A direct theory for the perturbed unstable nonlinear Schrodinger equation

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Nian-Ning Huang, Sien Chi, B. L. Lou, and Xiang-Jun Chen

Citation: Journal of Mathematical Physics 41, 2931 (2000); doi: 10.1063/1.533281 View online: http://dx.doi.org/10.1063/1.533281

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/41/5?ver=pdfcov Published by the AIP Publishing

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A direct theory for the perturbed unstable nonlinear

Schro

¨ dinger equation

Nian-Ning Huang,a) Sien Chi, and B. L. Lou

Institute of Electro-optical Engineering, National Chiao Tung University, Hsinchu, Taiwan 30049, Republic of China

Xiang-Jun Chenb)

Department of Physics, Jinan University, Guangzhou 510632, People’s Republic of China

共Received 28 December 1999; accepted for publication 19 January 2000兲

A direct perturbation theory for the unstable nonlinear Schro¨dinger equation with perturbations is developed. The linearized operator is derived and the squared Jost functions are shown to be its eigenfunctions. Then the equation of linearized op-erator is transformed into an equivalent 4⫻4 matrix form with first order derivative in t and the eigenfunctions into a four-component row. Adjoint functions and the inner product are defined. Orthogonality relations of these functions are derived and the expansion of the unity in terms of the four-component eigenfunctions is im-plied. The effect of damping is discussed as an example. © 2000 American

In-stitute of Physics.关S0022-2488共00兲00405-9兴

I. INTRODUCTION

The unstable nonlinear Schro¨dinger 共UNLS, for short兲 equation was introduced in plasma physics1,2to describe the nonlinear modulation of a high frequency mode in electron beam plasma such as a system where an electron beam is injected under high frequency electric field. This equation may be considered as a prototype amplitude equation for the soliton phenomena in an unstable system. It also describes the nonlinear modulation of waves in Rayleigh-Taylor problem.3 The UNLS equation can be expressed as

iu_x⫹u_tt⫹2兩u兩2_u_⫽0, _共1兲

where t and x are time and space coordinate.

Interchange of x and t in共1兲 leads to the conventional stable nonlinear Schro¨dinger 共SNLS, for short兲 equation. Since the SNLS equation has been proved to be a completely integrable system,4,5it has been solved by using the inverse scattering transform共IST兲. Soliton solutions for the UNLS equation can be obtained by simply interchanging x and t from the soliton solutions for the SNLS equation. The UNLS equation has also been generally solved, by a similar IST, includ-ing the contribution of continuous spectrum of the spectral parameter,2,6 which is necessary in developing a perturbation theory for the UNLS equation with perturbations. To have some insight into the physical significance of the UNLS equation and to have an effective method to study practical problems, it is necessary to consider the UNLS equation with perturbations,7

ivx⫹vtt⫹2兩v兩2v⫽⑀r关v兴, 共2兲

where⑀is a small positive parameter and r关v兴 is a functional of v. Since 共2兲 has a second order derivative in t, the initial conditions must include one aboutvt(x,0) in addition to the one about v(x,0). We choose

a兲_{Permanent address: Department of Physics, Wuhan University, Wuhan 430072, People’s Republic of China.}

b兲_{Electronic mail: xiangjun-chen@21cn.com}

2931

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v共x,0兲⫽u共x,0兲, vt共x,0兲⫽ut共x,0兲. 共3兲

However,共1兲 and 共2兲 are second order partial differential equations in time. The initial value problem under the condition共3兲, which is very different from that for the SNLS equation,7 has never appeared in the literature. The purpose of this work is to find the perturbed solution of共2兲 under the initial condition共3兲. This work is arranged as follows:

共1兲 The linearized equation for 共2兲 is derived, and the squared Jost functions are shown to be solutions of this linearized equation by means of the Lax equations.

共2兲 A 4⫻4-matrix form of the linearized equation which has only a first order derivative in t is introduced to replace its original 2⫻2-matrix form with a second derivative in t;

共3兲 The two-component squared Jost functions are transformed into four-component ones. The four-component adjoint functions and the inner product are introduced. The orthogonality relations are then derived.

共4兲 The expansion of the unity in terms of the four-component squared Jost functions is implied. 共5兲 The secularity conditions are found and the adiabatic solution can be determined with them. 共6兲 Finally, the effect of damping is discussed as an example.

A brief review of the inverse scattering transform for共1兲 is given in the Appendix.2,6 II. THE LINEARIZED EQUATION

Suppose8–11

v⫽ua⫹⑀q, 共4兲

where uais the so-called adiabatic solution which has the same functional form as that of the exact soliton solution but the parameters involved may depend on t of the order of ⑀, which will be discussed in detail later. Here⑀q is the remaining term up to the order of⑀. Substitution of共4兲 into 共2兲 yields

iq_x⫹q_tt⫹4兩u兩2q⫹2u2¯q⫽R关u兴, 共5兲

R关u兴⫽r关u兴⫺s关u兴, s关u兴⫽1

⑀ 兵iux⫹utt⫹2兩u兩2u其. 共6兲

Equation共5兲 is an equation up to the order of ⑀, u in the left hand side and in r关u兴 is the exact solution, and u in s关u兴 is the adiabatic solution. Here the bars denote complex conjugates.

Equation共5兲 and its complex conjugate can be combined as

冉

i⳵x⫹⳵t 2_⫹4兩u兩2 _⫺2u2 ⫺2u¯2 _⫺i_⳵ x⫹⳵t 2_⫹4兩u兩2

冊

冉

q ⫺q¯

冊

⫽

冉

R ⫺R¯

冊

. 共7兲

The initial condition共3兲 turns to

q共x,t⫽0兲⫽0, q_t共x,t⫽0兲⫽0. 共8兲

To find the perturbed solution of 共2兲 under the initial condition 共3兲 is equivalent to solving 共7兲 under the initial condition共8兲.

In order to solve共7兲, we need to find a complete set of solutions for its homogeneous version, i.e., 共7兲 with a vanishing right hand side. From 共A2兲 and 共A3兲, the Lax equations of 共1兲, we obtain8–11

冉

i⳵_x⫹⳵_t2⫹4兩u兩2 ⫺2u2 ⫺2u¯2 _⫺i_⳵ x⫹⳵t 2_⫹4兩u兩2

冊

W⫽

冉

0 0

冊

. 共9兲

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Here

W⫽

冉

w1 2

w₂2

冊

, 共10兲

in which w₁ and w₂are components of a solution of the Lax equations, w, which can be chosen as those Jost functions, h(t,␭)⫺1␺(x,␭), h(t,␭)␺˜ (x,␭), h(t,␭)␾(x,␭), or h(t,␭)⫺1␾˜ (x,␭) 共see the Appendix兲. That is, like the case of the SNLS equation,8solutions of the homogeneous version of 共7兲 can be constructed with those so-called squared Jost functions W. We denote W⫽wⴰw.

III. A TRICK TO TREAT THE SECOND ORDER DERIVATIVE INT

If one can find a complete set of the squared Jost functions, solutions of共7兲 can be expanded in the complete set. However, owing to the fact that共7兲 and 共9兲 have second derivatives in t, like the case of sine-Gordon equation,10 it is more convenient to transform them into an equation having only a first derivative in t. Thus, equivalently, we rewrite共7兲 as

冉

⫺i⳵t 0 1 0

0 ⫺i⳵t 0 1

i⳵x⫹4兩u兩2 ⫺2u2 ⫺i⳵t 0

⫺2u¯2 _⫺i_⳵ x⫹4兩u兩2 0 ⫺i⳵t

冊

冊

⫺2u¯2 _⫺i_⳵ x⫹4兩u兩2 0 ⫺i⳵t

冊

冉

⌿共x,␭兲1 ⌿共x,␭兲2 ⌿共x,␭兲3 ⌿共x,␭兲4

冊

⫽2␭

冉

⌿共x,␭兲1 ⌿共x,␭兲2 ⌿共x,␭兲3 ⌿共x,␭兲4

冊

, 共13兲 where ⌿共x,␭兲⫽共⌿共x,␭兲1⌿共x,␭兲2⌿共x,␭兲3⌿共x,␭兲4兲T 共14兲

is a four-component squared Jost function with the additional third and fourth components: ⌿共x,␭兲3⫽i2共i␭␺共x,␭兲1⫺u␺共x,␭兲2兲␺共x,␭兲1,

共15兲 ⌿共x,␭兲4⫽i2共⫺i␭␺共x,␭兲2⫹u¯␺共x,␭兲1兲␺共x,␭兲2.

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⫺L共u兲⫽

冉

0 0 1 0 0 0 0 1 i⳵x⫹4兩u兩2 ⫺2u2 0 0 ⫺2u¯2 _⫺i_⳵ x⫹4兩u兩2 0 0

冊

. 共16兲 Equation共15兲 becomes 兵⫺i⳵t⫺L共u兲其⌿共x,␭兲⫽2␭⌿共x,␭兲. 共17兲

It is obvious that at␭_n, one of the zeros of a(␭),

兵⫺i⳵t⫺L共u兲其⌿共x,␭n兲⫽2␭n⌿共x,␭n兲, 共18兲

and

兵⫺i⳵t⫺L共u兲其⌿˙共x,␭n兲⫽2␭n⌿˙共x,␭n兲⫹2⌿共x,␭n兲, 共19兲

where⌿˙(x,␭n)⫽ (d/d␭) ⌿(x,␭)兩␭⫽␭_n.

Similarly, we have equations for other four-component squared Jost functions, ⌽(x,␭), ⌿˜ (x,␭) and ⌽˜ (x,␭), similar to 共17兲–共19兲.

Introducing

q⫽共q⫺q¯iq_t⫺iq¯_t兲T_, _R_{⫽共0 0 R ⫺R¯兲}T_, _共20兲

共11兲 can be rewritten as

兵⫺i⳵t⫺L共u兲其q⫽R. 共21兲

IV. ADJOINT FUNCTIONS AND INNER PRODUCTS

We now introduce adjoint functions and inner products. The essential point is that the inner product of a squared Jost function with its adjoint function is proportional to the ␦(␭⫺␭

⬘

) function in the continuous spectrum.8–11 Definition of the inner product is given by

具

⌿共␭

⬘

兲兩⌿共␭兲

典

⫽

冕

⫺⬁ ⬁

dx⌿共x,␭

⬘

兲A⌿共x,␭兲. 共22兲

We choose the adjoint function to be ⌿共x,␭兲A_{⫽共⫺⌽共x,␭兲}

4⫺⌽共x,␭兲3⌽共x,␭兲2⌽共x,␭兲1兲, 共23兲

where⌽₃ and⌽₄ are as in 共15兲, replacing components of␺with those of␾. From the Lax equation共A2兲 we obtain

d

dxW关␸共x,␭

⬘

兲,␺共x,␭兲兴⫽⫺i2共␭

⬘

2_⫺␭2_兲共_␸_共x,␭

_⬘

_兲

1␺共x,␭兲2⫹␸共x,␭

⬘

兲2␺共x,␭兲1兲⫹2共␭

兲␺1共x,␭兲兴. 共25兲

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Hence we find

d

dx兵W关␸共x,␭

⬘

兲,␺共x,␭兲兴其

2_{⫽⫺i2共␭}

_⬘

_{⫺␭兲⌿共x,␭}

_⬘

_兲A_{⌿共x,␭兲.} _共26兲

⬘

should be considered as those approaching the real or the imaginary axis from the first or the third quadrant. The limit is considered as the Cauchy principal value

lim L→⬁ P 1 ⫺i2共␭

⬘

⫺␭兲e⫺i4(␭⬘ 2_⫺␭2_)L ⫽␲␭␦共␭2_⫺␭

_⬘

2_兲. _共28兲

Hence the values of共27兲 at the upper and at the lower limits can be found. We thus find

具

⌿共␭

⬘

兲兩⌿共␭兲

典

⫽␲a共␭兲22␭␦共␭2⫺␭

⬘

d2 d␭2

冎

兵W关␸共x,␭

⬘

兲,␺共x,␭兲兴其 2_兩 ␭⫽␭_⬘⫽␭n兩⫺L L _⫽i12

_具

_⌿˙共␭ n兲兩⌿˙ 共␭n兲

典

. 共33兲 Finally we obtain

具

⌿˙共␭m兲兩⌿˙ 共␭n兲

典

具

⌿˜˙_共␭¯_m_兲兩⌿˜˙_共␭¯_n_兲

_典

_⫽i1 2a˜˙共␭¯n兲a˜¨共␭¯n兲␦mn. 共37兲

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V. THE EXPANSION OF THE UNITY

If the above squared Jost functions form a complete set, like the case of the SNLS equation,8 a stateq(x) can be expanded in terms of them:

q共x兲⫽1

␲⌫d␭兵f共␭兲⌿共x,␭兲⫹ f˜共␭兲⌿˜共x,␭兲其⫹

兺

_n 兵fn⌿共x,␭n兲⫹gn⌿˙共x,␭n兲其⫹

兺

n 兵

f˜n⌿˜共␭¯n兲

⌿˜_共␭¯_n_兲兩q

_典

_. _共42兲

Substituting them into共38兲, we obtain

␦共x⫺y兲⫽1 ␲

冕

⌫d␭ 1 a共␭兲2⌿共x,␭兲⌿共y,␭兲 A_⫹

兺

n i2a¨共␭n兲 a˙共␭n兲3 ⌿共x,␭n兲⌿共y,␭n兲A ⫺

兺

n i 2 a˙共␭n兲2 兵⌿˙共x,␭n兲⌿共y,␭n兲A⫹⌿共x,␭n兲⌿˙共y,␭n兲A其 ⫺1 ␲

冕

⌫d␭ 1 a ˜共␭兲2⌿ ˜ _{共x,␭兲⌿}˜_共y,␭兲A_⫹

兺

n i2a˜¨共␭¯n兲 a ˜˙共␭¯n兲 ⌿˜ _共x,␭¯ n兲⌿˜共y,␭¯n兲A ⫺

兺

n i 2 a ˜˙共␭¯n兲兵⌿˜˙_共x,␭¯_n_兲⌿˜_共y,␭¯_n_兲A_⫹⌿_˜_共x,␭¯ n兲⌿˜˙共y,␭¯n兲A其. 共43兲

This is the expansion of the unity in terms of the squared Jost functions. VI. SECULARITY CONDITIONS

Suppose q in共21兲 can be expanded in the form of 共38兲共the coefficients may be dependent on

t). Substituting it into 共21兲 and performing the inner product with ⌿(x,␭)A_, _⌿(x,␭

m)A and

⌿˙(x,␭m)A from the left, respectively, by using the orthogonality relations, we obtain

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⫽0, 共50兲

and

具

⌿˜_共␭¯_n_兲兩R

_典

_⫽0,

_具

_⌿˜˙_共␭¯_n_兲兩R

_典

_⫽0. _共51兲

It is easy to show that共51兲 are just complex conjugates of 共50兲 and are not independent of them. The so-called secularity conditions共50兲 become

冕

⫺⬁ ⬁ dx兵⌽2共x,␭n兲R关u兴⫺⌽1共x,␭n兲R关u兴其⫽0, 共52兲 and

冕

⫺⬁ ⬁ dx兵⌽˙2共x,␭n兲R关u兴⫺⌽˙1共x,␭n兲R关u兴其⫽0. 共53兲

They give 4N real conditions for the N-soliton case. In the N-soliton case, we have just 4N parameters. By means of these secularity conditions we can determine the time dependence of the parameters up to the order of⑀in the adiabatic solution. After determining the adiabatic solution, from共44兲 we can determine f (␭) as a function of t. Finally, we can findq.

VII. A SINGLE SOLITON CASE

The secularity conditions共52兲 and 共53兲 can be rewritten as

S1⫽R1 共54兲

and

S2⫽R2, 共55兲

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S1⫽

冕

⫺⬁ ⬁

dx兵⌽₂共X,␭₁兲ei2␦s关u兴⫺⌽₁共X,␭₁兲e⫺i2␦s关u兴其, 共56兲

S2⫽

冕

⫺⬁ ⬁

dx兵⌽˙2共X,␭1兲ei2␦s关u兴⫺⌽˙1共X,␭1兲e⫺i2␦s关u兴其, 共57兲 andR1andR2are obtained simply by replacing s关u兴 with r关u兴 from 共56兲 and 共57兲, respectively.

For the single soliton solution,

u⫽2␯sech Xe⫺i␸, 共58兲

where the parameter ␭1⫽␮⫹i␯. We assume ␭1 lies within the first quadrant without loss of generality, hence ␮⬎0 and␯⬎0:

X⫽2␯关⫺t⫹4␮共x⫺x₁兲兴, ␸⫽⫺2␮t⫹4共␮2⫺␯2兲x⫹␸₀, 共59兲 where x1 and␸0 are real constants.

For the adiabatic solution,␮,␯,x1,␸0 may be dependent on t of the order of ⑀. We write X⫽8␮␯z, z⫽x⫺xˆ, d dtˆx⫽ 1 4␮, 共60兲 and ␸⫽4共␮2_⫺_␯2_兲z⫹2_␦_, d dt2␦⫽⫺ ␮2_⫹_␯2 ␮ . 共61兲

Simple algebra yields

s关u₁兴⫽16␯␯_␶sech XthXe⫺i␸⫺8␯2_关8共_␯_␮_兲

␶z⫺8共␯␮兲xˆ␶兴关sech X⫺2 sech3X兴e⫺i␸

⫹8␯␮关4共␮2_⫺_␯2_兲

␶z⫺4共␮2⫺␯2兲xˆ␶⫹2␦␶兴 sech Xe⫺i␸⫹i8共␯␮兲␶sech Xe⫺i␸

⫺i8␯2_关4共_␮2_⫺_␯2_兲

␶z⫺4共␮2⫺␯2兲xˆ␶⫹2␦␶兴 sech XthXe⫺i␸

⫺i8␯␮关8共␯␮兲␶z⫺8共␯␮兲xˆ␶兴 sech XthXe⫺i␸. 共62兲

Except unimportant factors共see Appendix兲 which can be dropped from both sides of 共54兲 and 共55兲, ⌽(x,␭1) and⌽˙(x,␭1) can be replaced by

⌽共X,␭1兲e⫺i2␦␴3, ⌽˙共X,␭1兲e⫺i2␦␴3, 共63兲

r关u1兴⫽⫺iu1⫽⫺i2␯sech Xe⫺i␸. 共68兲

We have R1⫽⫺i␯

冕

⫺⬁ ⬁ dz sech2X⫽⫺i 1 4␮ 共69兲 and R2⫽0. 共70兲

The secularity conditions共54兲 and 共55兲 become

␮␶⫽0, ␮ 2_⫹_␯2 ␮2_{␯ ␯}␶⫽⫺ 1 4␮, 共71兲 and xˆ_␶⫽0, ␦_␶⫽0. 共72兲

Hence, up to the order of⑀, we have

d dt␮⫽0, ␮2_⫹_␯2 ␮2_␯ d dt␯⫽⫺⌫ 1 4␮, 共73兲 and d dtˆx⫽⫺⌫ 1 4␮, d dt␦⫽ ␮2_⫹_␯2 2␮ . 共74兲

Equations共73兲 and 共74兲 yield

␮⫽␮0, log

冉

␯ ␯0

冊

⫹ 1 2␮₀2共␯ 2_⫺_␯ 0 2_兲⫽⫺⌫ 1 4␮t, 共75兲 and xˆ⫽x1⫺⌫ 1 4␮t, ␦⫽␦0⫹ 1 2␮t⫹ 1 2␮

冕

0 t dt␯2. 共76兲

Here␮0, ␯0, x1 and␦0 are constants.

After determination of the adiabatic solution, the right hand side of共47兲 is given, and we can find f (t,␭) and then q(x,t). Finally, we obtain q(x,t).

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IX. DISCUSSION

We have developed a direct perturbation theory for the perturbed UNLS equation. Because of the second order derivative in t, the perturbation theory is essentially different from that for the perturbed SNLS equation involving only the first derivative in t.

In a single soliton case, by substituting the explicit expressions of the Jost solutions into the right hand side of共43兲, like the case of dark solitons of SNLS,12we can see that it is indeed equal to␦(x⫺y). Hence the completeness relation 共43兲 is shown in this case. However, for the multi-soliton case the explicit expressions of the Jost solutions are very complicated so that it is impos-sible to substitute them into the right hand side of共43兲 and to show it is equal to ␦(x⫺y). This problem will be discussed separately.

ACKNOWLEDGMENTS

One of authors, N. N. Huang, acknowledges the sponsorship by National Science Council of R. O. China and National Chiao Tung University, Hsinchu, Taiwan. This work was supported by National Science Council of R. O. China under Grant No. NSC87-2215-E009-014. The authors would like to express their thanks to Professor Q. S. Liu for discussion.

APPENDIX: A REVIEW OF THE INVERSE SCATTERING TRANSFORM FOR THE UNLS EQUATION

We review the inverse scattering transform2,6for the unperturbed equation共1兲 with the bound-ary condition

u→0, as 兩x兩→⬁. 共A1兲

Two Lax equations for the UNLS equation are obtained from those for the SNLS equation2 by interchanging their roles. Starting from the first Lax equation

⳵xw共x,t,␭兲⫽

冉

⫺i2␭2_⫹兩u兩2 ₂_␭u⫺iu

t

⫺2␭u¯⫺iu¯t i2␭2⫺兩u兩2

冊

w共x,t,␭兲, 共A2兲

and by using the boundary conditions共A1兲, the analyticity of the Jost functions can be found and the equation of IST can be derived. Then, by using the second Lax equation,

⳵tw共x,t,␭兲⫽

冉

i␭ ⫺u u

¯ ⫺i␭

冊

w共x,t,␭兲, 共A3兲

the t dependence of the scattering data can be determined.

From the Lax equation共A2兲 and the boundary condition 共A1兲, the asymptotic solution in the limit of兩x兩→⬁ of 共A2兲 is

E共x,␭兲⫽e⫺i2␭2x␴3_. 共A4兲

In comparison with the asymptotic solution for the SNLS equation, e⫺i␭x␴3_{, one can see that the}

parameter in the exponential,␭, is replaced by 2␭2 in the UNLS case. This leads to the followo-ing.

共1兲 The domain of definition of the asymptotic solution for the SNLS equation is for real ␭, namely, on the real axis in the complex ␭-plane. The domain of definition of the asymptotic solution for the UNLS equation is for real␭2, namely, on the real axis in the complex␭-plane where␭2⬎0, as well as on the imaginary axis where ␭2⬍0.

共2兲 Jost functions are defined by

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and

共␸ ␸˜兲共x,␭兲→E共x,␭兲 as x→⫺⬁; 共A6兲

the monodromy matrix is introduced as well:

共␸␸˜兲共x,␭兲⫽共␺˜␺兲共x,␭兲

冉

a共␭兲 ⫺b˜共␭兲

b共␭兲 a˜共␭兲

冊

, 共A7兲

similarly in both cases. In the SNLS case␺(x,␭), ␸(x,␭) and a(␭) are analytic in the upper half plane of complex ␭-plane, and␺˜ (x,␭), ␸˜ (x,␭) and a˜(␭) are analytic in the lower plane.

More-over, b(␭) and b˜(␭) cannot be analytically continued out of the real axis. The zeros of a(␭) lie in the upper plane. On the other hand, in the UNLS case,␺(x,␭), ␸(x,␭) and a(␭) are analytic in the first and third quadrants, and␺˜ (x,␭), ␸˜ (x,␭) and a˜(␭) are analytic in the second and fourth

quadrants. Moreover, b(␭) and b˜(␭) cannot be analytically continued out of the real and the imaginary axes. The zeros of a(␭) lie in the first or the third quadrants.

兲ei2␭⬘ 2_x . 共A10兲

Here cn and r(␭

⬘

) are the usual symbols.

2

The path of integration is

⌫⫽共0,⫹⬁兲艛共0,⫺⬁兲艛共i⬁,i0兲艛共⫺i⬁,i0兲. 共A11兲

共4兲 By using the Lax equation 共6兲, we can obtain the t dependence of scattering data in 共12兲, Simply, the Jost functions␺(x,␭), etc., which are determined by only one of the Lax equations, can be extended to those to satisfy simultaneously the two Lax equations. For example,

h共t,␭兲␺˜共x,␭兲, h共t,␭兲⫺1␺共x,␭兲, h共t,␭兲⫽ei␭t. 共A12兲 The scattering data are replaced by

r共␭兲→r共␭兲h共t,␭兲⫺2, cn→cnh共t,␭n兲⫺2, 共A13兲

etc.

The soliton solutions correspond to a reflectionless potential and in this case the continuous spectrum disappears. The poles of the transmission coefficient a(␭)⫺1 lie within the first or the third quadrants. However, it has been shown13that the forms of the soliton solutions depend on the absolute values of the imaginary part of these poles and the values of the real parts. Thus the soliton solutions of the UNLS equation can be obtained from those of the SNLS equation by simply interchanging x and t.

1_{T. Yajima and M. Wadati, J. Phys. Soc. Jpn. 59, 41}_共1990兲. 2_{T. Yajima and M. Wadati, J. Phys. Soc. Jpn. 59, 3237}_共1990兲. 3

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4_{V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62}_共1972兲. 5

P. Lax, Commun. Pure Appl. Math. 21, 467共1968兲.

6

Z.-D. Chen, X.-J. Chen, and N.-N. Huang, Commun. Theor. Phys.共to be published兲

7_{Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 794}_{共1989兲 and references therein.} 8_{D. J. Kaup, J. Math. Anal. Appl. 54, 489}_共1976兲.

9_{J. P. Keener and D. W. McLaughlin, J. Math. Phys. 18, 2008}_共1977兲. 10

D. W. McLaughlin and A. C. Scott, Phys. Rev. A 18, 1652共1978兲.

11_{R. L. Herman, J. Phys. A 23, 1063}_共1990兲.

12_{X.-J. Chen, Z.-D. Chen, and N.-N. Huang, J. Phys. A 31, 6929}_共1998兲. 13_{N.-N. Huang and G.-J. Liao, Phys. Lett. A 154, 373}_共1991兲.