ISSN 1749-3889 (print), 1749-3897 (online)
International Journal of Nonlinear Science
Vol.4(2007) No.2,pp.147-150
Exact Solutions to the Dispersive Long Wave Equation
1
2
Yuzhen Chen1 , Xuelin Yong2 ∗
Department of Mathematics, Henan Institute of Science and Technology
Xinxiang 453003,China
School of Mathematical Science, Graduate University of Chinese Academy of Sciences
Beijing 100049, China
(Received 9 April 2007, accepted 6 September 2007)
Abstract: Based upon the extended projective Riccati equations method, we investigate the
dispersive long wave equation (DLWE) using the symbolic computation system. Several families of analytical solutions are obtained, including some new and more general ones.
Key words:Extended projective Riccati equations method; Exact solutions; Dispersive long
wave equations (DLWE)
1
Introduction
The (2+1)-dimensional dispersive long wave equation(DLWE)
(
uyt + vxx + ux uy + uuxy = 0,
vt + ux + vux + uvx + uxxy = 0,
(1)
was first obtained by Boiti as a compatibility condition for a weak Lax pairs[1]. There are lots of papers
discussing its possible applications and exact solutions. In [2], Paquin and Winternitz showed that the symmetry algebra of DLWE possesses the infinite-dimensional Kac-Moody-Virasoro structure. Some special
similarity solutions are given in [3] by using symmetry method and the classical group analysis. The more
general symmetry algebra, W∞ symmetry algebra, is given in [4]. Recently, Fan et al. obtained some exact
solutions of the equation by the ansatz-based method, including traveling-wave solutions, multiple-soliton
solutions, soliton-like solutions, periodic solutions and Weierstrass function solutions[5-9]. In this paper,
we aim to seek more general analytical solutions of the dispersive long wave equation by the extended
projective Riccati equations method.
2
Summary of the extended method
The key idea of the extended projective Riccati equations method is to take full advantages of the following
projective system[10]
( 0
f (ξ) = pf (ξ)g(ξ),
(2)
0
g (ξ) = 1 + pg 2 (ξ) − rf (ξ)
which has a first integral
1
g 2 (ξ) = − [1 − 2rf (ξ) + (r2 + δ)f 2 (ξ)],
p
(3)
where p = ±1, r is a real constant and δ is also a real constant which depends on the concrete expressions
of f (ξ), g(ξ). On the basis of [11], we can get following solutions to this system:
∗
Corresponding author.
E-mail address: [email protected]
c
Copyright°World
Academic Press, World Academic Union
IJNS.2007.10.15/105
148
International Journal of Nonlinear Science,Vol.4(2007),No.2,pp. 147-150
(i) When p = −1,
f (ξ) =
1
,
r + ksinh(ξ) + lcosh(ξ)
g(ξ) =
kcosh(ξ) + lsinh(ξ)
,
r + ksinh(ξ) + lcosh(ξ)
(4)
mcos(ξ) − nsin(ξ)
,
r + msin(ξ) + ncos(ξ)
(5)
where k and l are real constants.
(ii) When p = 1,
f (ξ) =
1
,
r + msin(ξ) + ncos(ξ)
g(ξ) = −
where m and n are real constants.
Now we establish the extended method as follows. Suppose we are given a partial differential equation
for a function u(x, y, t) :
H(u, ut , ux , uy , uxx , uxy , uxt , uyt ...) = 0.
(6)
Step 1. We assume that (6) has the following solutions:
u(x, y, t) = a0 +
N
X
[ai f i (ξ) + bi f i−1 (ξ)g(ξ)],
(7)
i=1
where a0 , ai , bi , ξ are all unknown functions of x, y, t, f (ξ) and g(ξ) satisfy (2). The parameter N can be
found by balancing the highest order derivative term and the nonlinear terms in (6).
Step 2. Substituting (7) along with(2)and (3) into (6) yields a set of algebraic polynomials for
i
f (ξ)g j (ξ)(i = 0, 1, ..., j = 0, 1). Setting the coefficients of these terms f i (ξ)g j (ξ) to zero, we’ll get a
system of over-determined PDEs with respect to unknown functions a0 , ai , bi and ξ.
Step 3. Solving the above system by using the symbolic computation system Maple, we would end up
with the explicit expressions for a0 , ai , bi and ξ or the constraints among them. Sometimes in order to get
analytical results we need to make a prior ansatz.
Step 4. According to the solutions (4) and (5) of (2) and the results in last step, we can obtain many
families of exact solutions for the given PDE.
3
Exact solutions to DLWE
By balancing the highest-order contributions from both the linear and nonlinear terms in (1), we can assume
the solutions in the form:
(
u(x, y, t) = a0 + a1 f (ξ) + b1 g(ξ),
(8)
v(x, y, t) = A0 + A1 f (ξ) + B1 g(ξ) + A2 f 2 (ξ) + B2 f (ξ)g(ξ),
where a0 = a0 (y, t), a1 = a1 (y, t), b1 = b1 (y, t), A0 = A0 (y, t), A1 = A1 (y, t), A2 = A2 (y, t), B1 =
B1 (y, t), B2 = B2 (y, t) and ξ = xα(y, t) + β(y, t) are all differential functions. The aim of choosing these
functions to be special forms, i.e, the x independence of a0 , a1 etc., is to make calculation feasible.
Substituting (8) along with (2,3) into (1), collecting coefficients of monomials of f (ξ), g(ξ), x of the
resulting system’s numerator(Notice that a0 , a1 , b1 , A0 , A1 , A2 , B1 , B2 , α, β are independent of x), then
setting each coefficient to zero, we obtain an over-determined PDE system. Solving the PDE system by
means of Maple gives the explicit expressions of the unknowns. Then we have the following exact solutions to EQ.(1):(Note: In the rest of this paper, Fi (i = 1, 2, 3, 4, 5) are arbitrary functions with respect to
d
corresponding independent variables, and 0 denotes dy
.)
Case 1. When p = −1 we have:
subcase 1.1:
√

r2 + δF2 (t)


,
 u = F1 (t) ± 2

r + l cosh (ξ) + k sinh (ξ)
(9)
¡ 2
¢2
¡
¢

r + δ [1 + F3 (y)]
2r r2 + δ [1 + F3 (y)]


 v = F3 (y) −
+2
,
δ [r + l cosh (ξ) + k sinh (ξ)]
δ [r + l cosh (ξ) + k sinh (ξ)]2
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X. Yong, Y. Chen: Exact Solutions to the Dispersive Long Wave Equation
R
2
3 (y)
2
2
where ξ = F2 (t) x − r δ+δ 1+F
F2 (t) dy + F4 (t) and δ = k − l .
subcase 1.2:

√

F3 (t) [l sinh (ξ) + k cosh (ξ)]
F3 (t) r2 + δ


+
,
u = F1 (t) ∓


r + l cosh (ξ) + k sinh (ξ)
r + l cosh (ξ) + k sinh (ξ)



¡
¢

F2 (y) r2 + δ
F2 (y) r
v = −1 ± √
∓√

r2 + δ[r + l cosh (ξ) + k sinh (ξ)]
r2 + δ[r + l cosh (ξ) + k sinh (ξ)]2





F2 (y) (l sinh (ξ) + k cosh (ξ))


+
,

[r + l cosh (ξ) + k sinh (ξ)]2
R
where ξ = −F3 (t) x + F3 (t) F1 (t) dt + F4 (y) and δ = k 2 − l2 .
subcase 1.3:

√

F3 (t) [l sinh (ξ) + k cosh (ξ)]
r2 + δF3 (t)


u = F2 (y) + F1 (t) ±
+
,


r + l cosh (ξ) + k sinh (ξ)
r + l cosh (ξ) + k sinh (ξ)


√

R
0


[ r2 + δF32 (t) F2 (y) dt + F5 (y)]r
0



 v = −1 − F2 (y) ∓ √ 2
r + δ[r + l cosh (ξ) + k sinh (ξ)]
√
R√
0
2
2

[ r + δF3 (t) F2 (y) dt + F5 (y)] r2 + δ



 ±

[r + l cosh (ξ) + k sinh (ξ)]2


√
R

0


[ r2 + δF32 (t) F2 (y) dt + F5 (y)][l sinh (ξ) + k cosh (ξ)]


,
 +
[r + l cosh (ξ) + k sinh (ξ)]2
R
where ξ = −F3 (t) x + F3 (t) [F2 (y) + F1 (t)]dt + F4 (y) and δ = k 2 − l2 .
Case 2. When p = 1 we have:
subcase 2.1:
√

−r2 − δF2 (t)


,
u
=
F
(t)
±
2

1

r + n cos (ξ) + m sin (ξ)
¡
¢
¡ 2
¢2

r r2 + δ [1 + F3 (y)]
r + δ [1 + F3 (y)]


 v = F3 (y) − 2
+2
,
δ [r + l cosh (ξ) + k sinh (ξ)]
δ [r + l cosh (ξ) + k sinh (ξ)]2
R
2
3 (y)
2
2
where ξ = F2 (t) x + r δ+δ 1+F
F2 (t) dy + F4 (t) and δ = −m − n .
subcase 2.2:

√

F3 (t) [n sin (ξ) − m cos (ξ)]
F3 (t) −δ − r2

 u = F1 (t) ±
+
,


r + n cos (ξ) + m sin (ξ)
r + n cos (ξ) + m sin (ξ)



¡
¢

F2 (y) r2 + δ
F2 (y) r
v = −1 ± √
∓√

−δ − r2 [r + n cos (ξ) + m sin (ξ)]
−δ − r2 [r + n cos (ξ) + m sin (ξ)]2





F2 (y) [n sin (ξ) − m cos (ξ)]


+

[r + n cos (ξ) + m sin (ξ)]2
R
where ξ = F3 (t) x − F3 (t) F1 (t) dt + F4 (y) and δ = −m2 − n2 .
subcase 2.3:
√

−r2 − δF3 (t)
F3 (t) [n sin (ξ) − m cos (ξ)]


+
,
u
=
F
(y)
+
F
(t)
∓

2
1


r + n cos (ξ) + m sin (ξ)
r + n cos (ξ) + m sin (ξ)


R √

0


[ − −r2 − δF32 (t) F2 (y) dt + F5 (y)]r
0



 v = −1 − F2 (y) ∓ √−r2 − δ[r + n cos (ξ) + m sin (ξ)]
¡
¢
R √
0

[ − −r2 − δF32 (t) F2 (y) dt + F5 (y)] r2 + δ


√

±


−r2 − δ[r + n cos (ξ) + m sin (ξ)]2



R √
0


[ − −r2 − δF32 (t) F2 (y) dt + F5 (y)][n sin (ξ) − m cos (ξ)]


 +
,
[r + n cos (ξ) + m sin (ξ)]2
R
where ξ = F3 (t) x − [F2 (y) + F1 (t)]F3 (t) dt + F4 (y) and δ = −m2 − n2 .
IJNS homepage:http://www.nonlinearscience.org.uk/
149
(10)
(11)
(12)
(13)
(14)
150
4
International Journal of Nonlinear Science,Vol.4(2007),No.2,pp. 147-150
Discussion
In this paper, various exact solutions of the dispersive long wave equation are derived with the aid of the
coupled projective Riccati equations and symbolic computation. We hope that the approach taken in our
present work may be further extended to other nonlinear systems.
Acknowledgements
Y. Chen is supported by the Natural Science Foundation of Henan Institute of Science and Technology under
the grant number 06057.
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