, m(t ) = 0.1, l(t ) = t; (d) f (t ) = 0.3 sin(t ), m(t ) = 0.1, l(t ) = sin(t ).
3. For possible physical interests and applications, let us now see the features of the solutions we have obtained. It is easy to see from the one-solitonic solution (51) that the amplitude and the velocity of the corresponding single solitonic wave can be respectively given by A = 3C1 p21 e− l(t )dt ,
V = (Vx , Vy ) =
(61)
p41 g (t ) + p21 q(t ) + p1 r1 n(t ) + r12 m(t ) p41 r1 g (t ) + p21 r1 q(t ) + p1 r12 n(t ) + r13 m(t )
,
p21 + r12
p31 + p1 r12
.
(62)
Since the amplitude only depends on the variable-coefficient function l(t ), we can reveal that when l(t ) = 0, the amplitude of the wave will keep invariant (Fig. 1(a)), when l(t ) is taken to be a non-zero constant C , the amplitude will decay exponentially as time converges at infinity (positive infinity for C < 0 while negative infinity for C > 0) (Fig. 1(b)), when l(t ) = Ct with C > 0, the amplitude will decay exponentially at all directions of a localized region (Fig. 1(c)), and the amplitude will change periodically when l(t ) is chosen to be a bounded periodic function (Fig. 1(d)). Fig. 2 illustrates us another two types of one-solitonic waves with periodic properties when m(t ), n(t ) and q(t ) are chosen to be periodic functions. When a12 > 0, the limit of the two-solitonic solution (52) is
2 − l(t )dt 2 η2 3c p e sech , η1 → −∞, 1 2 η2 3c1 p21 e− l(t )dt sech2 1 , η2 → −∞, 2 u(x, y, t ) → η2 + A12 2 − l(t )dt 2 3c1 p2 e sech , η1 → +∞, 2 3c1 p2 e− l(t )dt sech2 η2 + A12 , η2 → +∞, 1
(63)
2
from which we can conclude that after the collision, the two solitary waves will experience amplitude changes which are determined by l(t ), velocity changes determined by f (t ), n(t ) and q(t ), and phase shifts that have no relationship with these variable coefficients. In Fig. 3(a), the amplitudes and the velocities of the two solitary waves undergo no change during the collision except for a phase shift −7.052. In comparison, although the phase shift is also −7.052, the amplitudes and the velocities of the two solitary waves in Fig. 3(b) change after the collision because of the inhomogeneity effect of l(t ).
Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
3275
Fig. 2. The profile of the one-solitonic solution (51) with C1 = 0.3, p1 = 0.3, r1 = 0.5, η10 = 0, y = 0: (a) f (t ) = 30 sin(0.1t ), m(t ) = 0.5 sin(0.2t ), l(t ) = 0, n(t ) = 0.3, q(t ) = 0.1; (b) f (t ) = 0.3, m(t ) = 0.1, l(t ) = 0, n(t ) = 0.3 sin(0.08t ), q(t ) = 1.5 sin(0.2t ).
40
u
2 1.5 1 0.5
40 2 1.5 1 0.5 u 0
20 0
0
x
20 0
x
40
40 20
20
-20
-20 0
0 -20
t
t -40
-20 -40
-40
-40
Fig. 3. The interaction profile of the two-solitonic solution (52) with C1 = 0.2, C2 = 0.2, p1 = 1.6, p2 = 0.6, r1 = 1.5, r2 = 1.6, η10 = η20 = 0, f (t ) = 0.3, n(t ) = 0.3, q(t ) = 0.1, y = 0: (a) l(t ) = 0; (b) l(t ) = 0.01.
b 40
a
20
t
-40 2 1.5 1 u 0.5 0
0
-20 -20
0
x
-40 -20
-40
20
0 t
20 40
-40
40
-20
0 x
20
40
Fig. 4. (a) The interaction profile of the two-solitonic solution (52) with C1 = 0.2, C2 = 0.2, p1 = 1.6, r1 = 1.5, r2 = 1.6, η10 = η20 = 0, y = 0, p2 = 0.59894, f (t ) = 0.3, l(t ) = 0, n(t ) = 0.3, q(t ) = 0.1; (b) the corresponding contourplot.
If a12 becomes zero in the two-solitonic solution (52), the limit can be given by
η 2 , η1 → −∞, 3c1 p22 e− l(t )dt sech2 2 η 1 2 − l(t )dt sech2 , η2 → −∞, u(x, y, t ) → 3c1 p1 e 2 η1 − η2 3c1 (p1 − p2 )2 e− l(t )dt sech2 , η1 → +∞, η2 → +∞.
(64)
2
In this case after the amalgamation of the two solitary waves during the interaction, instead of separating into two waves as in Fig. 3, they will merge into a single larger solitary wave (see Fig. 4). Two periodic solitary waves without collision in the process of propagation are shown in Fig. 5. When f (t ) is chosen to be Ct with C a non-zero constant, the solitary wave will be reflected at a certain location and move towards the opposite direction. It can be seen from Fig. 6(a) that before t = 0 two solitary waves run towards the negative x axis, but their propagation direction changes to the positive x axis after the collision at t = 0. The collision between two localized decaying solitary waves can be seen in Fig. 6(b). Amplitudes of the two waves experience increase before t = 0 and exponential attenuation after the collision at t = 0. It can be concluded from above that by proper choice of the relative variable-coefficient functions in the generalized variable-coefficient KP equation (9), many physical and engineering practical nonlinear phenomena can be explained accordingly.
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Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
b 40
a
20
0
t
--20 2 1.5 1 u 0.5 0
-10 -1 0
-20
x 10
-40 -20
-40
20
0 20
t
40
-20
30
-10
0
10
20
30
x
Fig. 5. (a) The interaction profile of the two-solitonic solution (52) with C1 = 0.2, C2 = 0.2, p1 = 1.6, p2 = 0.6, r1 = 1.5, r2 = 1.6, f (t ) = 0.3 sin(2t ), l(t ) = 0, n(t ) = 0.3 sin(t ), q(t ) = 0.1 sin(3t ), η10 = η20 = 0, y = 0; (b) the corresponding contourplot.
a
b -20
--40
4
4 -10
u
2
-20 u
0 -10
0 -5
10
0 t
5
x
2 0 -4
0 -2 0 t
10 20
x
20 2 4 40
Fig. 6. The interaction profile of the two-solitonic solution (52) with C1 = 1.03, C2 = 1.66, p1 = 0.65, r1 = 0.7, p2 = 0.73, r2 = 2.34, η10 = η20 = 0, y = 0: (a) f (t ) = 0.3t , l(t ) = 0, n(t ) = sin(t ), q(t ) = sin(3t ); (b) f (t ) = 1, l(t ) = t , n(t ) = 0.3 sin(t ), q(t ) = 0.1 cos(2t ).
Acknowledgements This work has been supported by the National Natural Science Foundation of China under Grant No. 61072145 and the Beijing Natural Science Foundation under Grant No. 1102018. References [1] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. [2] L.L. Yadav, R.S. Tiwari, K.P. Maheshwari, S.R. Sharma, Ion-acoustic nonlinear periodic waves in a two-electron-temperature plasma, Phys. Rev. E 52 (1995) 3045–3052. [3] H.S. Hong, H.J. Lee, Korteweg–de Vries equation of ion acoustic surface waves, Phys. Plasmas 16 (1999) 3422–3424. [4] N.C. Lee, Small amplitude electron-acoustic double layers and solitons in fully relativistic plasmas of two-temperature electrons, Phys. Plasmas 16 (2009) 042316. [5] L.A. Ostrovsky, Yu.A. Stepanyants, Do internal solitons exist in the ocean? Rev. Geophys. 27 (1989) 293–310. [6] H.X. Ge, R.J. Cheng, S.Q. Dai, KdV and Kink–Antikink solitons in car-following models, Physica A 357 (2005) 466–476. [7] B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl. 15 (1970) 539–541. [8] P.J. Bryant, Two-dimensional periodic permanent waves in shallow water, J. Fluid Mech. 115 (1982) 525–532. [9] W.S. Duan, The Kadomtsev–Petviashvili (KP) equation of dust acoustic waves for hot dust plasmas, Chaos Solitons Fractals 14 (2002) 503–506. [10] M.M. Lin, W.S. Duan, The Kadomtsev–Petviashvili (KP), MKP, and coupled KP equations for two-ion-temperature dusty plasmas, Chaos Solitons Fractals 23 (2005) 929–937. [11] Y.L. Wang, Z.X. Zhou, X.Q. Jiang, C.F. Hou, Y.Y. Jiang, X.D. Sun, R.H. Qin, H.F. Zhang, Interaction of a weakly relativistic soliton in the magnetized plasma, Phys. Plasmas 13 (2006) 052307. [12] H. Ur-Rehman, S.A. Khan, W. Masood, M. Siddiq, Solitary waves with weak transverse perturbations in quantum dusty plasmas, Phys. Plasmas 15 (2008) 124501. [13] A. Biswas, Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients, Nonlinear Dyn. 58 (2009) 345–348. [14] S. Maxon, J. Viecelli, Cylindrical solitons, Phys. Fluids 39 (1974) 1614–1616. [15] B. Tian, Y.T. Gao, On the solitonic structures of the cylindrical dust-acoustic and dust-ion-acoustic waves with symbolic computation, Phys. Lett. A 340 (2005) 449–455. [16] M.W. Coffey, Nonlinear dynamics of vortices in ultraclean type-II superconductors: integrable wave equations in cylindrical geometry, Phys. Rev. B 54 (1996) 1279–1285. [17] B. Sahu, R. Roychoudhury, Exact solutions of cylindrical and spherical dust ion acoustic waves, Phys. Plasmas 10 (2003) 4162–4165. [18] K.L. Jones, X.G. He, Y.K. Chen, Existence of periodic travelling wave solution to the forced generalized nearly concentric Korteweg–de Vries equation, Internat. J. Math. Math. Sci. 24 (2000) 371–377. [19] R. Johnson, Water waves and Korteweg–de Vries equations, J. Fluid Mech. 97 (1980) 701–719. [20] A.P. Misra, P.K. Shukla, C. Bhowmik, Electron-acoustic solitary waves in dense quantum electron-ion plasmas, Phys. Plasmas 14 (2007) 082309. [21] Y.T. Gao, B. Tian, Some two-dimensional and non-travelling-wave observable effects of the shallow-water waves, Phys. Lett. A 301 (2002) 74–82. [22] Y.T. Gao, B. Tian, Cylindrical Kadomtsev–Petviashvili model, nebulons and symbolic computation for cosmic dust ion-acoustic waves, Phys. Lett. A 349 (2006) 314–319. [23] B. Tian, Y.T. Gao, Cylindrical nebulons, symbolic computation and Bäcklund transformation for the cosmic dust acoustic waves, Phys. Plasmas 12 (2005) 070703. [24] B. Tian, G.M. Wei, C.Y. Zhang, W.R. Shan, Y.T. Gao, Transformations for a generalized variable-coefficient Korteweg–de Vries model from blood vessels, Bose–Einstein condensates, rods and positons with symbolic computation, Phys. Lett. A 356 (2006) 8–16.
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Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Transformations and multi-solitonic solutions for a generalized variable-coefficient Kadomtsev–Petviashvili equation Yueqian Liang ∗ , Guangmei Wei, Xiaonan Li LMIB and School of Mathematics and System Sciences, Beihang University, Beijing 100191, China
article
abstract
info
Article history: Received 17 June 2010 Received in revised form 7 April 2011 Accepted 7 April 2011
Kadomtsev–Petviashvili equations with variable coefficients can be used to characterize many nonlinear phenomena in fluid dynamics and plasma physics more realistically than the equations with constant coefficients. Hereby, a generalized variable-coefficient Kadomtsev–Petviashvili equation with nonlinearity, dispersion and perturbed terms is investigated. Transformations, of which the consistency conditions are exactly the Painlevé integrability conditions, to the Korteweg–de Vries equation, cylindrical Korteweg–de Vries equation, Kadomtsev–Petviashvili equation and cylindrical Kadomtsev–Petviashvili equation are presented by formal dependent variable transformation assumptions. Using the Hirota bilinear method, from the variable-coefficient bilinear equation, the multisolitonic solution, auto-Bäcklund transformation and Lax pair for the variable-coefficient Kadomtsev–Petviashvili equation are obtained. Moreover, the influence of inhomogeneity coefficients on solitonic structures and interaction properties is discussed for physical interest and possible applications. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Generalized variable-coefficient Kadomtsev–Petviashvili equation Dependent variable transformation Hirota bilinear method Multi-solitonic solution
1. Introduction The celebrated, historic Korteweg–de Vries (KdV) equation UT + 6UUX + UXXX = 0,
(1)
has been found to model many physical, mechanical and engineering phenomena, such as ion-acoustic waves, stratified internal waves, geophysical fluid dynamics, collision-free hydromagnetic waves, lattice dynamics, the jams in the congested traffic and so on (see Refs. [1–6] and references therein). As a two-spatial dimensional analog of the KdV equation (1), the Kadomtsev–Petviashvili (KP) equation
(UT + 6UUX + UXXX )X + σ UYY = 0,
(2)
where σ = ±1 has been discovered to describe the evolution of long water waves and small-amplitude surface waves with weak nonlinearity, weak dispersion, and weak perturbation in the y direction, weakly relativistic soliton interactions in the magnetized plasma, dust acoustic waves for hot dust plasmas and some other nonlinear models [1,7–12]. As a special case of the KdV equation with power-law nonlinearity and time-dependent coefficients [13], the cylindrical KdV equation UT + 6UUX + UXXX +
∗
1 2T
U = 0,
Corresponding author. E-mail addresses: [email protected] (Y. Liang), [email protected] (G. Wei).
0898-1221/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.04.007
(3)
Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
3269
which was first proposed by Maxon and Viecelli in 1974 when they studied propagation of radically ingoing acoustic waves in cylindrical geometry [14], has been given out in some different situations by using the stretched coordinates [14–18]. And its counterpart in (2 + 1)-dimensional, the cylindrical KP equation 1
σ
UYY = 0, (4) 2T T2 where σ = ±1, has also been constructed to describe the nearly straight wave propagation which varies in a very small angular region, the cylindrical shallow-water waves subject to certain kinds of the small transversal disturbances, electronacoustic solitary waves in dense quantum electron–ion plasmas, the dust ion-acoustic waves with azimuthal perturbation and so on [18–23]. Recently, when the inhomogeneous media and/or the nonuniform boundaries are considered, nonlinear evolution equations (NLEEs) with variable coefficients, which describe nonlinear phenomena of the complex world more realistically than their constant-coefficient forms, have been paid much attention [24–27]. Hereby, a new generalized variablecoefficient KP equation
(UT + 6UUX + UXXX )X +
UX +
(ut + χ1 (t )uux + χ2 (t )uxxx )x + χ3 (t )ux + χ4 (t )uyy + χ5 (t )uxy + χ6 (t )uxx = χ7 (t ),
(5)
where χ1 (t ) ̸= 0, χ2 (t ) ̸= 0, χ3 (t ), χ4 (t ) ̸= 0, χ5 (t ), χ6 (t ) and χ7 (t ) are all real analytic functions, is investigated. Of mathematical and physical interests, some examples of Eq. (5) are given below: 1. When χ5 (t ) = χ6 (t ) = 0, Eq. (5) becomes
(ut + χ1 (t )uux + χ2 (t )uxxx )x + χ3 (t )ux + χ4 (t )uyy = χ7 (t ),
(6)
where χ1 (t ) ̸= 0, χ2 (t ) ̸= 0, χ3 (t ), χ4 (t ) ̸= 0 and χ7 (t ) are all analytic functions and represent the nonlinearity, dispersion, perturbed effect, disturbed wave velocity along the y direction and external force, respectively. Eq. (6) can describe surface waves propagation in shallow seas and marine straits with varying depth and width [28,29], long waves in turbulent flow over a sloping bottom [30], long waves on the surface of a three-dimensional fluid domain bounded below by slowly varying topography when the initial data is modulated slowly in the direction parallel to the wavefronts [31]. Exact solutions for Eq. (6) have been obtained using computer algebra [32]. Painlevé property and analytic solutions for Eq. (6) without consideration of the external-force term χ7 (t ) have been discussed in Ref. [25], and pfaffianization procedure has been performed on it when χ3 (t ) = χ7 (t ) = 0 [33]. Besides, much attention has been paid to exact solutions for Eq. (6) when χ7 (t ) = 0, and both χ1 (t ) and χ2 (t ) are chosen to be constants [26,34–36]. 2. When χ3 (t ) = χ5 (t ) = χ7 (t ) = 0, Eq. (5) reduces to the following equation
(ut + χ1 (t )uux + χ2 (t )uxxx )x + χ4 (t )uyy + χ6 (t )uxx = 0,
(7)
which can be used to describe nonlinear waves with a weakly diffracted wave beam [8,37], internal waves propagating along the interface of two fluid layers [38], internal solitary waves in Alboran sea [39] and so on. 3. Integrability property and Jacobi elliptic function solutions for the following variable-coefficient KP equation
(ut + f1 (t )uux + f2 (t )uxxx )x + 6f (t )ux + g 2 (t )uyy + f3 (t ) = 0,
(8)
have been presented in Refs. [40,41]. Taking no account of external-force term in Eq. (5), we will mainly study the following generalized variable-coefficient KP equation
(ut + f (t )uux + g (t )uxxx )x + l(t )ux + m(t )uyy + n(t )uxy + q(t )uxx = 0,
(9)
in this paper. Grammian solutions have been presented in Ref. [27]. This paper is structured as follows. In Section 2, making use of computerized symbolic computation, four transformations from the generalized variable-coefficient KP equation (9) to KP equation (2), cylindrical KP equation (4), KdV equation (1) and cylindrical KdV equation (3) are given out. Analytic multi-solitonic solution, auto-Bäcklund transformation in bilinear form and Lax pair for Eq. (9) are derived from the variable-coefficient bilinear equation in Section 3. Conclusions and discussions of our results will be presented in the last section. 2. Transformations to Eqs. (1)–(4) Considering the complexity of the analysis of Eq. (9) due to the existence of six variable coefficients and abundant valuable results given for the completely integrable models (1)–(4), we aim at finding the transformations from Eq. (9) to Eqs. (1)–(4) and indirectly studying the variable-coefficient equation making use of the existing results of Eqs. (1)–(4). To seek transformations from Eq. (9) to Eqs. (2) and (4), we can assume u(x, y, t ) to be of the form u(x, y, t ) = A(x, y, t )U [X (x, y, t ), Y (x, y, t ), T (x, y, t )] + B(x, y, t ),
(10)
while to determine transformations from Eq. (9) to Eqs. (1) and (3), the following assumption can be taken u(x, y, t ) = A(x, y, t )U [X (x, y, t ), T (x, y, t )] + B(x, y, t ),
(11)
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Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
with A(x, y, t ), B(x, y, t ), X (x, y, t ), Y (x, y, t ) and T (x, y, t ) being analytic functions with respect to {x, y, t } to be determined [24,42]. By substituting Eq. (10) into Eq. (9) and utilizing computerized symbolic computation, we can find that if 4fglm − 4ft gm + 3fgt m + fgmt = 0, 3gt2 m2
− 3g
2
m2t
(12)
− 2ggtt m + 2g mmtt = 0, 2
2
(13)
Eq. (9) can be transformed into Eq. (2) through the dependent variable transformation 6gF 2
u(x, y, t ) =
f
4gmFt + gt mF − gmt F
X (x, y, t ) = F (t )x −
Y (y, t ) = yF 2 T (t ) =
∫
H11 (t ) =
σg
8gm2
H13 (t ) =
fF
x + H11 (t )y2 + H12 (t )y + H13 (t ),
2
y −
nF 2m
Qt
+
2mF
(14)
σg
y + P (t ),
(15)
m
+ Q (t ),
m
(16)
gF 3 dt ,
1
l
2m
(17) Ft
−f
fF
Ft
2
+
fF
mnFt − mt nF + mnt F n2 F − 4mqF − 4mPt 4fmF
σg m
−
2fm2 F
Ft
,
fF
H12 (t ) =
Ft
U [X (x, y, t ), Y (y, t ), T (t )] −
(18)
t
(3gFt Qt + gt FQt − gFQtt ) 2σ fg 2 F 3
,
(19)
Qt2
−
(20)
4σ fgF 4
where F (t ) ̸= 0, P (t ), Q (t ) are all arbitrary functions of t, while through the transformation 6gF 2
u(x, y, t ) =
f
U [X (x, y, t ), Y (y, t ), T (t )] −
gF 4
X (x, y, t ) = F (t )x +
Y (y, t ) = y T (t ) =
∫
H21 (t ) =
F
2
T (t )
−
4mT (t )
σg
H23 (t ) =
x + H21 (t )y2 + H22 (t )y + H23 (t ),
4gmFt + gt mF − gmt F 8gm2
2
y −
nF 2m
+
T (t )Qt 2mF
(21)
y + P (t ),
σg
(23)
gF 3 dt , 1
2m
l
(22)
m
+ Q (t ),
m
(24) Ft
fF
−f
Ft
2
+
fF
H22 (t ) =
Ft fF
mnFt − mt nF + mnt F
−
2fm2 F n2 F − 4mqF − 4mPt 4fmF
−
Ft
,
fF σg m
(25)
t
(3gFt Qt + gt FQt − gFQtt )T (t ) 2σ
T 2 (t )Qt2 4σ fgF 4
fg 2 F 3
FQt
+
σf
σg m
,
,
(26)
(27)
where F (t ) ̸= 0, P (t ), Q (t ) are still all arbitrary functions of t, Eq. (9) can be converted to cylindrical KP equation (4). From Eqs. (12)–(13), we can derive that
g (t ) = C1 f (t )e
− l(t )dt
λ1 + λ2
∫
m(t ) = C2 f (t )e
− l(t )dt
λ1 + λ2
∫
f (t )e
− l(t )dt
dt ,
(28)
−3 f (t )e
− l(t )dt
dt
,
(29)
Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
3271
where C1 ̸= 0, C2 ̸= 0, λ1 and λ2 are all arbitrary constants with λ21 + λ22 ̸= 0, which coincide with the integrability conditions of Eq. (9). Following similar procedures, we can deduce the following transformation 6gF 2
u(x, y, t ) =
f
U [X (x, y, t ), T (t )] −
X (x, y, t ) = F (t )x − T (t ) =
∫
H31 (t ) =
Ft 2m
+
Ft fF
x + H31 (t )y2 + H32 (t )y + H33 (t ),
(30)
F 2fgm
(fgl − ft g + fgt ) y2 + P (t )y + Q (t ),
(31)
gF 3 dt ,
1
l
2m
H32 (t ) =
n f 2 gm
H33 (t ) = −
(32) Ft
−f
fF
2
+
fF
Ft
,
fF
(fgl − ft g + fgt ) +
f 2 gF
qF + nP + Qt
−
fF
mP 2 fF 2
(33)
t
2P
+
Ft
1 fmF 2
(nFFt + 2mFt P − mFPt ),
(34)
,
(35)
that can be used to transform Eq. (9) into KdV equation (1) with F (t ) ̸= 0, P (t ), Q (t ) being arbitrary functions of t and conditions (28)–(29) holding, and another dependent variable transformation 6gF 2
u(x, y, t ) =
f
U [X (x, y, t ), T (t )] −
X (x, y, t ) = F (t )x − T (t ) =
∫
H41 (t ) =
Ft 2m
+
F 2fgm
Ft fF
x + H41 (t )y2 + H42 (t )y + H43 (t ),
(fgl − ft g + fgt ) −
gF 4 4mT (t )
(36)
y2 + P (t )y + Q (t ),
(37)
gF 3 dt ,
1
l
2m
H42 (t ) = H43 (t ) = −
n f 2 gm
(38) Ft fF
+
−f
2
+
fF
Ft
fF
,
f 2 gF
qF + nP + Qt
(fgl − ft g + fgt ) + −
mP 2 fF 2
(39)
t
2P
fF
Ft
1 fmF 2
(nFFt + 2mFt P − mFPt ) −
gF 2 2fmT (t )
(nF + 2mP ),
,
(40)
(41)
through which Eq. (9) can be transformed into cylindrical KdV equation (3) with F (t ) ̸= 0, P (t ), Q (t ) being arbitrary functions of t under conditions (28)–(29). Making use of the afore-obtained transformations and abundant solutions of Eqs. (1)–(4) (See Refs. [15,17,18,21–23, 43–50] and related references therein), a large number of corresponding analytic solutions for Eq. (9), such as similarity solutions, positon-like solutions, breather-like solutions, complexiton solutions, travelling wave solutions and multisolitonic solutions, can be obtained. 3. Analytic multi-solitonic solutions It is straightforward to verify that Eq. (9) can be converted to the variable-coefficient bilinear equation
[Dx Dt + g (t )D4x + q(t )D2x + n(t )Dx Dy + m(t )D2y ]Φ · Φ = 0,
(42)
through the dependent variable transformation u(x, y, t ) =
12g (t ) ∂ 2 f ( t ) ∂ x2
log(Φ (x, y, t )),
(43)
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under the condition g (t ) = C1 f (t )e− l(t )dt ,
(44)
with C1 a non-zero arbitrary constant. The Hirota bilinear operators in Eq. (42) are defined by [50,51] n l Dm x Dy Dt a(x, y, t ) · b(x, y, t ) =
∂m ∂n ∂l a(x + x′ , y + y′ , t + t ′ )b(x − x′ , y − y′ , t − t ′ )|x′ =0,y′ =0,t ′ =0 . ∂ x′ m ∂ y′ n ∂ t ′ l
(45)
Following the usual procedure of the formal parameter expansion method, analytic multi-solitonic solutions of Eq. (9) can be given by ∂2 uN (x, y, t ) = 12C1 e− l(t )dt 2 log(Φ (x, y, t )), ∂x
−
Φ (x, y, t ) =
exp
N −
µ=0,1
(46)
−
µi ηi +
µi µj Aij ,
(47)
1≤i
i =1
where
ηi = pi x + ri y − p3i aij = eAij =
∫
g (t )dt −
ri2
∫
pi
m(t )dt − ri
∫
n(t )dt − pi
3g (t )p2i p2j (pi − pj )2 − m(t )(pi rj − pj ri )2 3g (t )p2i p2j (pi + pj )2 − m(t )(pi rj − pj ri )2
∫
q(t )dt + ηi,0 ,
(48)
,
(49)
for i, j = 1, 2, . . . , N with N being the number of solitons. The notation µ=0,1 denotes the summation over all possible ∑ combinations of µi = 0, 1 (i = 1, 2, . . . , N ) and 1≤i
∑
m(t ) = C2 f (t )e− l(t )dt ,
(50)
where C2 is a non-zero arbitrary constant, has to be satisfied for N ≥ 2. In particular, we have the one-solitonic solution for Eq. (9) which reads
u1 ( x , y , t ) =
3C1 p21 e−
l(t )dt
2
sech
η1 2
= 3C1 p21 e−
l(t )dt
sech2
p1 x + r1 y −
p31 g (t ) + p1 q(t ) + r1 n(t ) +
r12 p1
m(t ) dt + η10
2
,
(51)
and the two-solitonic solution
u2 (x, y, t ) = 12C1 e
− l(t )dt
2e
η1 +η2 2
√
eA12 cosh
η1 + η2 + A12 2
+ cosh
η1 − η2 2
xx
2 η1 +η2 √ η −η + log ( p / p ) + A 1 2 1 2 12 + 2 p1 p2 cosh (p1 + p2 )e 2 2 − l(t )dt = 12C1 e − √ η +η + A η −η 1 2 12 1 2 A12 cosh 2 e + cosh 2 2 2 (p1 + p2 )2 e 2 +A12 + 2p1 p2 cosh η1 −η + log(p1 /p2 ) 2 + . √ η +η + A12 η −η 1 2 1 2 A 2 e 12 cosh + cosh 2 2 η1 +η2
(52)
Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
3273
Bäcklund transformation, which originated from differential geometry, is a transformation that relates pairs of solutions of NLEEs. Especially, auto-Bäcklund transformation in bilinear form performed on the bilinear equation derived by Hirota bilinear method, can provide analytic soliton solutions, Lax pair, an infinite number of conservation laws and the inverse scattering transformation for the original physical equation [51]. If Φ (x, y, t ) and Φ ′ (x, y, t ) are considered to be two different solutions of the bilinear equation (42), then the following equation P ≡ {[Dx Dt + g (t )D4x + q(t )D2x + n(t )Dx Dy + m(t )D2y ]Φ ′ · Φ ′ }Φ 2 2
− Φ ′ {[Dx Dt + g (t )D4x + q(t )D2x + n(t )Dx Dy + m(t )D2y ]Φ · Φ } = 0,
(53)
can be rewritten as P = 2Dx (Dt Φ ′ · Φ ) · ΦΦ ′ + 2g (t )Dx [(D3x Φ ′ · Φ ) · ΦΦ ′ + 3(D2x Φ ′ · Φ ) · (Dx Φ · Φ ′ )]
+ 2q(t )Dx (Dx Φ ′ · Φ ) · ΦΦ ′ + 2n(t )Dx (Dy Φ ′ · Φ ) · ΦΦ ′ + 2m(t )Dy (Dy Φ ′ · Φ ) · ΦΦ ′ = [3a(t )D2x − m(t )Dy − λ(t )]Φ ′ · Φ + 2Dx
Dt + g (t )D3x + q(t )Dx + n(t )Dy
λ(t )g (t ) ′ + 3a(t )Dx Dy + Dx − ρ(t ) Φ · Φ · ΦΦ ′ = 0, a(t )
(54)
where λ(t ), ρ(t ) are both arbitrary functions of t, and 3a(t )2 = g (t )m(t ). Therefore the auto-Bäcklund transformation in bilinear form for Eq. (9) can be expressed by
[3a(t )D2x − m(t )Dy ]Φ ′ · Φ = λ(t )ΦΦ ′ , [ ] λ(t )g (t ) Dt + g (t )D3x + q(t )Dx + n(t )Dy + 3a(t )Dx Dy + Dx Φ ′ · Φ = ρ(t )ΦΦ ′ . a( t )
(55) (56)
If we take ρ(t ) = 0 and Φ ′ (x, y, t ) = Ψ (x, y, t )Φ (x, y, t ), we can deduce from Eqs. (55)–(56) the Lax pair of Eq. (9) L = 3a(t )∂x2 − m(t )∂y +
a(t )f (t ) 2g (t )
[
u,
M = −[4a(t )∂x ∂y + n(t )∂y ] − q(t ) +
(57) 4λ(t )g (t ) 3a(t )
+
f (t ) 3
] u +
f (t ) 6g (t )
[
g (t )
∂u − 3a(t ) ∂x
∫
∂u dx , ∂y ]
(58)
where u = u(x, y, t ) is a solution of Eq. (9), Ψ = Ψ (x, y, t ) is an arbitrary function and the operators L and M satisfy LΨ = λ(t )Ψ , MΨ =
∂Ψ . ∂t
(59) (60)
4. Conclusions and discussions Variable-coefficient KP equation is an important model of various nonlinear real situations in hydrodynamics, plasma physics and some other nonlinear science when the inhomogeneities of media and nonuniformities of boundaries are taken into consideration. A generalized variable-coefficient KP equation (9) with perturbed effect terms has been investigated here. Relations between the generalized variable-coefficient KP equation and the KdV equation (1), cylindrical KdV equation (3), KP equation (2) and cylindrical KP equation (4) have been presented using the transformation assumptions (10) and (11). Analytic multi-solitonic solutions for Eq. (9) have been given out explicitly by Hirota bilinear method. What is more, autoBäcklund transformation in bilinear form and the Lax pair have been deduced from the variable-coefficient bilinear equation. Conclusions and discussions on our results are given below. 1. The four transformations, i.e., transformation (14)–(20), transformation (21)–(27), transformation (30)–(35) and transformation (36)–(41), which can be used to convert Eq. (9) to KP equation (2), cylindrical KP equation (4), KdV equation (1) and cylindrical KdV equation (3) respectively, share the same constraint conditions, which are consistent with the Painlevé integrability conditions of Eq. (9). And making use of the known solutions of Eqs. (1)–(4) [15,17,18, 21–23,43–50], many different kinds of analytic solutions can be derived from these transformations. 2. Under conditions (44) and (50), we have constructed multi-solitonic solutions for Eq. (9) from the bilinear equation (42) and expansion of the format parameter ε . We can find that the conditions for multi-solitonic solutions with N ≥ 2 are a special case of the Painlevé integrability conditions (28)–(29), i.e., the case when λ1 = 1, λ2 = 0 in Eqs. (28)–(29).
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a
b -40
0.15 0.1 u
0 -40
0.2 0.1
-20 0 -50
u 0
x
-20
--40
0.3
-20
0.5
0
20
0
20
0
20
t
25
t
50
40 40
c u
x
-25
40
d
0.1 0.075 0.05 0.025
--20 -10 u
0
0
x
-4 -2
-20 -10 0 -5
10
0 t
0.3 0.2 0.1 0 -10
2
t 4
20
x
10
0 5
20 10
Fig. 1. The profile of the one-solitonic solution (51) with C1 = 0.3, n(t ) = 0.3, q(t ) = 0.1, p1 = 0.3, r1 = 0.5, η10 = 0, y = 0: (a) f (t ) = 0.3, m(t ) = 0.1, l(t ) = 0; (b) f (t ) = 1, m(t ) = 0.1 sin(0.2t ), l(t ) = 0.02; (c) f (t ) = 0.3, m(t ) = 0.1, l(t ) = t; (d) f (t ) = 0.3 sin(t ), m(t ) = 0.1, l(t ) = sin(t ).
3. For possible physical interests and applications, let us now see the features of the solutions we have obtained. It is easy to see from the one-solitonic solution (51) that the amplitude and the velocity of the corresponding single solitonic wave can be respectively given by A = 3C1 p21 e− l(t )dt ,
V = (Vx , Vy ) =
(61)
p41 g (t ) + p21 q(t ) + p1 r1 n(t ) + r12 m(t ) p41 r1 g (t ) + p21 r1 q(t ) + p1 r12 n(t ) + r13 m(t )
,
p21 + r12
p31 + p1 r12
.
(62)
Since the amplitude only depends on the variable-coefficient function l(t ), we can reveal that when l(t ) = 0, the amplitude of the wave will keep invariant (Fig. 1(a)), when l(t ) is taken to be a non-zero constant C , the amplitude will decay exponentially as time converges at infinity (positive infinity for C < 0 while negative infinity for C > 0) (Fig. 1(b)), when l(t ) = Ct with C > 0, the amplitude will decay exponentially at all directions of a localized region (Fig. 1(c)), and the amplitude will change periodically when l(t ) is chosen to be a bounded periodic function (Fig. 1(d)). Fig. 2 illustrates us another two types of one-solitonic waves with periodic properties when m(t ), n(t ) and q(t ) are chosen to be periodic functions. When a12 > 0, the limit of the two-solitonic solution (52) is
2 − l(t )dt 2 η2 3c p e sech , η1 → −∞, 1 2 η2 3c1 p21 e− l(t )dt sech2 1 , η2 → −∞, 2 u(x, y, t ) → η2 + A12 2 − l(t )dt 2 3c1 p2 e sech , η1 → +∞, 2 3c1 p2 e− l(t )dt sech2 η2 + A12 , η2 → +∞, 1
(63)
2
from which we can conclude that after the collision, the two solitary waves will experience amplitude changes which are determined by l(t ), velocity changes determined by f (t ), n(t ) and q(t ), and phase shifts that have no relationship with these variable coefficients. In Fig. 3(a), the amplitudes and the velocities of the two solitary waves undergo no change during the collision except for a phase shift −7.052. In comparison, although the phase shift is also −7.052, the amplitudes and the velocities of the two solitary waves in Fig. 3(b) change after the collision because of the inhomogeneity effect of l(t ).
Y. Liang et al. / Computers and Mathematics with Applications 61 (2011) 3268–3277
3275
Fig. 2. The profile of the one-solitonic solution (51) with C1 = 0.3, p1 = 0.3, r1 = 0.5, η10 = 0, y = 0: (a) f (t ) = 30 sin(0.1t ), m(t ) = 0.5 sin(0.2t ), l(t ) = 0, n(t ) = 0.3, q(t ) = 0.1; (b) f (t ) = 0.3, m(t ) = 0.1, l(t ) = 0, n(t ) = 0.3 sin(0.08t ), q(t ) = 1.5 sin(0.2t ).
40
u
2 1.5 1 0.5
40 2 1.5 1 0.5 u 0
20 0
0
x
20 0
x
40
40 20
20
-20
-20 0
0 -20
t
t -40
-20 -40
-40
-40
Fig. 3. The interaction profile of the two-solitonic solution (52) with C1 = 0.2, C2 = 0.2, p1 = 1.6, p2 = 0.6, r1 = 1.5, r2 = 1.6, η10 = η20 = 0, f (t ) = 0.3, n(t ) = 0.3, q(t ) = 0.1, y = 0: (a) l(t ) = 0; (b) l(t ) = 0.01.
b 40
a
20
t
-40 2 1.5 1 u 0.5 0
0
-20 -20
0
x
-40 -20
-40
20
0 t
20 40
-40
40
-20
0 x
20
40
Fig. 4. (a) The interaction profile of the two-solitonic solution (52) with C1 = 0.2, C2 = 0.2, p1 = 1.6, r1 = 1.5, r2 = 1.6, η10 = η20 = 0, y = 0, p2 = 0.59894, f (t ) = 0.3, l(t ) = 0, n(t ) = 0.3, q(t ) = 0.1; (b) the corresponding contourplot.
If a12 becomes zero in the two-solitonic solution (52), the limit can be given by
η 2 , η1 → −∞, 3c1 p22 e− l(t )dt sech2 2 η 1 2 − l(t )dt sech2 , η2 → −∞, u(x, y, t ) → 3c1 p1 e 2 η1 − η2 3c1 (p1 − p2 )2 e− l(t )dt sech2 , η1 → +∞, η2 → +∞.
(64)
2
In this case after the amalgamation of the two solitary waves during the interaction, instead of separating into two waves as in Fig. 3, they will merge into a single larger solitary wave (see Fig. 4). Two periodic solitary waves without collision in the process of propagation are shown in Fig. 5. When f (t ) is chosen to be Ct with C a non-zero constant, the solitary wave will be reflected at a certain location and move towards the opposite direction. It can be seen from Fig. 6(a) that before t = 0 two solitary waves run towards the negative x axis, but their propagation direction changes to the positive x axis after the collision at t = 0. The collision between two localized decaying solitary waves can be seen in Fig. 6(b). Amplitudes of the two waves experience increase before t = 0 and exponential attenuation after the collision at t = 0. It can be concluded from above that by proper choice of the relative variable-coefficient functions in the generalized variable-coefficient KP equation (9), many physical and engineering practical nonlinear phenomena can be explained accordingly.
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b 40
a
20
0
t
--20 2 1.5 1 u 0.5 0
-10 -1 0
-20
x 10
-40 -20
-40
20
0 20
t
40
-20
30
-10
0
10
20
30
x
Fig. 5. (a) The interaction profile of the two-solitonic solution (52) with C1 = 0.2, C2 = 0.2, p1 = 1.6, p2 = 0.6, r1 = 1.5, r2 = 1.6, f (t ) = 0.3 sin(2t ), l(t ) = 0, n(t ) = 0.3 sin(t ), q(t ) = 0.1 sin(3t ), η10 = η20 = 0, y = 0; (b) the corresponding contourplot.
a
b -20
--40
4
4 -10
u
2
-20 u
0 -10
0 -5
10
0 t
5
x
2 0 -4
0 -2 0 t
10 20
x
20 2 4 40
Fig. 6. The interaction profile of the two-solitonic solution (52) with C1 = 1.03, C2 = 1.66, p1 = 0.65, r1 = 0.7, p2 = 0.73, r2 = 2.34, η10 = η20 = 0, y = 0: (a) f (t ) = 0.3t , l(t ) = 0, n(t ) = sin(t ), q(t ) = sin(3t ); (b) f (t ) = 1, l(t ) = t , n(t ) = 0.3 sin(t ), q(t ) = 0.1 cos(2t ).
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