Bäcklund transformation and soliton solutions for KP equation

Chaos, Solitons and Fractals 25 (2005) 475–480 www.elsevier.com/locate/chaos

Ba¨cklund transformation and soliton solutions for KP equation Shu-fang Deng Institute of Mathematics, Fudan University, Shanghai 200433, PeopleÕs Republic of China Accepted 23 November 2004 Communicated by Prof. M. Wadati

Abstract A new representation of N soliton solution and novel N soliton solution for the KP equation are derived through a new form Ba¨cklund transformation. Ó 2005 Elsevier Ltd. All rights reserved.

1. Introduction The KP equation ut ¼ uxxx þ 6uux þ 3@
1 uyy

ð1:1Þ

was first introduced by Kadomtsev and Peteviashvili [1] in order to study the stability of one-dimensional soliton against transverse perturbations. The N soliton solution for the KP equation was obtained by various methods, for instance, the inverse scattering method [2], Hirota method [3], bilinear Ba¨cklund transformation [4], the trace method [5,6] and Wronskian technique [7] et al. The novel N soliton solution for the KP equation was derived by use of Hirota method [8]. Recently, Zhang and Chen [9,10] have obtained a modified BT by a dependent transformation for some soliton equations, from which some novel soliton can be derived through the Hirota method. In this paper, we would like to consider the solutions of the KP equation similar to Ref. [9,10]. First we present a new form BT in bilinear form through a transformation. Then a new representation of N soliton solution and novel N soliton solution can been derived from the Hirota expansion for special choices of parameter, where the novel N soliton solution was not obtained in Ref. [9,10]. The paper is organized as following. In Section 2, we write the BT in a new bilinear form. In Section 3, the exact solutions for the KP equation are derived by the new form bilinear BT. Finally, a conclusion is given.

E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.11.019

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S.-f. Deng / Chaos, Solitons and Fractals 25 (2005) 475–480

2. New form Ba¨cklund transformation for the KP equation The bilinear BT [4] for the KP Eq. (1.1) is Dy g  f ¼ D2x g  f ;

ð2:1aÞ

Dt g  f ¼ ðD3x þ 3Dx Dy Þg  f ;

ð2:1bÞ

where D is the well-known operator defined by Dmt Dnx a  b ¼ ð@ t
@ t0 Þm ð@ x
@ x0 Þn aðt; xÞbðt0 ; x0 Þjt0 ¼t;x0 ¼x :

ð2:2Þ

The soliton solutions for the KP equation can be denoted by [11] ð2:3Þ

u ¼ 2ðln f Þxx : n

g

Replacing f by e f and g by e g in Eq. (2.1), according to the formula Dmx Dly en f  eg g ¼ enþg ½Dx þ ðk
hÞm ½Dy þ ðp
qÞl f  g;

ð2:4aÞ

Dnt en f  eg g ¼ enþg ½Dt þ ðx
rÞn f  g;

ð2:4bÞ

n ¼ kx þ wt þ py þ nð0Þ ;

ð2:4cÞ

g ¼ hx þ rt þ qy þ gð0Þ ;

we can get the new form bilinear BT Dy g  f
D2x g  f
2KDx g  f ¼ 0;

ð2:5aÞ

Dt g  f
D3x g  f
3Dx Dy g  f
6KD2x g  f
12K 2 Dx g  f ¼ 0;

ð2:5bÞ

where K is a new parameter. Expanding f and g as f ¼ 1 þ f ð1Þ
þ f ð2Þ
2 þ f ð3Þ
3 þ    ;

ð2:6aÞ

g ¼ 1 þ gð1Þ
þ gð2Þ
2 þ gð3Þ
3 þ    :

ð2:6bÞ

Substituting Eq. (2.6) into (2.5) and equating coefficients of
yield ð1Þ ð1Þ ð1Þ ð1Þ gð1Þ
gð1Þ y
fy xx
fxx
2Kðg x
fx Þ ¼ 0;

ð2:7aÞ

ð2Þ ð2Þ ð2Þ ð2Þ ð1Þ
gð2Þ  f ð1Þ þ D2x gð1Þ  f ð1Þ þ 2KDx gð1Þ  f ð1Þ ; gð2Þ y
fy xx
fxx
2Kðg x
fx Þ ¼
Dy g

ð2:7bÞ

ð3Þ ð3Þ ð3Þ ð3Þ ð1Þ
gð3Þ  f ð2Þ þ gð2Þ  f ð1Þ Þ þ D2x ðgð1Þ  f ð2Þ þ gð2Þ  f ð1Þ Þ gð3Þ y
fy xx
fxx
2Kðg x
fx Þ ¼
Dy ðg

þ 2KDx ðgð1Þ  f ð2Þ þ gð2Þ  f ð1Þ Þ; . . .

...

ð2:7cÞ

and ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ 2 ð1Þ ð1Þ
gð1Þ gð1Þ t
ft xxx þ fxxx
3gxy
3fxy
6Kðg xx þ fxx Þ
12K ðgx
fx Þ ¼ 0;

ð2:8aÞ

ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ 2 ð2Þ ð2Þ
gð2Þ gð2Þ t
ft xxx þ fxxx
3gxy
3fxy
6Kðg xx þ fxx Þ
12K ðgx
fx Þ

¼
Dt gð1Þ  f ð1Þ þ D3x gð1Þ  f ð1Þ þ 3Dx Dy gð1Þ  f ð1Þ þ 6KD2x gð1Þ  f ð1Þ þ 12K 2 Dx gð1Þ  f ð1Þ ;

ð2:8bÞ

ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ ð3Þ 2 ð3Þ ð3Þ
gð3Þ gð3Þ t
ft xxx þ fxxx
3gxy
3fxy
6Kðg xx þ fxx Þ
12K ðgx
fx Þ

¼
Dt ðgð1Þ  f ð2Þ þ gð2Þ  f ð1Þ Þ þ D3x ðgð1Þ  f ð2Þ þ gð2Þ  f ð1Þ Þ þ 3Dx Dy ðgð1Þ  f ð2Þ þ gð2Þ  f ð1Þ Þ þ 6KD2x ðgð1Þ  f ð2Þ þ gð2Þ  f ð1Þ Þ þ 12K 2 Dx ðgð1Þ  f ð2Þ þ gð2Þ  f ð1Þ Þ; . . .

...

3. Solutions of the KP equation In this section we are going to derive some exact solutions for the KP equation from the new form BT.

ð2:8cÞ

S.-f. Deng / Chaos, Solitons and Fractals 25 (2005) 475–480

3.1. A new representation of N-soliton solution for K ¼

pj
k j 2

477

, j ¼ 1,2, . . . ,N

1 A zero soliton solution corresponds to f = 1 which yields u = 0. g then satisfies, by Eqs. (2.7)–(2.8) when K ¼ p1
k 2

ð0Þ

gð1Þ ¼ en1 ;

n1 ¼ k 1 ðx þ x1 t þ p1 yÞ þ n1 ;

ð2:9aÞ

gðjÞ ¼ 0; j ¼ 2; 3; . . . ;

x1 ¼ k 21 þ 3p21 ;

ð2:9b;cÞ

g ¼ g1 ¼ 1 þ en1 :

ð2:9dÞ

Therefore the one soliton solution for the KP equation is u ¼ 2ðln g1 Þxx ¼

k 21 n sech2 1 : 2 2

ð2:10Þ

2 If we take f = g1 given by (2.9) and K ¼ p2
k , then the solution generated by the new form bilinear BT is given by 2

gð1Þ ¼ b1 en1 þ b2 en2 ; gðjÞ ¼ 0; b1 ¼


ð0Þ

gð2Þ ¼ b3 en1 þn2 ;

nl ¼ k l ðx þ xl t þ pl yÞ þ nl ; l ¼ 1; 2;

j ¼ 3; 4; . . . ;

ð2:11aÞ ð2:11bÞ

k 1 þ k 2 þ p1
p2 ; k 1
k 2
p1 þ p2

b2 ¼

k 1 þ k 2
p1 þ p2 ; k 1
k 2
p1 þ p2

b3 ¼


k 1
k 2 þ p1
p2 k 1
k 2
p1 þ p2

ð2:11cÞ

and g ¼ g2 ¼ 1 þ b1 en1 þ b2 en2 þ b3 en1 þn2 :

ð2:11dÞ

The corresponding two soliton solution is obtained from u = 2(ln g2)xx. Generally, taking f = gN
1, we can get the N soliton solution where " # ( ) N N N X X X X g¼ exp lj nj þ ð1
ll ÞBjl þ lj ll ðC jl þ ipÞ ; l¼0;1

eBjl ¼

l¼1;l6¼j

j¼1

k l þ k j
pl þ pj ; k l
k j
pl þ pj

eCjl ¼

ð2:12aÞ

16j
k j
k l þ p j
pl : k j
k l
p j þ pl

ð2:12bÞ

Eqs. (2.12) can be further written as ( ) N N X X X 0 g¼ exp lj nj þ lj ll ðC jl
Bjl
Blj þ ipÞ ; l¼0;1

n0j ¼ nj þ

N X

ð2:13aÞ

16j
j¼1

eCjl
Bjl
Blj þip ¼

Bjl ;

l¼1;l6¼j

ðk l
k j Þ2
ðpl
pj Þ2 ðk l þ k j Þ2
ðpl
pj Þ2

¼ eAjl ;

ð2:13bÞ

which is coincidence with the N-soliton solution for the KP equation by Hirota method [11]. 3.2. Novel multi-soliton solutions for K ¼

pj
k j 2

, j ¼ 1,3,5, . . . ,2N
1 and K ¼

pj
1
k j
1 2

, j ¼ 2,4,6, . . . ,2N

1 1 2 , we can derive the one soliton solution (2.10). If we take f = g1 and K ¼ p1
k (not K ¼ P 2
k ), Taking f = 1, K ¼ p1
k 2 2 2 from (2.7)–(2.8), we have

gð1Þ ¼ g1 en1 ; gð2Þ ¼

a11 þ k 1 2n1 e ; k1

x1 ¼ k 31 þ 3p21 ;

ð0Þ

ð2:14aÞ

gðjÞ ¼ 0; j ¼ 3; 4; . . . ;

ð2:14bÞ

g1 ¼ a11 x þ a12 t þ a13 y þ g1 ;

a12 ¼ 3½4k 21 p1 þ a11 ðk 1 þ p1 Þ2 ;

g ¼ g2 ¼ 1 þ g1 en1 þ

a11 þ k 1 2n1 e : k1

a13 ¼ a11 k 1 þ 2k 21 þ a11 p1 ;

ð2:14cÞ ð2:15Þ

478

S.-f. Deng / Chaos, Solitons and Fractals 25 (2005) 475–480

The one soliton solution 
 a11 þ k 1 2n1 u ¼ 2ðln gÞxx ¼ 2 ln 1 þ g1 en1 þ e k1 xx

ð2:16Þ

is uniform with the novel one soliton solution for the KP equation by Hirota method [8]. 2 , by solving (2.7)–(2.8), we can get If f = g2, K ¼ p2
k 2 gð1Þ ¼ ðd 1 g1 þ d 2 Þen1 þ d 3 en2 ; d1 ¼


d3 ¼

k 1 þ k 2 þ p1
p2 ; k 1
k 2
p1 þ p2

ðk 1 þ k 2
p1 þ p2 Þ2 ðk 1
k 2
p1 þ p2 Þ2

ð2:17aÞ d2 ¼


d6 ¼

ðk 1
k 2
p1 þ p2 Þ2

ð2:17bÞ

;

ð2:17cÞ

;

gð2Þ ¼ ðd 4 g1 þ d 5 Þen1 þn2 þ d 6 e2n1 ; d4 ¼


  2 2k 21 þ a11 ðk 1
k 2
p1 þ p2 Þ

gð3Þ ¼ d 7 e2n1 þn2 ;

ðk 1
k 2 þ p1
p2 Þðk 1 þ k 2
p1 þ p2 Þ ðk 1
k 2
p1 þ p2 Þ2

ða11 þ k 1 Þðk 1 þ k 2 þ p1
p2 Þ2 k 1 ðk 1
k 2
p1 þ p2 Þ2

d5 ¼


;

d7 ¼

;

gðjÞ ¼ 0;

j ¼ 4; 5; . . . ;

ð2:18aÞ

2½2k 21 þ a11 ðk 1 þ k 2
p1 þ p2 Þ ðk 1
k 2
p1 þ p2 Þ2

;

ða11 þ k 1 Þðk 1
k 2 þ p1
p2 Þ2

ð2:18bÞ

ð2:18cÞ

k 1 ðk 1
k 2
p1 þ p2 Þ2

and g ¼ g3 ¼ ðd 1 g1 þ d 2 Þen1 þ d 3 en2 þ ðd 4 g1 þ d 5 Þen1 þn2 þ d 7 e2n1 þn2 :

ð2:19Þ

2 Let f = g3 and K ¼ p2
k , from (2.7)–(2.8) we have 2

gð1Þ ¼ ðd 8 g1 þ d 9 Þen1 þ ðd 10 g2 þ d 11 Þen2 ; xj ¼ k 2j þ 3p2j ; d8 ¼

aj2 ¼ 3½4k 2j pj þ aj1 ðk j þ pj Þ2 ;

ðk 1 þ k 2 þ p1
p2 Þ2 ðk 1
k 2
p1 þ p2 Þ

d 10 ¼

ð0Þ

gj ¼ aj1 x þ aj2 t þ aj3 y þ gj ;

2

ðk 1 þ k 2
p1 þ p2 Þ2 ðk 1
k 2
p1 þ p2 Þ

d9 ¼

;

2

;

ð2:20aÞ

aj3 ¼ aj1 k j þ 2k 2j þ aj1 pj ;

j ¼ 1; 2;

4ðk 1 þ k 2 þ p1
p2 Þ½2k 21 þ a11 ðk 1
k 2
p1 þ p2 Þ

d 11 ¼

ðk 1
k 2
p1 þ p2 Þ3

ð2:20bÞ ð2:20cÞ

;

4ðk 1 þ k 2
p1 þ p2 Þ½
2k 22 þ a21 ðk 1
k 2
p1 þ p2 Þ ðk 1
k 2
p1 þ p2 Þ3

;

ð2:20dÞ

gð2Þ ¼ ðd 12 g1 g2 þ d 13 g2 þ d 14 g1 þ d 15 Þen1 þn2 þ d 16 e2n1 þ d 17 e2n2 ;

ð2:21aÞ

gð3Þ ¼ ðd 18 g2 þ d 19 Þe2n1 þn2 þ ðd 20 g1 þ d 21 Þen1 þ2n2 ;

ð2:21bÞ

gð4Þ ¼ d 22 e2n1 þ2n2 ;

ð2:21cÞ

d 12 ¼

d 13 ¼

gðlÞ ¼ 0;

l ¼ 5; 6; . . . :

ðk 1
k 2 þ p1
p2 Þðk 1 þ k 2 þ p1
p2 Þðk 1 þ k 2
p1 þ p2 Þ ðk 1
k 2
p1 þ p2 Þ3

ð2:22aÞ

;

4f2k 21 ½k 21
k 22
ðp1
pÞ2  þ a11 ½k 21
k 22
2k 1 ðp1
p2 Þ þ ðp1
p2 Þ2 ðk 1 þ p1
p2 Þg

d 14 ¼


ðk 1
k 2
p1 þ p2 Þ4 4f2k 22 ½k 21
k 22 þ ðp1
pÞ2 
a21 ½
k 21 þ ðk 2 þ p1
p2 Þ2 ðk 2
p1 þ p2 Þg ðk 1
k 2
p1 þ p2 Þ4

;

;

ð2:22bÞ

ð2:22cÞ

S.-f. Deng / Chaos, Solitons and Fractals 25 (2005) 475–480

479

d 15 ¼ 4f
4k 21 ½2k 22 þ a21 ðk 2
p1 þ p2 Þ þ a11 ½
4k 22 ðk 1 þ p1
p2 Þ þ a21 ½k 21 þ k 22
2k 2 ðp1
p2 Þ
3ðp1
p2 Þ2
2k 1 ðk 2
p1 þ p2 Þg=ðk 1
k 2
p1 þ p2 Þ4 ; d 16 ¼

d 18 ¼

ða21 þ k 2 Þðk 1 þ k 2
p1 þ p2 Þ4 k 2 ðk 1
k 2
p1 þ p2 Þ

ða11 þ k 1 Þðk 1 þ k 2 þ p1
p2 Þ4 k 1 ðk 1
k 2
p1 þ p2 Þ4

k 1 ðk 1
k 2
p1 þ p2 Þ4

ð2:22eÞ

;

ð2:23aÞ

;

4ða11 þ k 1 Þ½k 21
k 22 þ 2k 1 ðp1
p2 Þ þ ðp1
p2 Þ2 ½2k 22 þ a21 ðk 1 þ k 2 þ p1
p2 Þ k 1 ðk 1
k 2
p1 þ p2 Þ4

ða21 þ k 2 Þ½k 21
ðk 2
p1 þ p2 Þ2 2 k 2 ðk 1
k 2
p1 þ p2 Þ4

d 21 ¼


d 22 ¼

d 17 ¼

;

ða11 þ k 1 Þ½k 21
k 22 þ 2k 1 ðp1
p2 Þ þ ðp1
p2 Þ2 2

d 19 ¼


d 20 ¼

4

ð2:22dÞ

;

ð2:23bÞ

ð2:23cÞ

;

4ða21 þ k 2 Þ½
k 21 þ ðk 2
p1 þ p2 Þ2 ½2k 21 þ a11 ðk 1 þ k 2
p1 þ p2 Þ k 2 ðk 1
k 2
p1 þ p2 Þ4

ð2:23dÞ

;

ða11 þ k 1 Þða21 þ k 2 Þðk 1
k 2 þ p1
p2 Þ4

ð2:24Þ

k 1 k 2 ðk 1
k 2
p1 þ p2 Þ4

and g ¼ g4 ¼ 1 þ ðd 8 g1 þ d 9 Þen1 þ ðd 10 g2 þ d 11 Þen2 þ ðd 12 g1 g2 þ d 13 g2 þ d 14 g1 þ d 15 Þen1 þn2 þ d 16 e2n1 þ d 17 e2n2 þ ðd 18 g2 þ d 19 Þe2n1 þn2 þ ðd 20 g1 þ d 21 Þen1 þ2n2 þ d 22 e2n1 þ2n2 ;

ð2:25Þ

from which we can get the two novel soliton solution for the KP equation. N Generally, taking f = g2(N
1), K ¼ pN
k , we have 2 ( " !#)( lj ðl2j
1Þ N
1
1
X X X NY aj1 þ k j exp lN nN þ 2 BNl ½aj1 ð@ kj þ @ pj Þ þ 2k j @ pj lj ð2
lj Þ g ¼ g2N
1 ¼ k j lN ¼0;1 l¼0;1;2 j¼1 l¼1 ! " #) N
1 N
1 N X X X   0 0  exp lj nj þ 2 Bjl þ BjN þ lj ll C jl
Bjl
Blj þ ip jB0 ¼Bjl ;B0 ¼BjN ; ð2:26Þ l¼1;l6¼j

j¼1

jl

16j
jN

N , we can get the N novel soliton solution where If f = g2N
1 and K ¼ pN
k 2

g¼

 N
X Y aj1 þ k j kj

l¼0;1;2 j¼1

"

 exp

N X

lj nj þ 2

½aj1 ð@ kj þ @ pj Þ þ 2k j @ pj lj ð2
lj Þ

N X

! B0jl

l¼1;l6¼j

j¼1 0

lj ðlj
1Þ 2

þ

N X

# lj ll ðC jl
Bjl
Blj þ ipÞ jB0 ¼Bjl ; jl

16j
ð2:27Þ

k 0 þk 0
p0 þp0

here eBjl ¼ k l0
kj0
pl0 þpj0 . l

j

l

j

Let aj1 =
2kj. Eqs. (2.26) and (2.27) can be rewritten as " # " #
1 N
1 N lj ðlj
1Þ X NY X X X lj ð2
lj Þ ~ ~ 2 expðlN nN Þ ð
1Þ ð
2k j @ k j Þ exp lj nj þ lj ll Ajl ; g¼ lN ¼0;1

~nN ¼ nN þ 2

l¼0;1;2 j¼1 N
1 X l¼1

BNl ;

~nj ¼ nj þ 2

j¼1 N
1 X l¼1;l6¼j

B0jl þ B0jN ;

ð2:28aÞ

16j
ð2:28bÞ

480

S.-f. Deng / Chaos, Solitons and Fractals 25 (2005) 475–480

g¼

N X Y

ð
1Þ

lj ðlj
1Þ 2

l¼0;1;2 j¼1


nj ¼ nj þ 2

N X

ð
2k j @ kj Þlj ð2
lj Þ exp

" N X

nj þ lj


j¼1

N X

# ð2:29aÞ

lj ll Ajl ;

16j
B0jl :

ð2:29bÞ

l¼1;l6¼j

In Ref. [8], if take f ð1Þ ¼

N
1 X

ð0Þ

ð0Þ

gj enj þ enN ; nj ¼ k j ðxj t þ x þ pj yÞ þ nj ; gj ¼ aj t þ bj x þ cj y þ gj :

ð2:30Þ

j¼1

We can get the N soliton solution which is uniform with (2.28). Eq. (2.29) is uniform with the N novel soliton solution for the KP equation [8].

4. Conclusion In this paper, the new form bilinear BT for the KP equation is derived. Through the new form BT we can not only get a new representation of N soliton solutions but also derive the novel N soliton solutions. The present method is also applicable to other nonlinear evolution equations.

Acknowledgment I am grateful to D.Y. Chen for his valuable discussion. I also would like to express my sincere thanks to the referees for their helpful suggestion. This project is supported by the National Science Foundation of China (10371070) and the Postdoctoral Science Foundation of China.

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