Method of meromorphic matrix of transformation for giving soliton solutions of the modified nonlinear Schrödinger equation

Volume 142, number 1

PHYSICS LETTERS A

27 November 1989

METHOD OF MEROMORPHIC MATRIX OF TRANSFORMATION FOR GIVING SOLITON SOLUTIONS OF THE MODIFIED NONLINEAR SCHRODINGER EQUATION Zong-Yun CHEN Department of Physics, Huazhong University of Scienceand Technology, Wuhan 430074, Hubei, PR China

and Nian-Ning HUANG Department of Physics, Wuhan University, Wuhan 430072, Hubei, PR China Received 16 June 1989; revised manuscript received 23 September 1989; accepted for publication 28 September 1989 Communicated by A.R. Bishop

A method based on the Darboux transformation in the form of a pole expansion is presented for finding soliton solutions ofthe modified nonlinear Schrodinger equation.

The modified nonlinear Schrödinger (MNLS) equation ofwhich the nonlinear terms are composed ofa usual cubic term and a derivative cubic term has been proposed [1,21 to describe the short pulse propagation in long monomode optical fibers in consideration of the inherent property of asymmetric output pulse spectrum [3,4]. The MNLS equation has been shown to be completely integrable [51,but it has never been solved except its 1 -soliton solution obtained by simply integrating in a moving coordinate system [61 so that some works have to be based on numerical analysis for suiting needs of practicethe [7,8]. The derivative nonlinear Schrodinger (DNLS) equation which involves also the derivative cubic term but not the usual cubic term has been solved by an appropriate inverse scattering method [9]. As compared with the DNLS equation, the Lax pair of the MNLS equation given in ref. [5] is more complicated show the analytic properties of so its that Jost to solutions is required hard. In this note, a new spectral parameter is introduced for writing the Lax pair of the MNLS equation in a simple form from which the so-called reduction transformation invariance [10] is easily seen. Instead of the usual form ofpower expansion, the Dar-

boux transformations are written in the form ofpole expansion and can be specified simply. A system of linear algebraic equations for giving the Jost solution in the N-soliton case is derived. To justify the present method, the Jost solutions obtained are shown to satisfy the Lax equations of the MNLS equation. The N-soliton solution of the MNLS equation is given finally in the form of determinants of the known quantities. The MNLS equation can be written as

)~ + 2p I UI 2U = 0, (1) iu, + u~ u 2uconstant. where p+ is ia(Ireal Its Lax pair of equations is i3~F(xtC)=L(xt~)F(xtC), 0 1F(xt~)=M(xt)F(xt~) 2_p)ii Lr_i( 3+~~U, 2—p)~’U 2—p)2a M= —2i(~— 3+2(~ iC2U2a 3 i U~o 3 +~ (U 3), / 0 u\

(2)

Ur

(6)

—

(3) (4)

—

a o)’

(5)

where the overbar denotes the complex conjugate,

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Volume 142, number 1

PHYSICS LETTERS A

and is a complex spectral parameter. The compatibility condition of (2) and (3) yields ~‘

U,+iU~a

3)~—2ipU3a 3—(U

3=0,

(7)

i.e. (1). It is easily seen from (2)—(6), for a general Jost solution, F(xt~),we have 2t)c F(x1~)—~exp[—i(—px+2p 3], when

KI—’~~~

27 November 1989

In consideration of (8), from (15) we have F1(~)=F~(C),G(~)=G~(C).

(16)

We then have G—’(~)=J+ ~

1

m=1 1Cm

a

—

3~a3.

m~1~‘+

(17)

(8)

and a particular Jost solution, F0(xt~),related to the 0-soliton solution, U0 = 0, is 2—p)x+2(~2_p)21]a F0(xt() =exp{—i[ ( 3}. (9) It is reasonable to try to find the Jost solution in the form F(xtc~)=G(xt~)F0(xt~),

(10)

where G(xt~)—J, when I~I—~x.

(11)

Consider a series of transformations, each of which has a pair of poles D~(xtC)=1+ —~-—B C~~fl

1

(18)

—

F~(xt?)=D~(xt?)F~_1(xt~),

(19)

G(xt~)=DN(xt~)DN_l(xt~)...D1(xt~).

(20)

The transformation D~(xt~) is referred to as the Darboux transformation, we have written it in the form of a pole expansion instead ofits usual form of power expansion. From (18)—(20), we have

(11) indicates that G(xt~)is conveniently written in the form of a pole expansion. From (4)—(6), we have the reduction transformation invariance (12)

L(c~)=a3L(—~)a3, M(~)=a3M(—~)a3,

and then

3, G(~)=a3G(—~)a3.

(13)

The latter of (13) indicates that is a pole of G( ~) as long as is, namely, poles of G(~)must be located in pairs symmetrically about the origin of the complex plane of the spectral parameter. We now assume that G(c) is meromorphic and further has only poles of order one. In the case of 2N poles, ~ C2 ~ ~N, G(~)is written as N1 N 1 G(~_—I+~ —An-~ —a3A~t~3 (14) — ~,,

~,,

~,

n=1 ~+~n

where A~is the residue of G(~)at

c,,, and —c3A~a3

—c,,, on account ofthelatter

L(~)= —L÷(C), M(~)=—M~(C),

it. Taking the limit as D’D —I, we have ‘~

B~D~—’ (~)= D ~‘

~

~,

(c,, )B~ 0. =

from DAD;’ = (24)

It means that B~is degenerate and may be written as

—~,

is the residue ofG(~)at of(13). From (4)—(6),we have

(15)

where the dagger denotes the Hermitian conjugate. 32

—

c,,,

F(~)=o~F(—~)a

n=i ~‘~n

—a3B~a3, (21) D~‘(xt~)=I+ B~ F;’(xtC)=F;’1(xt~)D;’(xtc), (22) 1(xt)=D(xt~)D~(xtO...D~’(xt~). (23) G B~is the residue of D~(~) at we shall determine

B~=

(g’n) (g~ ha), n

(25)

where ‘n

2+C~Ih~I2~

~2c2

g~ ~ ~ Ig~I h~=~ffnClI~+~.”IhI2.

(26) (27)

For the Jost solution, F~(xtc),(2) and (3) are of the form

Volume 142, number 1

PHYSICS LETTERS A

ôxFn(xt~)=Ln(xt~)Fn(x1~), ô,Fn(xt~)=Mn(xt~)Fn(xt~),

(28) (29)

27 November 1989

where the superscript T denotes the transpose. From (14), (18) and (20), we have A~=lim(~—c’~)G(~’)

where L~and M~are obtained from (4)—(6) by setting U,, which we shall determine. From (28), we have [8~F,,(xt~) ]F;’(xt~)=L~(xt~).

=DN(~fl)...Dfl+,(~fl)BflDfl_,(~fl)...D,(~fl)

(38)

(30) where

Taking the limit as (—~,,,we have a,,— —

(31) if ~,,is a regular point of L,,. In virtue of (26), we have

~

~

m(n)~n~m

2~’2 C2~

(39)

(14) can be rewritten as 1

N

G(~’)=I+~

äx[BnFn_i(Xt~n)]=...BnFn_ 1(Xt~n),

(32)

where the expression on the left of B,, in the righthand side is not explicitly written. Similarly, from (29), we have also 8t[BnFn_,(xt~n)1=...BnFn_,(xt~n).

—

On the other hand, from (19), (25) and (35), we can write

(33)

T

) (b,, b;’)F~’(~’,,), (41)

ci2=

\ YIn I

Substituting (25) into (31) and (32), we obtain (g,,

h~)—~(y~o,,)F;~,(xtç,,),

(34)

where y,, and ö,, are constants. Since the common constant factor in (34) is immaterial, we may write (g~ h,,)=(b b—’)F—’ (xt~,,), (35)

n

n

n—I

for later convenience, where b,, is a constant. B,, is completely specified. Although the above procedure can determine the Darboux transformation, and then the Jost solution and the multi-soliton solution, it is not convenient to discuss the general problem and to obtain the Nsoliton solution in the case oflarge N. We now derive a system of linear algebraic equations for determining the Jost solution for arbitrary N which is similar to that given by Zakharov and Shabat [11] in the case of the NLS equation. From (18), (21),and(25)—(27),wehave

(40)

N _J___a;1a3o.2G_1(~,,)To~a3. n~1 ~+~n

and then G (Cm) =

(

Wm’\ -

~

m1

—

(42)

Em)F ~ ‘(Cm)

~°mI

on account of (16). Here çô,,, etc. can be expressed in terms of the known quantities, but they may be directly determined in the following manner. Setting ~‘ Cm in (40), and taking account of (41) and (42), we obtain

( \

—

Vim’) ço,,, i

J)

(J;l

N

—_1 ‘~~‘

1

N —

,

Cm +

a;’

~‘n

—

a;’ ~

(

(v”) ~ YIn

~ — ~

(J~ —f;’),

(43)

/

where T a2B,,a2=

____

2~,,

D;’(~,,),

2—C2 =

~

n

(36) (37)

f,,=b~F;’(~,,),, =b,,

ço,, and

exp{i[ (~~2 —p)x+2(C;2 —p)2t] (44) ~ as well as ~,, and ~,,, can be determined }.

33

Volume 142, number 1

PHYSICS LETTERS A

27 November 1989

by solving the 2N linear algebraic equations (43) in the case of the N-soliton. Multiplying (43) by (I’m J~~)T from the right,

[8~F( ~)] F’ (~‘)

we have

[a,F(~)]F’(~)

=2Q2+~Q_, 4R

( I’m

=

‘\

3R_ 4+

+Qo+W,

+...,

2R_ 3+

2+~’&,+...,

~—f~I)

(53)

1 +

-_1

a;

n=I~m~n N n~l

where Q2,

a;’

~m+ ~

(

‘°“

‘~

(.fnfrn

f_If

~)

2+~’Q_,]

Liouville theorem, it is equal to ipa3. This yields

—

2Q_

(45) is the reduced form of (43), since çü,, as well as w~can be determined by solving these N linear algebraic equations in the N-soliton case. To justify the present method for giving soliton solutions of the MNLS equation, we ought to show the Jost solutions obtained by the above procedure to satisfy the corresponding Lax equations. From (10) and (9), we have ]F’

ô~F(~)=[~

2+ç’Q_,

Q, =

ipa3,

Qo =

3G~(~).

(47)

Owing to (34) or (35), and then C~, C~, n = 1, 2, ..., N, are all regular points of (46) and (47), so (46) and (47) are analytic everywhere except =0.WeexpandG(() andG’(() inaTaylor series about 0, — ~‘,,,

—

~‘=

~-~‘k

=

(48) (49)

ko

where

2a —2ip 3]F(ç).

(55)

Q2

=

Q,

=

~R_4= —iv0a3v~ —ia3, =

We have 34

(56)

—i(v, a3v0 + v0a3vj~) (57)

since Veven are diagonal matrices and thus commute with a3, and ~Odd are matrices with vanishing diagonal elements and thus anti-commute with a3. So we can identify with U, we then obtain (2). From (48) and (49), and taking account of (54), we obtain Q_,

2a

&2=4ipa3—iU

3,

(58)

3.

(59)

R_, = —2pU—iU~a3+ U Therefore, (55) is just (3). From (57), the N-soliton solution of the MNLS equation, UN, can be ex-

UN=2i(vI), 2(vo)22.

~ ~~‘[A,,+(—l)~a3A,,a3J,

~

(54)

pressed as N

j+k=i

0.

=—2iv,v~cr3,

2—p)2G(~)a

~ ~ G~(~)=~

...

From (48) and (49), we have

1F’(C) =G,(~)G’(~)

c,,,

Q2 =

Similarly, we obtain

(~)=G~(~)G~(~)

—2i(~

+ipa3]F(~),

and

(46) [ô~F()

R

2Q

[~

(45)

[ö~F()

...,

4, ... can be expressed in terms of v~. From (52) and (9), [ô~F(~)]F~(~)— is analytic in the whole cornplex c-plane and tends to ipa3 as I ~I—~, and by the

\WnJ

1

-

(~“‘)(fnJrn+f~JY)

=0.

~

(52)

(50) (51)

(60)

From (45), we have N

2 ~

n=,

a;’q,,f;’K,,~=fl,,,

(61)

Volume 142, number 1

PHYSICS LETTERS A

27 November 1989

(71)

K’~mKnm+C~.

N

2 ~ a;’w,,f;’(K~),,m=—l,

(62)

where Knm =

2

1 ~m (~nJ2nJ~,,+ Cm).

(63)

—

Taking account of (61) and (62), from (50), we obtain (v,),

(64)

2,

2=— n.m ~ fln(K’)mn~

As compared with the same method for the nonlinear Schrodinger equation [12], the procedure for the MNLS equation in this note seems circuitous. The main reason is the impossibility to directly solve the equations of the Darboux transformations obtained from the Lax pair in the case ofthe MNLS equation. This work is supported by the Chinese National Fund for Natural Science Research. Valuable suggestions by Professor C.H. Tsai and Professor Y.L.

(v 0)22=l+ ~

[(K~)’]mn~’,

(65)

C’C1~)mn.

(66)

Chen are gratefully acknowledged.

n,m

and then References

_______

(v0)22=l+ ~

[1] N. Tzoar and M. Jam, Phys. Rev. A 23 (1981)1266.

n.m

Since we have the known formula of linear algebra, det(p,q~+R,~)

[3]L.F. Mollenauer, R.M. Stolen and J.P.

Gordon, Phys.

Rev.

Lett.45 (1980) 1095. [4] H. Nakatsuka, D. Grischkowsky and A.C. Balant, Phys. Rev.

I

=det R ~l ~ p1q1(R ‘)~~ , (67) [ + i,j1 which is valid for a nonsingular Mx M matrix R and arbitrary rows p and q, (64) and (66) can be written as (v,),2=—[(detK)~detK’—l],

(68)

(v0)22 = (det K)~det K”,

(69)

where K’nmKnm+t~n—22’2 ./ m,

[2] D. Anderson and M. Lisak, Opt. LeIt. 7 (1982) 394.

[5] M.Wadati, Lett.47 (1981) K.KonnoandY.H. 910. Ichikawa,J. Phys. Soc.Japan 46(1979) 1965. [6] D. Anderson and M. Lisak, Phys. Rev. A 27 (1983)1393. [7] E.A. Golovchenko, E.M. Dianov, A.M. Prokhorov and V.N. Serkin, Pis’ma Zh. Eksp. Teor. Fiz. 42 (1985) 74 [JETP Lett.42 (1985) 88]. [8] K. Ohkuma, Y.H. Ichikawa and Y. Abe, Opt. Lett. 12 (1987)516. [9] D.J. Kaup and A.C. Newell, J. Math. Phys. 19 (1978) 798. [10] A.V. Mikhailov, PhysicaD 3 (1981) 73. [11] V.E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fiz. 61 (1971) 118 [Sov.Phys.JETP34 (1972) 62].

(70)

Chen, N.N. Huang and Y. Xiao, Phys. Rev. A 38 (1988) 4355.

[12]Z.Y.

35