Exact Solutions for the Nonisospectral Kadomtshev--Petviashvili

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The nonisospectral Kadomtshev–Petviashvili (KP) equation is solved by the ... KEYWORDS: nonisospectral KP equation, Hirota method, Wronskian technique.
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: [email protected]

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



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

2383

2384

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