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A generalized sub-equations rational expansion method for nonlinear evolution equations
Commun Nonlinear Sci Numer Simulat 15 (2010) 1454–1461
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A generalized sub-equations rational expansion method for nonlinear evolution equations Wenting Li *, Hongqing Zhang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 18 February 2009 Received in revised form 15 May 2009 Accepted 20 June 2009 Available online 7 July 2009 PACS: 02.30.Jr 05.45.Yv Keywords: Nonlinear evolution equations Generalized sub-equations rational expansion method (2 + 1)-Dimensional Burgers equations Complexiton solutions
a b s t r a c t In this paper, based on symbolic computation and the idea of rational expansion method, a generalized sub-equations rational expansion method (GSRE) is devised to uniformly construct a series of exact complexiton solutions for nonlinear evolution equations. Compared with most existing tanh function methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions which include many new types of complexiton solutions: the combination of hyperbolic (and square form) function and elliptic function, trigonometric (and square form) function and elliptic function. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equations. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Constructing exact solutions for nonlinear evolution equations (NEEs) has long been a major concern for both mathematicians and physicists. On the one hand, due to their occurrence in many fields of science, in physics as well as in chemistry or biology and the interesting features and rich variety of their solutions, on the other hand, due to the availability of computer systems like Maple or Mathematica which allow us to perform some complicated and tedious algebraic calculation and differential calculation on a computer, at the same time help us to find new exact solution of NEEs. Up to now, there exist many powerful methods to obtain exact solutions of NEEs related to nonlinear problems, such as inverse scattering method [1], Bäcklund transformation [2], Darboux transformation [3], Hirota bilinear method [4], similarity reductions method [5], variable separation approach [6], Painlevé analysis method [7], homogeneous balance method [8], various tanh function methods [9–16], sech-function method [17] and so on. Among them, the tanh function method is considered to be one of the most straightforward and effective algebraic algorithm to obtain exact solutions for lots of NEEs. As we know, when applying the tanh function method, the choice of an appropriate sub-equation is of great importance. Much work has been concentrated on solving more solutions of the sub-equation. Recently, Ma [18] found a novel class of explicit exact solutions to the Korteweg–de Vries equation through its bilinear form and defined the solutions as complexiton solutions. In Ref. [19], Lou et al. presented and answered the problem ‘‘Are there any exact explicit multiple periodic wave solutions and periodic–solitary wave solutions for the nSG equation” with help of the mapping relations among the sine-Gordon field equation and the cubic nonlinear Klein–Gordon fields. Such * Corresponding author. E-mail address: [email protected] (W. Li). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.06.030
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solutions possess singularities of combinations of trigonometric function waves and exponential function waves which have different travelling speeds of new type. Above mentioned solutions of nonlinear evolution equations have a common character: combination of trigonometric function waves and exponential function waves. For unification and conciseness, so we call the solutions obtained by Ma and the solutions obtained by Lou as complexiton solutions. More recently, Wang and Chen [15] presented that the solutions of two different Riccati equations with different parameters are used as two variables in the components of finite rational expansion, which further exceeded the applicability of the tanh methods. Much work has been concentrated on solving more solutions of the sub-equation. Here, we not only focus on improving the general formal travelling transformation, but also the multiform choice of sub-equation. The present work is motivated by the desire to present a more generalized sub-equations to construct more types and more general formal solutions which contain not only the results obtained by using the methods [9–16] but also other types of solutions including many new types of complexiton solutions: various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and Jacobi elliptic double periodic function solutions, various combination of hyperbolic and rational function solutions, etc. For illustration, we apply the new method GSRE to solve (2 + 1)-dimensional Burgers equation and successfully construct new and more complexiton solutions, which have not been found before. The rest of this paper is organized as follows. In Section 2, our method GSRE is summarized. In Section 3, it is applied to the (2 + 1)-dimensional Burgers equation and bring out rich complexiton solutions for this model. A short summary and discussion are given in final. 2. Summary of the generalized sub-equations rational expansion method In the following we would like to outline the main steps of our general method: Step 1. Given a system of polynomial nonlinear partial differential equations with some physical fields ui ðx1 ; . . . ; xi ; . . . ; xn ; tÞ ði ¼ 1; . . . ; nÞ in ðn þ 1Þ variables x1 ; . . . ; xn ; t,
Hi ðui ; uit ; uix1 ; uix2 ; . . . ; uixn ; uitt ; uix1 t ; uix2 t ; . . . ; uixn t ; uix1 x1 ; . . .Þ ¼ 0:
ð2:1Þ
Step 2. We introduce a new ansätz in terms of finite rational formal expansion in the following forms:
ðmÞ ui ðx1 ; . . . ; xn ; tÞ ¼ Rui /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ; /01 ðn1 Þ; . . . ; /0n ðnn Þ; . . . ; /1 ðn1 Þ; . . . ; /ðmÞ n ðnn Þ where Rui ðÞ is the rational formal function of /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ; P ni ¼ nj¼1 ci;j xj þ ci;0 t ¼ ci;1 x1 þ . . . þ ci;n xn þ ci;0 t ði ¼ 1; . . . ; nÞ are satisfying
0 c1;0 1 n1 B B . C B c2;0 B. C ¼ B @ . A B .. @. nn cn;0 0
10 1 c1;n t B C c2;n C C B x1 C CB C .. CB .. C; . A@ . A
c1;1 c2;1 .. .
.. .
cn;1
cn;n
ð2:2Þ
ðmÞ /01 ðn1 Þ; . . . ; /0n ðnn Þ; . . . ; /1 ðn1 Þ; . . . ; /ðmÞ n ðnn Þ,
and
ð2:3Þ
xn
and ci;j ði ¼ 1; . . . ; n; j ¼ 0; 1; . . . ; nÞ are arbitrary constants. And /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ satisfy: m
1. /ðmÞ ðnÞ ¼ d dn/ðnÞ m ; 2. /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ are arbitrary function; 3. /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ satisfy various sub-equation, such as Riccati equation, projective Riccati equation, elliptic equation, Bessel equation, Klein–Gordon equation and so on. That is, all derivatives of /i ðni Þ ði ¼ 1; . . . ; nÞ with respect to ni are the rational formal function of /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ. For example, (1) When we take n ¼ 1; m ¼ 0,
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X
aij ð/1 ðn1 ÞÞj
j¼1
ðb0 þ b1 /1 ðn1 ÞÞj
ð2:4Þ
;
where a0 ; a1 ; b0 and b1 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. (2) When we take n ¼ 1; m ¼ 1,
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X j¼1
P
ij r j1 þr 0j1 ¼j ar j1 r 0
j1
r0
ð/1 ðn1 ÞÞrj1 ð/01 ðn1 ÞÞ j1
j ; 0 0 1 1 b0 þ b1 /1 ðn1 Þ þ b1 /01 ðn1 Þ þ b1 /1 ðn þ b 0 1 Þ / ðn Þ 1 1
ð2:5Þ
1
0
0
where a0 ; a1 ; a01 ; b0 ; b1 ; b1 ; b1 and b1 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. (3) When we take n ¼ 1; m ¼ 2,
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ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X
P
r0j1 r 00j1 r j1 ij 0 00 r j1 þr0j1 þr 00j1 ¼j ar j1 r 0 r 00 ð/1 ðn1 ÞÞ ð/1 ðn1 ÞÞ ð/1 ðn1 ÞÞ j1 j1
j¼1
j 0 00 b0 þ b1 /1 ðn1 Þ þ b1 /01 ðn1 Þ þ b1 /001 ðn1 Þ
0
ð2:6Þ
;
00
where a0 ; a1 ; a01 ; a001 ; b0 ; b1 ; b1 and b1 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. (4) When we take n ¼ 2; m ¼ 0,
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X j¼1
P rj1 r j2 ij r j1 þr j2 ¼j ar j1 r j2 ð/1 ðn1 ÞÞ ð/2 ðn2 ÞÞ j ; 1 1 b0 þ b1 /1 ðn1 Þ þ b2 /2 ðn2 Þ þ b1 /1 ðn þ b 2 /2 ðn2 Þ 1Þ
ð2:7Þ
where a0 ; a1 ; a2 ; b0 ; b1 ; b2 ; b1 and b2 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. (5) When we take n ¼ 2; m ¼ 1,
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X
P
r0j1 r 0j2 r j1 r j2 ij 0 0 r j1 þr0j1 þr j2 þr0j2 ¼j ar j1 r j2 r0 r 0 ð/1 ðn1 ÞÞ ð/2 ðn2 ÞÞ ð/1 ðn1 ÞÞ ð/2 ðn2 ÞÞ j1 j2 0
0
ðb0 þ b1 /1 ðn1 Þ þ b2 /2 ðn2 Þ þ b1 /01 ðn1 Þ þ b2 /02 ðn2 ÞÞj
j¼1
0
;
ð2:8Þ
0
where a0 ; a1 ; a2 ; a01 ; a02 ; b0 ; b1 ; b2 ; b1 and b2 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. Step 3. To determine the balanced parameter M i . Determine the Mi of the rational formal polynomial solutions by respectively balancing the highest nonlinear terms and the highest-order partial derivative terms in the given equations, and then give the formal solutions. Step 4. Substitute (2.2) into (2.1) with /i ðni Þ ði ¼ 1; . . . ; nÞ satisfying the sub-equations which will be chosen in Step 6. ðpÞr Then set all coefficients of /i ðni Þ ði ¼ 1; 2; . . . ; n; p ¼ 0; 1; . . . ; m; r ¼ 0; 1; . . .Þ of the resulting system’s numerator to be zero to get an over-determined system of nonlinear algebraic equations with respect to ci;j ði ¼ 1; . . . ; n; j ¼ 0; 1; . . . ; nÞ, and the parameters of the function Rui ðÞ. Step 5. To solve the parameters. By solving the over-determined system of nonlinear algebraic equations by use of symbolic computation system Maple, we end up with the explicit expressions for ci;j ði ¼ 1; . . . ; n; j ¼ 0; 1; . . . ; nÞ, and the parameters of the function Rui ðÞ. Step 6. To choose sub-equations. According to system (2.2), the conclusions in Step 5 and some particular solutions of the chosen sub-equations, we can obtain rational formal exact solutions of system (2.1). Remark 1. To our knowledge, most of the tanh function methods can be summarized as above the generalized sub-equations rational expansion method. What’s more, solutions obtained by this method include the square form of hyperbolic (solitary) function and triangular periodic functions at the same time, which are different from the solutions in [15]. Remark 2. Eq. (2.2) can be changed into the following form:
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X j¼1
PP P n m s¼1
PP P n m t¼1
where ai0 ; aijQn Qm s¼1
ðpÞ r p¼0 js
ij ; bQn Qm t¼1
ðpÞ q p¼0 jt
ðpÞ r p¼0 js
ðpÞ q p¼0 jt
ij aQ n Qm s¼1
ij bQn Qm t¼1
ðpÞ
rj
Qn Qm
js ðpÞ p¼0 ð/s ðns ÞÞ
s¼1
p¼0 s
ðpÞ q p¼0 jt
Qn Qm t¼1
ðpÞ jt p¼0 ð/t ðnt ÞÞ
þ bj0
;
ð2:9Þ
; bj0 and nk ðp ¼ 0; 1; . . . ; m; rjs ; qjt ¼ 1; . . . ; j; s; t ¼ 1; . . . ; n; i ¼ 1; 2; . . . ; k ¼ 1; . . . ; nÞ
are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t or arbitrary constants to be determined later. Remark 3. We should point out that in Eq. (2.9) n Y m Y
ðmÞ
0 0 ðmÞ /ðpÞ s ðns Þ ¼ /1 ðn1 Þ/2 ðn2 Þ . . . /n ðnn Þ/1 ðn1 Þ . . . /n ðnn Þ . . . /1 ðn1 Þ . . . /n ðnn Þ;
s¼1 p¼0 n Y m Y
ðpÞ
ðmÞ
ðmÞ
r js ¼ r j1 r j2 . . . rjn r 0j1 . . . r 0jn . . . r j1 . . . r jn ;
s¼1 p¼0 n X m X
ðpÞ
ðmÞ
ðmÞ
r js ¼ r j1 þ rj2 þ þ rjn þ r0j1 þ þ r 0jn þ þ rj1 þ r jn :
ð2:10Þ
s¼1 p¼0 ðmÞ
ðmÞ
and r j1 ; r j2 ; . . . ; r jn ; r 0j1 ; . . . ; r 0jn ; . . . ; r j1 ; . . . ; rjn are not only positive integers, but also can be negative integers in Eq. (2.4).
W. Li, H. Zhang / Commun Nonlinear Sci Numer Simulat 15 (2010) 1454–1461
1457
ðpÞ
Remark 4. /i ðni Þ ði ¼ 1; . . . ; n; p ¼ 0; 1; . . . ; mÞ satisfy sub-equation, such as elliptic equation, Riccati equation, projective Riccati equations, Bessel equation, and so on. They not only satisfy the same sub-equation, but also satisfy the different sub-equations. The appeal and success of our method lie on the fact that we choose different sub-equations. Because each sub-equation has especial formal solutions, the combination of some solutions can construct various formal complexiton solutions. In particular, we do not take the previous travelling wave transformation, but take difference travelling wave ðpÞ transformation in /i ðni Þ ði ¼ 1; . . . ; n; p ¼ 0; 1; . . . ; mÞ, so the solutions have different wave speed. And our method is a unified straightforward and pure algebraic algorithm to integrable equations and non-integrable equations, which is implemented in a computer algebraic system and can be easily extended to other integrable systems and non-integrable systems. 3. Exact complexiton solutions of the (2 + 1)-dimensional Burgers equation In this section we would like to apply our method GSRE to obtain complexiton solutions of the (2 + 1)-dimensional Burgers equation which reads
ut þ uuy þ av ux þ buyy þ abuxx ¼ 0; ux v y ¼ 0;
ð3:1Þ
where u ¼ uðx; y; tÞ; v ¼ v ðx; y; tÞ, and a; b–0 are all constants. Balancing the highest nonlinear terms with the highest-order partial derivative terms in (3.1) and we suppose the directive order m ¼ 1, so (3.1) has the following rational formal exact solutions:
a1 /1 ðn1 Þ þ a2 /2 ðn2 Þ þ a3 /01 ðn1 Þ þ a4 /02 ðn2 Þ ; l0 þ l1 /1 ðn1 Þ þ l2 /2 ðn2 Þ þ l3 /01 ðn1 Þ þ l4 /02 ðn2 Þ b1 /1 ðn1 Þ þ b2 /2 ðn2 Þ þ b3 /01 ðn1 Þ þ b4 /02 ðn2 Þ ; v ðx; y; tÞ ¼ b0 þ l0 þ l1 /1 ðn1 Þ þ l2 /2 ðn2 Þ þ l3 /01 ðn1 Þ þ l4 /02 ðn2 Þ uðx; y; tÞ ¼ a0 þ
ð3:2Þ
where n1 ¼ k1 x þ l1 y þ k1 t; n2 ¼ k2 x þ l2 y þ k2 t; l0 ; l1 ; l2 ; l3 ; l4 ; a0 ; a1 ; a2 ; a3 ; a4 ; b0 ; b1 ; b2 ; b3 and b4 are constants to be determined later. For obtaining our ideal complexiton solutions, we choose the celebrated Riccati equation (1) and the general elliptic equation (2) as sub-equations. (1)
/0 ðnÞ ðR þ /2 ðnÞÞ ¼ 0;
ð3:3Þ
(2)
/0 ðnÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh0 þ h1 /ðnÞ þ h2 /2 ðnÞ þ h3 /3 ðnÞ þ h4 /4 ðnÞÞ ¼ 0;
ð3:4Þ
where R; h0 ; h1 ; h2 ; h3 and h4 are arbitrary constants. Then we let the new variables /1 ðn1 Þ and /2 ðn2 Þ satisfy (3.3) and (3.4), respectively. With the aid of Maple, substituting (3.2) along with (3.3) and (3.4) into (3.1) and setting the coefficients of these terms /i1 ðn1 Þ/j2 ðn2 Þ ði; j ¼ 0; 1; . . .Þ to be zero yields a set of over-determined algebraic equations with respect to k1 ; l1 ; k1 ; k2 ; l2 ; k2 ; a0 ; a1 ; a2 ; a3 ; a4 ; b0 ; b1 ; b2 ; b3 ; b4 ; l0 ; l1 ; l2 ; l3 and l4 . By use of the Maple soft package ‘‘Charsets” by Dongming Wang, which is based on the Wu-elimination method, solving the over-determined algebraic equations, we get the following cases: Case 1
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi k2 l3 ð aa3 þ ab3 Þk2 aa0 k2 l3 ; l2 ¼ ak2 ; k2 ¼ k2 ; ak2 l3 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a2 ¼ ab2 þ ða3 b3 aÞl2 l1 a1 ¼ ab1 þ ða3 b3 aÞl1 l1 3 ; 3 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 1 a4 ¼ ab4 þ ða3 b3 aÞl4 l3 ; l0 ¼ 0; k1 ¼ k1 k2 k2 ; l1 ¼ ak1 ; b0 ¼
where k1 ; k2 ; a0 ; a3 ; b1 ; b2 ; b3 ; b4 ; Case 2
b0 ¼
k1
k1 ¼ k 1 ;
pffiffiffiffiffiffiffi aa0 k1 ; ak1 1
l1 ; l2 ; l3 and l4 are all arbitrary constants.
pffiffiffiffiffiffiffi l1 ¼ ak1 ;
k2 ¼ k1 k2 k1 ;
a1 a2 b1 ¼ pffiffiffiffiffiffiffi ; b2 ¼ pffiffiffiffiffiffiffi ; a a pffiffiffiffiffiffiffi a3 a4 l2 ¼ ak2 ; b3 ¼ pffiffiffiffiffiffiffi ; b4 ¼ pffiffiffiffiffiffiffi ; a a
where k1 ; k2 ; a0 ; b0 ; a1 ; a2 ; a3 ; a4 ;
l0 ; l1 ; l2 ; l3 and l4 are all arbitrary constants.
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W. Li, H. Zhang / Commun Nonlinear Sci Numer Simulat 15 (2010) 1454–1461
Case 3
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi k1 ¼ ða1 b1 aÞl1 l1 a4 ¼ ab4 þ ða1 b1 aÞl4 l1 1 þ ða0 b0 aÞl1 ; 1 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 k2 ¼ ða1 b1 aÞl2 l1 þ ða b a ; a ¼ a þ ða b a l l Þl b Þ 0 0 2 2 2 1 1 1 2 1 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a3 ¼ ab3 ; l1 ¼ ak1 ; l2 ¼ ak2 ; l0 ¼ 0; l3 ¼ 0;
where a0 ; a1 ; b0 ; b1 ; b2 ; b3 ; b4 ; k1 ; k2 ; Case 4
l1 ; l2 and l4 are all arbitrary constants.
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi k1 ¼ ða2 b2 aÞl1 l1 a1 ¼ ab1 þ ða2 b2 aÞl1 l1 2 þ ða0 b0 aÞl1 ; 2 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 k2 ¼ ða2 b2 aÞl2 l1 þ ða b a ; a ¼ a þ ða b a l l Þl b Þ 0 0 2 3 3 2 2 2 3 2 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a4 ¼ ab4 ; l1 ¼ ak1 ; l2 ¼ ak2 ; l0 ¼ 0; l4 ¼ 0; where a0 ; a2 ; b0 ; b1 ; b2 ; b3 ; b4 ; k1 ; k2 ;
l1 ; l2 and l3 are all arbitrary constants.
Note. Since the solutions obtained here are so many, we just list a new type of solutions corresponding to Case 1 for the (2 + 1)-dimensional Burgers equation to illustrate the efficiency of our method. Case 1. k1 ; k2 ; a0 ; a3 ; b1 ; b2 ; b3 ; b4 ; l1 ; l2 ; l3 and l4 are all arbitrary constants, and
8 > n1 > > > > > n2 > > > < b0 > > a1 > > > > > a2 > > : a4
pffiffiffiffiffiffiffi 1 ¼ k1 ðx ay þ k2 k2 tÞ; pffiffiffiffiffiffiffi ¼ k2 ðx ayÞ þ k2 t; pffiffiffiffiffi pffiffiffiffiffi ¼ k2 l3 ð aa3aþka2bl3 Þk2 aa0 k2 l3 ; 3 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ¼ ab1 þ ða3 b3 aÞl1 l1 3 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ¼ ab2 þ ða3 b3 aÞl2 l1 3 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ¼ ab4 þ ða3 b3 aÞl4 l1 3 ;
According to (3.2) and the particular solutions of (3.3) and (3.4) listed in Appendix, we can obtain the following complexiton solutions for (2 + 1)-dimensional Burgers equation. Family 1 When R < 0 and h0 ¼ 1; h2 ¼ ðm2 þ 1Þ; h4 ¼ m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 cdðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 ðm2 1Þsnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 cdðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 ðm2 1Þsnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 cdðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 ðm2 1Þsnðn2 Þ : ¼ b0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 cdðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 ðm2 1Þsnðn2 Þ
u1;1 ¼ a0 þ
v 1;1
ð3:5Þ
Family 2 When R < 0 and h0 ¼ 1 m2 ; h2 ¼ 2m2 1; h4 ¼ m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 cnðn2 Þ þ a3 Rcsch ð Rn1 Þ þ a4 snðn2 Þdnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 cnðn2 Þ þ l3 Rcsch2 ð Rn1 Þ þ l4 snðn2 Þdnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 cnðn2 Þ þ b3 Rcsch ð Rn1 Þ þ b4 snðn2 Þdnðn2 Þ ¼ b0 þ pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 cnðn2 Þ þ l3 Rcsch2 ð Rn1 Þ þ l4 snðn2 Þdnðn2 Þ
u1;2 ¼ a0 þ
v 1;2
ð3:6Þ
Family 3 When R < 0 and h0 ¼ m2 1; h2 ¼ 2 m2 ; h4 ¼ 1; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 dnðn2 Þ þ a3 Rcsch ð Rn1 Þ þ a4 m2 cnðn2 Þsnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 dnðn2 Þ þ l3 Rcsch2 ð Rn1 Þ þ l4 m2 cnðn2 Þsnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 dnðn2 Þ þ b3 Rcsch ð Rn1 Þ þ b4 m2 cnðn2 Þsnðn2 Þ ¼ b0 þ pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 dnðn2 Þ þ l3 Rcsch2 ð Rn1 Þ þ l4 m2 cnðn2 ÞSNðn2 Þ
u1;3 ¼ a0 þ
v 1;3
ð3:7Þ
Family 4 When R < 0 and h0 ¼ m2 ; h2 ¼ ðm2 þ 1Þ; h4 ¼ 1; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 dcðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 ð1 m2 Þscðn2 Þ ; u1;4 ¼ a0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 dcðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 ð1 m2 Þscðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi 2 b R cothð Rn1 Þ b2 dcðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 ð1 m2 Þscðn2 Þ : v 1;4 ¼ b0 þ 1pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 dcðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 ð1 m2 Þscðn2 Þ
ð3:8Þ
W. Li, H. Zhang / Commun Nonlinear Sci Numer Simulat 15 (2010) 1454–1461
1459
Family 5 When R < 0 and h0 ¼ m2 ; h2 ¼ 2m2 1; h4 ¼ 1 m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 ncðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 snðn2 Þdnðn2 Þ ; u1;5 ¼ a0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 l1 pR Rn1 Þ l4 snðn2 Þdnðn2 Þ ffiffiffiffiffiffifficothð pR ffiffiffiffiffiffiffin1 Þ l2 ncðn2 Þ þ l3 Rcsch2 ðpffiffiffiffiffiffiffi b R cothð Rn1 Þ b2 ncðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 snðn2 Þdnðn2 Þ : v 1;5 ¼ b0 þ 1pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ncðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 snðn2 Þdnðn2 Þ
ð3:9Þ
Family 6 When R < 0 and h0 ¼ 1; h2 ¼ 2 m2 ; h4 ¼ 1 m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 ndðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 m2 cnðn2 Þsnðn2 Þ u1;6 ¼ a0 þ pffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ndðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 m2 cnðn2 Þsnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi 2 b R cothð Rn1 Þ b2 ndðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 m2 cnðn2 Þsnðn2 Þ v 1;6 ¼ b0 þ 1pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ndðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 m2 cnðn2 Þsnðn2 Þ
ð3:10Þ
Family 7 When R < 0 and h0 ¼ 1; h2 ¼ ðm2 þ 1Þ; h4 ¼ 1 m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 scðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 dnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 scðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 dnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 scðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 dnðn2 Þ ¼ b0 þ pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 scðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 dnðn2 Þ
u1;7 ¼ a0 þ
v 1;7
ð3:11Þ
Family 8 When R < 0 and h0 ¼ 1; h2 ¼ 2m2 1; h4 ¼ m2 ðm2 1Þ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 sdðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 cnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 sdðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 cnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 sdðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 cnðn2 Þ : ¼ b0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 sdðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 cnðn2 Þ
u1;8 ¼ a0 þ
v 1;8
ð3:12Þ
Family 9 2 2 2 ; h2 ¼ 1þm ; h4 ¼ 1m ; h1 ¼ h3 ¼ 0, we obtain following solutions: When R < 0 and h0 ¼ 1m 4 2 4
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 ðncðn2 Þ scðn2 ÞÞ þ a3 Rcsch ð Rn1 Þ a4 ðsnðn2 Þdnðn2 Þ dnðn2 ÞÞ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ðncðn2 Þ scðn2 ÞÞ þ l3 Rcsch2 ð Rn1 Þ l4 ðsnðn2 Þdnðn2 Þ dnðn2 ÞÞ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 ðncðn2 Þ scðn2 ÞÞ þ b3 Rcsch ð Rn1 Þ b4 ðsnðn2 Þdnðn2 Þ dnðn2 ÞÞ : ¼ b0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ðncðn2 Þ scðn2 ÞÞ þ l3 Rcsch2 ð Rn1 Þ l4 ðsnðn2 Þdnðn2 Þ dnðn2 ÞÞ
u1;9 ¼ a0 þ
v 1;9
ð3:13Þ
Family 10 2 2 2 When R < 0 and h0 ¼ m4 ; h2 ¼ m 22 ; h4 ¼ m4 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 ðsnðn2 Þ icnðn2 ÞÞ þ a3 Rcsch ð Rn1 Þ a4 ðcnðn2 Þdnðn2 Þ isnðn2 Þdnðn2 ÞÞ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ðsnðn2 Þ icnðn2 ÞÞ þ l3 Rcsch2 ð Rn1 Þ l4 ðcnðn2 Þdnðn2 Þ isnðn2 Þdnðn2 ÞÞ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b R cothð Rn1 Þ b2 ðsnðn2 Þ icnðn2 ÞÞ þ b3 Rcsch ð Rn1 Þ b4 ðcnðn2 Þdnðn2 Þ isnðn2 Þdnðn2 ÞÞ : ð3:14Þ v 1;10 ¼ b0 þ 1pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ðsnðn2 Þ icnðn2 ÞÞ þ l3 Rcsch2 ð Rn1 Þ l4 ðcnðn2 Þdnðn2 Þ isnðn2 Þdnðn2 ÞÞ u1;10 ¼ a0 þ
Remark 5. We should point out that the solutions obtained in this paper are not only the above (3.5)–(3.14). From Tables 1 and 2, we can obtain 5 32 24 types of complexiton solutions of the (2 + 1)-dimensional Burgers equation. We only list some of them to show this methods efficiency in constructing complexiton solutions of NEEs. Remark 6. Introducing two independent sub-equations, a celebrated Riccati equation and a general elliptic equation, the formal solutions to the given NEEs (3.1) can be written by different combinations of solutions to (3.3) and solutions to (3.4). It should be pointed out that general solutions to (3.3) we used in the paper were first systematically presented by Ma and Fuchssteiner [20]. From Tables 1 and 2, we can easily see that hyperbolic function and elliptic function, trigonometric function, elliptic function and the square form of hyperbolic (solitary) function can appear in one solution at the same time which are also complexiton solutions. Although our method cannot recover complexiton solutions obtained by Ma’s method
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[18] and Lou’s method [19], new types of complexiton solutions obtained by us cannot be found by Ma’s method and Lou’s method. In particular, our solutions have not been obtained by any other tanh function methods. 4. Summary and discussion In summary, a new generalized sub-equations rational expansion method (GSRE) is presented to find new exact complexiton solutions of nonlinear evolution equations. The (2 + 1)-dimensional Burgers equation is chosen to illustrate the method such that some novel solutions are found which include novel types of complexiton solutions. Of course, the algorithm proposed above can also be applied to many other nonlinear evolution equations in mathematical physics. In this paper, we naturally present a more general ansätz. Therefore, for some nonlinear equations, more types of nontravelling complexiton solutions would be expected. This will be completed in following papers. Acknowledgement The work is partially supported by the National Key Basic Research Project of China under the Grant No. 2004CB318000. Appendix A See Tables 1 and 2.
Table 1 The general solutions to the celebrated Riccati equation (3.3). R
The solutions to (3.3) pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi R tanhð RnÞ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi R cothð RnÞ 1n pffiffiffi pffiffiffi R tanð RnÞ pffiffiffi pffiffiffi R cotð RnÞ
<0 <0 =0 >0 >0
Table 2 The general solutions to the general elliptic equation (3.4). The solutions to (3.4) pffiffiffiffiffiffiffiffiffi h1 h1 2h þ 2h sinð h2 nÞ 2 2 pffiffiffiffiffiffi h1 h1 2h2 þ 2h2 sinhð h2 nÞ qffiffiffiffiffiffiffiffi ffi ffiffiffiffiffi ffi p hh24 sechð h2 nÞ qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi h2 2h tanhð h22 nÞ 4 qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi hh24 secð h2 nÞ qffiffiffiffi qffiffiffiffiffiffi h2 tanð h22 nÞ 2h4
h0
h1
h2
h3
h4
0
h1
<0
0
0
0
h1
>0
0
0
0
0
>0
0
<0
h2 4h4
0
<0
0
>0
0
0
<0
0
>0
h22 4h4
0
>0
0
>0
1
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
m2 m2 1 1
0
1m2 4 1 4
ncðnÞ scðnÞ
0
m2 4
snðnÞ icnðnÞ pffiffiffiffi h 2 hh23 sech ð 2 2 ðnÞÞ pffiffiffiffiffiffiffi h hh23 sec2 ð 2 2 ðnÞÞ pffiffiffiffi h }ð 2 3 ; 4 hh23 ; 4 hh03 Þ
2
1 m2 m2 1 m2 m2 1 1 m2 1 1 m2 ðm2 1Þ 1 4
2
ðm þ 1Þ 2m2 1 2 m2 2 m2 2m2 1 2 m2 2 m2 2 m2 2m2 1 2m2 1 12m2 2 1þm2 2 m2 2 2 m2 2 2
1 m2 1 m2 1 1 m2 m2 ðm2 1Þ 1 1 4
1m2 4 m2 4 m2 4
0
0
0
>0
h3
0
0
0
<0
h3
0
h0
h1
0
>0
0
0 0
0
snðnÞ or cdðnÞ cnðnÞ dnðnÞ nsðnÞ or dcðnÞ ncðnÞ ndðnÞ csðnÞ scðnÞ sdðnÞ dsðnÞ nsðnÞ csðnÞ nsðnÞ dsðnÞ
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1461
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Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. New York: Cambridge University Press; 1991. Wahlquist HD, Estabrook FB. Bäcklund transformations for solitons of the Korteweg–de Vriu equation. Phys Rev Lett 1973;31:1386. Matveev VB, Salle MA. Darbooux transformation and soliton. Berlin: Springer; 1991. Hirota R. The direct method in soliton theory. New York: Cambridge University Press; 2004. Clarkson PA, Kruskal MD. New similarity reductions of Boussinesq equation. J Math Phys 1989;30:2202. Lou SY, Lu JZ. Special solutions from variable separation approach: Davey–Stewartson equation. J Phys A Math Gen 1996;29:4029. Conte R, Musette M. Painlevé analysis and Bäcklund transformation in the Kuramoto–Sivashinsky equation. J Phys A Math Gen 1989;22:169. Wang ML. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys Lett A 1996;216:67. Fan EG. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212. Fan EG, Chao L. Soliton solutions for the new complex version of a coupled KdV equation and a coupled MKdV equation. Phys Lett A 2001;285:373. Yan ZY. New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations. Phys Lett A 2001;292:100. Chen Y, Wang QZ. A new Riccati equation rational expansion method and its application. Naturforschung 2005;60a:1. Yan ZY, Zhang HQ. New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water. Phys Lett A 2001;285:355. Wang Q, Chen Y, Zhang HQ. Rational form solitary wave solutions and doubly periodic wave solutions to (1 + 1)-dimensional dispersive long wave equation. Commun Theor Phys (Beijing, China) 2005;43:975. Wang Q, Chen Y. A multiple Riccati equations rational expansion method and novel solutions of the Broer–Kaup–Kupershmidt system. Chaos, Solitons & Fractals 2006;30:197. Song LN, Wang Q, Zheng Y, Zhang HQ. A new extended Riccati equation rational expansion method and its application. Chaos, Solitons & Fractals 2007;31:548. Ma WX. Travelling wave solutions to a seventh order generalized KdV equation. Phys Lett A 1993;180:221. Ma WX. Complexiton solutions to the Korteweg–de Vries equation. Phys Lett A 2002;301:35. Lou SY, Hu HC, Tang XY. Interactions among periodic waves and solitary waves of the (n + 1)-dimensional sine-Gordon field. Phys Rev E 2005;71:036604. Ma WX, Fuchssteiner B. Explicit and exact solutions to a Kolmogorov–PetrovskII–Piskunov equation. Int J Nonlinear Mech 1996;31:329.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
A generalized sub-equations rational expansion method for nonlinear evolution equations Wenting Li *, Hongqing Zhang Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 18 February 2009 Received in revised form 15 May 2009 Accepted 20 June 2009 Available online 7 July 2009 PACS: 02.30.Jr 05.45.Yv Keywords: Nonlinear evolution equations Generalized sub-equations rational expansion method (2 + 1)-Dimensional Burgers equations Complexiton solutions
a b s t r a c t In this paper, based on symbolic computation and the idea of rational expansion method, a generalized sub-equations rational expansion method (GSRE) is devised to uniformly construct a series of exact complexiton solutions for nonlinear evolution equations. Compared with most existing tanh function methods and other sophisticated methods, the proposed method not only recover some known solutions, but also find some new and general solutions which include many new types of complexiton solutions: the combination of hyperbolic (and square form) function and elliptic function, trigonometric (and square form) function and elliptic function. The efficiency of the method can be demonstrated on (2 + 1)-dimensional Burgers equations. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Constructing exact solutions for nonlinear evolution equations (NEEs) has long been a major concern for both mathematicians and physicists. On the one hand, due to their occurrence in many fields of science, in physics as well as in chemistry or biology and the interesting features and rich variety of their solutions, on the other hand, due to the availability of computer systems like Maple or Mathematica which allow us to perform some complicated and tedious algebraic calculation and differential calculation on a computer, at the same time help us to find new exact solution of NEEs. Up to now, there exist many powerful methods to obtain exact solutions of NEEs related to nonlinear problems, such as inverse scattering method [1], Bäcklund transformation [2], Darboux transformation [3], Hirota bilinear method [4], similarity reductions method [5], variable separation approach [6], Painlevé analysis method [7], homogeneous balance method [8], various tanh function methods [9–16], sech-function method [17] and so on. Among them, the tanh function method is considered to be one of the most straightforward and effective algebraic algorithm to obtain exact solutions for lots of NEEs. As we know, when applying the tanh function method, the choice of an appropriate sub-equation is of great importance. Much work has been concentrated on solving more solutions of the sub-equation. Recently, Ma [18] found a novel class of explicit exact solutions to the Korteweg–de Vries equation through its bilinear form and defined the solutions as complexiton solutions. In Ref. [19], Lou et al. presented and answered the problem ‘‘Are there any exact explicit multiple periodic wave solutions and periodic–solitary wave solutions for the nSG equation” with help of the mapping relations among the sine-Gordon field equation and the cubic nonlinear Klein–Gordon fields. Such * Corresponding author. E-mail address: [email protected] (W. Li). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.06.030
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solutions possess singularities of combinations of trigonometric function waves and exponential function waves which have different travelling speeds of new type. Above mentioned solutions of nonlinear evolution equations have a common character: combination of trigonometric function waves and exponential function waves. For unification and conciseness, so we call the solutions obtained by Ma and the solutions obtained by Lou as complexiton solutions. More recently, Wang and Chen [15] presented that the solutions of two different Riccati equations with different parameters are used as two variables in the components of finite rational expansion, which further exceeded the applicability of the tanh methods. Much work has been concentrated on solving more solutions of the sub-equation. Here, we not only focus on improving the general formal travelling transformation, but also the multiform choice of sub-equation. The present work is motivated by the desire to present a more generalized sub-equations to construct more types and more general formal solutions which contain not only the results obtained by using the methods [9–16] but also other types of solutions including many new types of complexiton solutions: various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and Jacobi elliptic double periodic function solutions, various combination of hyperbolic and rational function solutions, etc. For illustration, we apply the new method GSRE to solve (2 + 1)-dimensional Burgers equation and successfully construct new and more complexiton solutions, which have not been found before. The rest of this paper is organized as follows. In Section 2, our method GSRE is summarized. In Section 3, it is applied to the (2 + 1)-dimensional Burgers equation and bring out rich complexiton solutions for this model. A short summary and discussion are given in final. 2. Summary of the generalized sub-equations rational expansion method In the following we would like to outline the main steps of our general method: Step 1. Given a system of polynomial nonlinear partial differential equations with some physical fields ui ðx1 ; . . . ; xi ; . . . ; xn ; tÞ ði ¼ 1; . . . ; nÞ in ðn þ 1Þ variables x1 ; . . . ; xn ; t,
Hi ðui ; uit ; uix1 ; uix2 ; . . . ; uixn ; uitt ; uix1 t ; uix2 t ; . . . ; uixn t ; uix1 x1 ; . . .Þ ¼ 0:
ð2:1Þ
Step 2. We introduce a new ansätz in terms of finite rational formal expansion in the following forms:
ðmÞ ui ðx1 ; . . . ; xn ; tÞ ¼ Rui /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ; /01 ðn1 Þ; . . . ; /0n ðnn Þ; . . . ; /1 ðn1 Þ; . . . ; /ðmÞ n ðnn Þ where Rui ðÞ is the rational formal function of /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ; P ni ¼ nj¼1 ci;j xj þ ci;0 t ¼ ci;1 x1 þ . . . þ ci;n xn þ ci;0 t ði ¼ 1; . . . ; nÞ are satisfying
0 c1;0 1 n1 B B . C B c2;0 B. C ¼ B @ . A B .. @. nn cn;0 0
10 1 c1;n t B C c2;n C C B x1 C CB C .. CB .. C; . A@ . A
c1;1 c2;1 .. .
.. .
cn;1
cn;n
ð2:2Þ
ðmÞ /01 ðn1 Þ; . . . ; /0n ðnn Þ; . . . ; /1 ðn1 Þ; . . . ; /ðmÞ n ðnn Þ,
and
ð2:3Þ
xn
and ci;j ði ¼ 1; . . . ; n; j ¼ 0; 1; . . . ; nÞ are arbitrary constants. And /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ satisfy: m
1. /ðmÞ ðnÞ ¼ d dn/ðnÞ m ; 2. /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ are arbitrary function; 3. /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ satisfy various sub-equation, such as Riccati equation, projective Riccati equation, elliptic equation, Bessel equation, Klein–Gordon equation and so on. That is, all derivatives of /i ðni Þ ði ¼ 1; . . . ; nÞ with respect to ni are the rational formal function of /1 ðn1 Þ; /2 ðn2 Þ; . . . ; /n ðnn Þ. For example, (1) When we take n ¼ 1; m ¼ 0,
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X
aij ð/1 ðn1 ÞÞj
j¼1
ðb0 þ b1 /1 ðn1 ÞÞj
ð2:4Þ
;
where a0 ; a1 ; b0 and b1 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. (2) When we take n ¼ 1; m ¼ 1,
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X j¼1
P
ij r j1 þr 0j1 ¼j ar j1 r 0
j1
r0
ð/1 ðn1 ÞÞrj1 ð/01 ðn1 ÞÞ j1
j ; 0 0 1 1 b0 þ b1 /1 ðn1 Þ þ b1 /01 ðn1 Þ þ b1 /1 ðn þ b 0 1 Þ / ðn Þ 1 1
ð2:5Þ
1
0
0
where a0 ; a1 ; a01 ; b0 ; b1 ; b1 ; b1 and b1 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. (3) When we take n ¼ 1; m ¼ 2,
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ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X
P
r0j1 r 00j1 r j1 ij 0 00 r j1 þr0j1 þr 00j1 ¼j ar j1 r 0 r 00 ð/1 ðn1 ÞÞ ð/1 ðn1 ÞÞ ð/1 ðn1 ÞÞ j1 j1
j¼1
j 0 00 b0 þ b1 /1 ðn1 Þ þ b1 /01 ðn1 Þ þ b1 /001 ðn1 Þ
0
ð2:6Þ
;
00
where a0 ; a1 ; a01 ; a001 ; b0 ; b1 ; b1 and b1 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. (4) When we take n ¼ 2; m ¼ 0,
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X j¼1
P rj1 r j2 ij r j1 þr j2 ¼j ar j1 r j2 ð/1 ðn1 ÞÞ ð/2 ðn2 ÞÞ j ; 1 1 b0 þ b1 /1 ðn1 Þ þ b2 /2 ðn2 Þ þ b1 /1 ðn þ b 2 /2 ðn2 Þ 1Þ
ð2:7Þ
where a0 ; a1 ; a2 ; b0 ; b1 ; b2 ; b1 and b2 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. (5) When we take n ¼ 2; m ¼ 1,
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X
P
r0j1 r 0j2 r j1 r j2 ij 0 0 r j1 þr0j1 þr j2 þr0j2 ¼j ar j1 r j2 r0 r 0 ð/1 ðn1 ÞÞ ð/2 ðn2 ÞÞ ð/1 ðn1 ÞÞ ð/2 ðn2 ÞÞ j1 j2 0
0
ðb0 þ b1 /1 ðn1 Þ þ b2 /2 ðn2 Þ þ b1 /01 ðn1 Þ þ b2 /02 ðn2 ÞÞj
j¼1
0
;
ð2:8Þ
0
where a0 ; a1 ; a2 ; a01 ; a02 ; b0 ; b1 ; b2 ; b1 and b2 are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t, or arbitrary constants to be determined later. Step 3. To determine the balanced parameter M i . Determine the Mi of the rational formal polynomial solutions by respectively balancing the highest nonlinear terms and the highest-order partial derivative terms in the given equations, and then give the formal solutions. Step 4. Substitute (2.2) into (2.1) with /i ðni Þ ði ¼ 1; . . . ; nÞ satisfying the sub-equations which will be chosen in Step 6. ðpÞr Then set all coefficients of /i ðni Þ ði ¼ 1; 2; . . . ; n; p ¼ 0; 1; . . . ; m; r ¼ 0; 1; . . .Þ of the resulting system’s numerator to be zero to get an over-determined system of nonlinear algebraic equations with respect to ci;j ði ¼ 1; . . . ; n; j ¼ 0; 1; . . . ; nÞ, and the parameters of the function Rui ðÞ. Step 5. To solve the parameters. By solving the over-determined system of nonlinear algebraic equations by use of symbolic computation system Maple, we end up with the explicit expressions for ci;j ði ¼ 1; . . . ; n; j ¼ 0; 1; . . . ; nÞ, and the parameters of the function Rui ðÞ. Step 6. To choose sub-equations. According to system (2.2), the conclusions in Step 5 and some particular solutions of the chosen sub-equations, we can obtain rational formal exact solutions of system (2.1). Remark 1. To our knowledge, most of the tanh function methods can be summarized as above the generalized sub-equations rational expansion method. What’s more, solutions obtained by this method include the square form of hyperbolic (solitary) function and triangular periodic functions at the same time, which are different from the solutions in [15]. Remark 2. Eq. (2.2) can be changed into the following form:
ui ðx1 ; . . . ; xn ; tÞ ¼ ai0 þ
Mi X j¼1
PP P n m s¼1
PP P n m t¼1
where ai0 ; aijQn Qm s¼1
ðpÞ r p¼0 js
ij ; bQn Qm t¼1
ðpÞ q p¼0 jt
ðpÞ r p¼0 js
ðpÞ q p¼0 jt
ij aQ n Qm s¼1
ij bQn Qm t¼1
ðpÞ
rj
Qn Qm
js ðpÞ p¼0 ð/s ðns ÞÞ
s¼1
p¼0 s
ðpÞ q p¼0 jt
Qn Qm t¼1
ðpÞ jt p¼0 ð/t ðnt ÞÞ
þ bj0
;
ð2:9Þ
; bj0 and nk ðp ¼ 0; 1; . . . ; m; rjs ; qjt ¼ 1; . . . ; j; s; t ¼ 1; . . . ; n; i ¼ 1; 2; . . . ; k ¼ 1; . . . ; nÞ
are differentiable functions in ðn þ 1Þ variables x1 ; . . . ; xn ; t or arbitrary constants to be determined later. Remark 3. We should point out that in Eq. (2.9) n Y m Y
ðmÞ
0 0 ðmÞ /ðpÞ s ðns Þ ¼ /1 ðn1 Þ/2 ðn2 Þ . . . /n ðnn Þ/1 ðn1 Þ . . . /n ðnn Þ . . . /1 ðn1 Þ . . . /n ðnn Þ;
s¼1 p¼0 n Y m Y
ðpÞ
ðmÞ
ðmÞ
r js ¼ r j1 r j2 . . . rjn r 0j1 . . . r 0jn . . . r j1 . . . r jn ;
s¼1 p¼0 n X m X
ðpÞ
ðmÞ
ðmÞ
r js ¼ r j1 þ rj2 þ þ rjn þ r0j1 þ þ r 0jn þ þ rj1 þ r jn :
ð2:10Þ
s¼1 p¼0 ðmÞ
ðmÞ
and r j1 ; r j2 ; . . . ; r jn ; r 0j1 ; . . . ; r 0jn ; . . . ; r j1 ; . . . ; rjn are not only positive integers, but also can be negative integers in Eq. (2.4).
W. Li, H. Zhang / Commun Nonlinear Sci Numer Simulat 15 (2010) 1454–1461
1457
ðpÞ
Remark 4. /i ðni Þ ði ¼ 1; . . . ; n; p ¼ 0; 1; . . . ; mÞ satisfy sub-equation, such as elliptic equation, Riccati equation, projective Riccati equations, Bessel equation, and so on. They not only satisfy the same sub-equation, but also satisfy the different sub-equations. The appeal and success of our method lie on the fact that we choose different sub-equations. Because each sub-equation has especial formal solutions, the combination of some solutions can construct various formal complexiton solutions. In particular, we do not take the previous travelling wave transformation, but take difference travelling wave ðpÞ transformation in /i ðni Þ ði ¼ 1; . . . ; n; p ¼ 0; 1; . . . ; mÞ, so the solutions have different wave speed. And our method is a unified straightforward and pure algebraic algorithm to integrable equations and non-integrable equations, which is implemented in a computer algebraic system and can be easily extended to other integrable systems and non-integrable systems. 3. Exact complexiton solutions of the (2 + 1)-dimensional Burgers equation In this section we would like to apply our method GSRE to obtain complexiton solutions of the (2 + 1)-dimensional Burgers equation which reads
ut þ uuy þ av ux þ buyy þ abuxx ¼ 0; ux v y ¼ 0;
ð3:1Þ
where u ¼ uðx; y; tÞ; v ¼ v ðx; y; tÞ, and a; b–0 are all constants. Balancing the highest nonlinear terms with the highest-order partial derivative terms in (3.1) and we suppose the directive order m ¼ 1, so (3.1) has the following rational formal exact solutions:
a1 /1 ðn1 Þ þ a2 /2 ðn2 Þ þ a3 /01 ðn1 Þ þ a4 /02 ðn2 Þ ; l0 þ l1 /1 ðn1 Þ þ l2 /2 ðn2 Þ þ l3 /01 ðn1 Þ þ l4 /02 ðn2 Þ b1 /1 ðn1 Þ þ b2 /2 ðn2 Þ þ b3 /01 ðn1 Þ þ b4 /02 ðn2 Þ ; v ðx; y; tÞ ¼ b0 þ l0 þ l1 /1 ðn1 Þ þ l2 /2 ðn2 Þ þ l3 /01 ðn1 Þ þ l4 /02 ðn2 Þ uðx; y; tÞ ¼ a0 þ
ð3:2Þ
where n1 ¼ k1 x þ l1 y þ k1 t; n2 ¼ k2 x þ l2 y þ k2 t; l0 ; l1 ; l2 ; l3 ; l4 ; a0 ; a1 ; a2 ; a3 ; a4 ; b0 ; b1 ; b2 ; b3 and b4 are constants to be determined later. For obtaining our ideal complexiton solutions, we choose the celebrated Riccati equation (1) and the general elliptic equation (2) as sub-equations. (1)
/0 ðnÞ ðR þ /2 ðnÞÞ ¼ 0;
ð3:3Þ
(2)
/0 ðnÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh0 þ h1 /ðnÞ þ h2 /2 ðnÞ þ h3 /3 ðnÞ þ h4 /4 ðnÞÞ ¼ 0;
ð3:4Þ
where R; h0 ; h1 ; h2 ; h3 and h4 are arbitrary constants. Then we let the new variables /1 ðn1 Þ and /2 ðn2 Þ satisfy (3.3) and (3.4), respectively. With the aid of Maple, substituting (3.2) along with (3.3) and (3.4) into (3.1) and setting the coefficients of these terms /i1 ðn1 Þ/j2 ðn2 Þ ði; j ¼ 0; 1; . . .Þ to be zero yields a set of over-determined algebraic equations with respect to k1 ; l1 ; k1 ; k2 ; l2 ; k2 ; a0 ; a1 ; a2 ; a3 ; a4 ; b0 ; b1 ; b2 ; b3 ; b4 ; l0 ; l1 ; l2 ; l3 and l4 . By use of the Maple soft package ‘‘Charsets” by Dongming Wang, which is based on the Wu-elimination method, solving the over-determined algebraic equations, we get the following cases: Case 1
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi k2 l3 ð aa3 þ ab3 Þk2 aa0 k2 l3 ; l2 ¼ ak2 ; k2 ¼ k2 ; ak2 l3 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a2 ¼ ab2 þ ða3 b3 aÞl2 l1 a1 ¼ ab1 þ ða3 b3 aÞl1 l1 3 ; 3 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 1 a4 ¼ ab4 þ ða3 b3 aÞl4 l3 ; l0 ¼ 0; k1 ¼ k1 k2 k2 ; l1 ¼ ak1 ; b0 ¼
where k1 ; k2 ; a0 ; a3 ; b1 ; b2 ; b3 ; b4 ; Case 2
b0 ¼
k1
k1 ¼ k 1 ;
pffiffiffiffiffiffiffi aa0 k1 ; ak1 1
l1 ; l2 ; l3 and l4 are all arbitrary constants.
pffiffiffiffiffiffiffi l1 ¼ ak1 ;
k2 ¼ k1 k2 k1 ;
a1 a2 b1 ¼ pffiffiffiffiffiffiffi ; b2 ¼ pffiffiffiffiffiffiffi ; a a pffiffiffiffiffiffiffi a3 a4 l2 ¼ ak2 ; b3 ¼ pffiffiffiffiffiffiffi ; b4 ¼ pffiffiffiffiffiffiffi ; a a
where k1 ; k2 ; a0 ; b0 ; a1 ; a2 ; a3 ; a4 ;
l0 ; l1 ; l2 ; l3 and l4 are all arbitrary constants.
1458
W. Li, H. Zhang / Commun Nonlinear Sci Numer Simulat 15 (2010) 1454–1461
Case 3
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi k1 ¼ ða1 b1 aÞl1 l1 a4 ¼ ab4 þ ða1 b1 aÞl4 l1 1 þ ða0 b0 aÞl1 ; 1 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 k2 ¼ ða1 b1 aÞl2 l1 þ ða b a ; a ¼ a þ ða b a l l Þl b Þ 0 0 2 2 2 1 1 1 2 1 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a3 ¼ ab3 ; l1 ¼ ak1 ; l2 ¼ ak2 ; l0 ¼ 0; l3 ¼ 0;
where a0 ; a1 ; b0 ; b1 ; b2 ; b3 ; b4 ; k1 ; k2 ; Case 4
l1 ; l2 and l4 are all arbitrary constants.
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi k1 ¼ ða2 b2 aÞl1 l1 a1 ¼ ab1 þ ða2 b2 aÞl1 l1 2 þ ða0 b0 aÞl1 ; 2 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 k2 ¼ ða2 b2 aÞl2 l1 þ ða b a ; a ¼ a þ ða b a l l Þl b Þ 0 0 2 3 3 2 2 2 3 2 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi a4 ¼ ab4 ; l1 ¼ ak1 ; l2 ¼ ak2 ; l0 ¼ 0; l4 ¼ 0; where a0 ; a2 ; b0 ; b1 ; b2 ; b3 ; b4 ; k1 ; k2 ;
l1 ; l2 and l3 are all arbitrary constants.
Note. Since the solutions obtained here are so many, we just list a new type of solutions corresponding to Case 1 for the (2 + 1)-dimensional Burgers equation to illustrate the efficiency of our method. Case 1. k1 ; k2 ; a0 ; a3 ; b1 ; b2 ; b3 ; b4 ; l1 ; l2 ; l3 and l4 are all arbitrary constants, and
8 > n1 > > > > > n2 > > > < b0 > > a1 > > > > > a2 > > : a4
pffiffiffiffiffiffiffi 1 ¼ k1 ðx ay þ k2 k2 tÞ; pffiffiffiffiffiffiffi ¼ k2 ðx ayÞ þ k2 t; pffiffiffiffiffi pffiffiffiffiffi ¼ k2 l3 ð aa3aþka2bl3 Þk2 aa0 k2 l3 ; 3 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ¼ ab1 þ ða3 b3 aÞl1 l1 3 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ¼ ab2 þ ða3 b3 aÞl2 l1 3 ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ¼ ab4 þ ða3 b3 aÞl4 l1 3 ;
According to (3.2) and the particular solutions of (3.3) and (3.4) listed in Appendix, we can obtain the following complexiton solutions for (2 + 1)-dimensional Burgers equation. Family 1 When R < 0 and h0 ¼ 1; h2 ¼ ðm2 þ 1Þ; h4 ¼ m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 cdðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 ðm2 1Þsnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 cdðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 ðm2 1Þsnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 cdðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 ðm2 1Þsnðn2 Þ : ¼ b0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 cdðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 ðm2 1Þsnðn2 Þ
u1;1 ¼ a0 þ
v 1;1
ð3:5Þ
Family 2 When R < 0 and h0 ¼ 1 m2 ; h2 ¼ 2m2 1; h4 ¼ m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 cnðn2 Þ þ a3 Rcsch ð Rn1 Þ þ a4 snðn2 Þdnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 cnðn2 Þ þ l3 Rcsch2 ð Rn1 Þ þ l4 snðn2 Þdnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 cnðn2 Þ þ b3 Rcsch ð Rn1 Þ þ b4 snðn2 Þdnðn2 Þ ¼ b0 þ pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 cnðn2 Þ þ l3 Rcsch2 ð Rn1 Þ þ l4 snðn2 Þdnðn2 Þ
u1;2 ¼ a0 þ
v 1;2
ð3:6Þ
Family 3 When R < 0 and h0 ¼ m2 1; h2 ¼ 2 m2 ; h4 ¼ 1; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 dnðn2 Þ þ a3 Rcsch ð Rn1 Þ þ a4 m2 cnðn2 Þsnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 dnðn2 Þ þ l3 Rcsch2 ð Rn1 Þ þ l4 m2 cnðn2 Þsnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 dnðn2 Þ þ b3 Rcsch ð Rn1 Þ þ b4 m2 cnðn2 Þsnðn2 Þ ¼ b0 þ pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 dnðn2 Þ þ l3 Rcsch2 ð Rn1 Þ þ l4 m2 cnðn2 ÞSNðn2 Þ
u1;3 ¼ a0 þ
v 1;3
ð3:7Þ
Family 4 When R < 0 and h0 ¼ m2 ; h2 ¼ ðm2 þ 1Þ; h4 ¼ 1; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 dcðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 ð1 m2 Þscðn2 Þ ; u1;4 ¼ a0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 dcðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 ð1 m2 Þscðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi 2 b R cothð Rn1 Þ b2 dcðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 ð1 m2 Þscðn2 Þ : v 1;4 ¼ b0 þ 1pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 dcðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 ð1 m2 Þscðn2 Þ
ð3:8Þ
W. Li, H. Zhang / Commun Nonlinear Sci Numer Simulat 15 (2010) 1454–1461
1459
Family 5 When R < 0 and h0 ¼ m2 ; h2 ¼ 2m2 1; h4 ¼ 1 m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 ncðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 snðn2 Þdnðn2 Þ ; u1;5 ¼ a0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 l1 pR Rn1 Þ l4 snðn2 Þdnðn2 Þ ffiffiffiffiffiffifficothð pR ffiffiffiffiffiffiffin1 Þ l2 ncðn2 Þ þ l3 Rcsch2 ðpffiffiffiffiffiffiffi b R cothð Rn1 Þ b2 ncðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 snðn2 Þdnðn2 Þ : v 1;5 ¼ b0 þ 1pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ncðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 snðn2 Þdnðn2 Þ
ð3:9Þ
Family 6 When R < 0 and h0 ¼ 1; h2 ¼ 2 m2 ; h4 ¼ 1 m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 ndðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 m2 cnðn2 Þsnðn2 Þ u1;6 ¼ a0 þ pffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ndðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 m2 cnðn2 Þsnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi 2 b R cothð Rn1 Þ b2 ndðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 m2 cnðn2 Þsnðn2 Þ v 1;6 ¼ b0 þ 1pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ndðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 m2 cnðn2 Þsnðn2 Þ
ð3:10Þ
Family 7 When R < 0 and h0 ¼ 1; h2 ¼ ðm2 þ 1Þ; h4 ¼ 1 m2 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 scðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 dnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 scðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 dnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 scðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 dnðn2 Þ ¼ b0 þ pffiffiffiffiffiffiffi : pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 scðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 dnðn2 Þ
u1;7 ¼ a0 þ
v 1;7
ð3:11Þ
Family 8 When R < 0 and h0 ¼ 1; h2 ¼ 2m2 1; h4 ¼ m2 ðm2 1Þ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 sdðn2 Þ þ a3 Rcsch ð Rn1 Þ a4 cnðn2 Þ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 sdðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 cnðn2 Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 sdðn2 Þ þ b3 Rcsch ð Rn1 Þ b4 cnðn2 Þ : ¼ b0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 sdðn2 Þ þ l3 Rcsch2 ð Rn1 Þ l4 cnðn2 Þ
u1;8 ¼ a0 þ
v 1;8
ð3:12Þ
Family 9 2 2 2 ; h2 ¼ 1þm ; h4 ¼ 1m ; h1 ¼ h3 ¼ 0, we obtain following solutions: When R < 0 and h0 ¼ 1m 4 2 4
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 ðncðn2 Þ scðn2 ÞÞ þ a3 Rcsch ð Rn1 Þ a4 ðsnðn2 Þdnðn2 Þ dnðn2 ÞÞ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ðncðn2 Þ scðn2 ÞÞ þ l3 Rcsch2 ð Rn1 Þ l4 ðsnðn2 Þdnðn2 Þ dnðn2 ÞÞ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b1 R cothð Rn1 Þ b2 ðncðn2 Þ scðn2 ÞÞ þ b3 Rcsch ð Rn1 Þ b4 ðsnðn2 Þdnðn2 Þ dnðn2 ÞÞ : ¼ b0 þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ðncðn2 Þ scðn2 ÞÞ þ l3 Rcsch2 ð Rn1 Þ l4 ðsnðn2 Þdnðn2 Þ dnðn2 ÞÞ
u1;9 ¼ a0 þ
v 1;9
ð3:13Þ
Family 10 2 2 2 When R < 0 and h0 ¼ m4 ; h2 ¼ m 22 ; h4 ¼ m4 ; h1 ¼ h3 ¼ 0, we obtain following solutions:
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 a1 R cothð Rn1 Þ a2 ðsnðn2 Þ icnðn2 ÞÞ þ a3 Rcsch ð Rn1 Þ a4 ðcnðn2 Þdnðn2 Þ isnðn2 Þdnðn2 ÞÞ ; pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ðsnðn2 Þ icnðn2 ÞÞ þ l3 Rcsch2 ð Rn1 Þ l4 ðcnðn2 Þdnðn2 Þ isnðn2 Þdnðn2 ÞÞ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2 b R cothð Rn1 Þ b2 ðsnðn2 Þ icnðn2 ÞÞ þ b3 Rcsch ð Rn1 Þ b4 ðcnðn2 Þdnðn2 Þ isnðn2 Þdnðn2 ÞÞ : ð3:14Þ v 1;10 ¼ b0 þ 1pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi l1 R cothð Rn1 Þ l2 ðsnðn2 Þ icnðn2 ÞÞ þ l3 Rcsch2 ð Rn1 Þ l4 ðcnðn2 Þdnðn2 Þ isnðn2 Þdnðn2 ÞÞ u1;10 ¼ a0 þ
Remark 5. We should point out that the solutions obtained in this paper are not only the above (3.5)–(3.14). From Tables 1 and 2, we can obtain 5 32 24 types of complexiton solutions of the (2 + 1)-dimensional Burgers equation. We only list some of them to show this methods efficiency in constructing complexiton solutions of NEEs. Remark 6. Introducing two independent sub-equations, a celebrated Riccati equation and a general elliptic equation, the formal solutions to the given NEEs (3.1) can be written by different combinations of solutions to (3.3) and solutions to (3.4). It should be pointed out that general solutions to (3.3) we used in the paper were first systematically presented by Ma and Fuchssteiner [20]. From Tables 1 and 2, we can easily see that hyperbolic function and elliptic function, trigonometric function, elliptic function and the square form of hyperbolic (solitary) function can appear in one solution at the same time which are also complexiton solutions. Although our method cannot recover complexiton solutions obtained by Ma’s method
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[18] and Lou’s method [19], new types of complexiton solutions obtained by us cannot be found by Ma’s method and Lou’s method. In particular, our solutions have not been obtained by any other tanh function methods. 4. Summary and discussion In summary, a new generalized sub-equations rational expansion method (GSRE) is presented to find new exact complexiton solutions of nonlinear evolution equations. The (2 + 1)-dimensional Burgers equation is chosen to illustrate the method such that some novel solutions are found which include novel types of complexiton solutions. Of course, the algorithm proposed above can also be applied to many other nonlinear evolution equations in mathematical physics. In this paper, we naturally present a more general ansätz. Therefore, for some nonlinear equations, more types of nontravelling complexiton solutions would be expected. This will be completed in following papers. Acknowledgement The work is partially supported by the National Key Basic Research Project of China under the Grant No. 2004CB318000. Appendix A See Tables 1 and 2.
Table 1 The general solutions to the celebrated Riccati equation (3.3). R
The solutions to (3.3) pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi R tanhð RnÞ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi R cothð RnÞ 1n pffiffiffi pffiffiffi R tanð RnÞ pffiffiffi pffiffiffi R cotð RnÞ
<0 <0 =0 >0 >0
Table 2 The general solutions to the general elliptic equation (3.4). The solutions to (3.4) pffiffiffiffiffiffiffiffiffi h1 h1 2h þ 2h sinð h2 nÞ 2 2 pffiffiffiffiffiffi h1 h1 2h2 þ 2h2 sinhð h2 nÞ qffiffiffiffiffiffiffiffi ffi ffiffiffiffiffi ffi p hh24 sechð h2 nÞ qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi h2 2h tanhð h22 nÞ 4 qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi hh24 secð h2 nÞ qffiffiffiffi qffiffiffiffiffiffi h2 tanð h22 nÞ 2h4
h0
h1
h2
h3
h4
0
h1
<0
0
0
0
h1
>0
0
0
0
0
>0
0
<0
h2 4h4
0
<0
0
>0
0
0
<0
0
>0
h22 4h4
0
>0
0
>0
1
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
m2 m2 1 1
0
1m2 4 1 4
ncðnÞ scðnÞ
0
m2 4
snðnÞ icnðnÞ pffiffiffiffi h 2 hh23 sech ð 2 2 ðnÞÞ pffiffiffiffiffiffiffi h hh23 sec2 ð 2 2 ðnÞÞ pffiffiffiffi h }ð 2 3 ; 4 hh23 ; 4 hh03 Þ
2
1 m2 m2 1 m2 m2 1 1 m2 1 1 m2 ðm2 1Þ 1 4
2
ðm þ 1Þ 2m2 1 2 m2 2 m2 2m2 1 2 m2 2 m2 2 m2 2m2 1 2m2 1 12m2 2 1þm2 2 m2 2 2 m2 2 2
1 m2 1 m2 1 1 m2 m2 ðm2 1Þ 1 1 4
1m2 4 m2 4 m2 4
0
0
0
>0
h3
0
0
0
<0
h3
0
h0
h1
0
>0
0
0 0
0
snðnÞ or cdðnÞ cnðnÞ dnðnÞ nsðnÞ or dcðnÞ ncðnÞ ndðnÞ csðnÞ scðnÞ sdðnÞ dsðnÞ nsðnÞ csðnÞ nsðnÞ dsðnÞ
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