Journal of the Physical Society of Japan Vol. 74, No. 9, September, 2005, pp. 2383–2385 #2005 The Physical Society of Japan
LETTERS
Exact Solutions for the Nonisospectral Kadomtshev–Petviashvili Equation Shu-fang D ENG, Da-jun Z HANG1 y and Deng-yuan C HEN1 1
Institute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China Department of Mathematics, Shanghai University, Shanghai 200436, People’s Republic of China (Received August 4, 2003; accepted December 15, 2004)
The nonisospectral Kadomtshev–Petviashvili (KP) equation is solved by the Hirota method and Wronskian technique. Exact solutions that possess soliton characters with nonisospectral properties are obtained. In addition, rational and mixed solutions are derived. We also obtain a new molecular equation that admits a solution in the Wronskian form. KEYWORDS: nonisospectral KP equation, Hirota method, Wronskian technique DOI: 10.1143/JPSJ.74.2383
In 1971, Hirota1) first proposed the formal perturbation technique, called the Hirota method, to obtain N-soliton solutions of the KdV equation. The solution obtained in this manner is written as a polynomial of exponential functions. The soliton solution can also be written in the Wronskian form, which was first introduced by Satsuma2) in 1979. Further more, Freeman and Nimmo3) developed a general verification procedure, which we call the Wronskian technique. Many soliton equations have been revealed to be exactly solvable by these two direct methods.4–7) The Hirota method can also be applied to equations with nonisospectral properties, for example, the KdV equation with loss and nonuniformity terms.8) In this letter, we would like to consider the nonisospectral Kadomtshev–Petviashvili (KP) equation9) in the above two direct methods. The bilinear form of the nonisospectral KP equation is given, and one- and two-soliton solutions are obtained by the standard Hirota method. A general formula that denotes higher order solutions is also given. We provide some figures to show the shapes and motions of some solutions we have obtained. We can observe in these figures the soliton characters with nonisospectral properties. We also derive new solutions in the Wronskian form. Further more we obtain rational and mixed rational-soliton solutions. It is interesting that, as a by-product, a new molecular equation that admits a solution in the Wronskian form is found. Here, we present the bilinear form of the nonisospectral KP equation and derive the soliton solutions by the Hirota method. The nonisospectral KP equation9) is 1
1
4ut þ yðuxxx þ 6uux þ 3@ uyy Þ þ 2xuy þ 4@ uy ¼ 0;
ð1Þ
eq. (1) can be transformed into the bilinear form 4Dx Dt f f þ yðD4x f f þ 3D2y f f Þ þ 2xDx Dy f f þ 4fy f ¼ 0; where D is the well-known Hirota bilinear operator n l m n Dlx Dm y Dt a b ¼ ð@x @x0 Þ ð@y @y0 Þ ð@t @t0 Þ
aðx; y; tÞbðx0 ; y0 ; t0 Þjx0 ¼x;y0 ¼y;t0 ¼t : This bilinear equation further suggests ð1Þ 4fxtð1Þ þ yð fxxxx þ 3fyyð1Þ Þ þ 2xfxyð1Þ þ 2fyð1Þ ¼ 0;
¼ 4Dx Dt f ð1Þ f ð1Þ yðD4x f ð1Þ f ð1Þ þ 3D2y f ð1Þ f ð1Þ Þ 2xDx Dy f ð1Þ f ð1Þ 4fyð1Þ f ð1Þ ;
Corresponding author. E-mail:
[email protected] E-mail:
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y
ð5bÞ
; under the perturbation expansion f ðx; y; tÞ ¼ 1 þ f ð1Þ þ f ð2Þ 2 þ f ð3Þ 3 þ :
ð6Þ
Taking f ð1Þ ¼ e1 ; 1 ¼ K1 ðtÞ½x þ P1 ðtÞy þ 1ð0Þ ;
ð7aÞ
from eq. (5), we obtain 1 K1;t ðtÞ ¼ K1 ðtÞP1 ðtÞ; 2 1 1 P1;t ðtÞ ¼ K12 ðtÞ P21 ðtÞ; 4 4
ð7bÞ
and f ð jÞ ¼ 0;
j ¼ 2; 3; . . . :
ð7cÞ
Further more, eq. (7b) give the solutions K1 ðtÞ ¼
ð2aÞ
8c1 ; 4c21 t2
P1 ðtÞ ¼
4t : 4c21 t2
ð7dÞ
Thus, the one-soliton solution for the nonisospectral KP equation is ð2bÞ
With the help of the dependent variable transformation u ¼ 2ðln f Þxx ;
ð5aÞ
ð2Þ 8fxtð2Þ þ yð2fxxxx þ 6fyyð2Þ Þ þ 4xfxyð2Þ þ 4fyð2Þ
and its Lax pair is y ¼ xx þ 2u; 3 1 t ¼ y xxx þ 3ux þ ð@ uy þ ux Þ 2 1 1 1 þ xðxx þ 2uÞ þ x þ ð@1 uÞ: 2 2 2
ð4Þ
ð3Þ
u¼
K12 ðtÞ 1 sech2 : 2 2
ð8Þ
Its shape and motion is shown in Fig. 1. For any fixed time t, the wave shows the characteristic of u ! 0 as x; y ! 1. The amplitude of the wave, i.e, K12 ðtÞ=2, tends to be zero as t ! 1. This nonisospectral characteristic is caused directly by the dominant term K1 ðtÞ ¼ 8c1 =ð4c21 t2 Þ in
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J. Phys. Soc. Jpn., Vol. 74, No. 9, September, 2005
LETTERS
S. DENG et al.
1.8 0.1
1.6 1.4
0.08
1.2 0.06
1 0.8
0.04
0.6 0.4
0.02
0.2 0 15
50 30 20 0
10
30
5
20 0
10
10
0
–5
–10 –50
y
0 –10
–10
–20
y
x
(a)
–15
–20 x
(b)
Fig. 1. Shape and motion of the one-soliton solution for c1 ¼ 5 and 1ð0Þ ¼ 1: (a) t ¼ 1 and (b) y ¼ 1.
eq. (8) as t ! 1. Similar to the one-soliton solution, if we take f ð1Þ ¼ e1 þ e2 ; j ¼ Kj ðtÞ½x þ Pj ðtÞy þ jð0Þ ;
ð j ¼ 1; 2Þ
ð9aÞ
then f ð2Þ ¼ e1 þ2 þA12 ; 1 Kj;t ðtÞ ¼ Kj ðtÞPj ðtÞ; 2 8c j Kj ðtÞ ¼ 2 ; 4cj t2
ð9bÞ
1 1 Pj;t ðtÞ ¼ Kj2 ðtÞ P2j ðtÞ; 4 4 4t Pj ðtÞ ¼ 2 ; ð j ¼ 1; 2Þ 4cj t2 ð9cÞ
where e
A12
½K2 ðtÞ K1 ðtÞ2 ½P2 ðtÞ P1 ðtÞ2 ¼ ; ½K2 ðtÞ þ K1 ðtÞ2 ½P2 ðtÞ P1 ðtÞ2
Fig. 2. Shape and motion of the two-soliton solution.
ð9dÞ
and f
ð jÞ
¼ 0;
j ¼ 3; 4; . . . :
ð9eÞ
Therefore, the two-soliton solution is obtained from eq. (3), where f ¼ 1 þ e1 þ e2 þ e1 þ2 þA12 :
ð10Þ
This process can be extended to the three-soliton solution, four-soliton solution and so on. Generally, we obtain " # N N X X X f ¼ exp j j þ j l Ajl ; ð11aÞ ¼0;1
j¼1
1 j
