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Semi-Commutative Differential Operators Associated with

the Dirac Opetator and Darboux Transformation

Masatomo Matsushima, Mayumi Ohmiya* Doshisha University, Kyoto, Japan

Email: *momiya@mail.doshisha.ac.jp

Received November 20, 2012; revised December 21, 2012; accepted December 29,2012

ABSTRACT

In the present paper, the semi-commutative differential oparators associated with the 1-dimensional Dirac operator are constructed. Using this results, the hierarchy of the mKdV (−) polynomials are expressed in terms of the KdV polyno-mials. These formulas give a new interpretation of the classical Darboux transformation and the Miura transformation. Moreover, the recursion operator associated with the hierarchy of the mKdV (−) polynomials is constructed by the al-gebraic method.

Keywords: KdV Polynomials; mKdV (−) Polynomials; Schrödinger Operator; Dirac Operator

 

n

1. Introduction

A

The main purpose of the present paper1 is to construct the semi-commutative differential operators associated with the 1-dimensional Dirac operator

 

0 1

 

0

 

, d ,

d v x

P v D D

0

1 0 v x x

 

 

    

 

 

 

,

L v   D v x

 

0

 

 

.

0 L v P v

L v

 

 

 

 

 

vv x

(1)

where the potential is the infinitely differen-tiable function. Define the 1st order ordinary differential operartors by

(2) then, the oprtator P v

 

is expressed as

 (3) Note that the variable x can be regarded as both real or complex throughout the paper.

The differential operators A, B are said to be semi-commutative, if the commutator

A B,

ABBA

 

 

,

is the multiplicative operator. As for the 1-dimensional Schrödinger operator

2

H u  Du x

   

(4) the identities

 

1

d

, ,

d

n n

A u H u Z u n

x

 

 

  N (5)

are well known for the differential operator u

2n 1

of the order  difined by

 

 

   

0

1 1 d

,

2 2 d

n

n j

n j j

j

A u Z u D Z u H u

x

 

 

(6)

 

j

Z

where u

 

, vv x t

2

6 0,

t x xxx

v v v v

are the KdV polynomials which will be explained precisely in Section 2.

On the other hand, in [1], R. M. Miura discovered the following interesting fact; if solves the mKdV (−) equation

  

 

, uux t

 

   

2

, x , ,

ux t  v x tv x t

6 0,

t x xxx

u uu u

then both functions defined by

(7) solve the KdV equation

  

2

(8) where the subscript denotes the partial differentiation. The transformation defined by (7) is the Miura transfor-mation which plays the crucial role in the soliton theory. By (7), we have immediately the relation

.

x

uu v

u u

(9)

The transformation defined by (9) is

noth-ing but the Darboux transformation. Usnoth-ing the KdV polynomial, the relation (9) can be expressed as

   

1 u Z u1 vx.

Z (10) Considering the above facts, we investigate the prob-lem to express the differential polynomials

*Corresponding author.

(2)

 

 

j j

Z u Z u in terms of the function vv x

for the stationary case, i.e., when the function v is independ-ent of the time variable t, i.e., in what follows

 

j

Z

 

ux

   

2 . v x v x are defined by

 

ux  

As a result, we obtain the new formulation of the stationary mKdV (−) hierarchy.

In our previous works [2,3], it is clarified that the semi-commutative operators and Darboux transformation are deeply related to the spectral theory of the differential operator H u

 

uu x

 

when the potential

is algebro- geometric. The aim of our work is to extend these results concerned with the 1-dimensional Schrödinger operator

H u to the 1-dimensional Dirac operator P v

 

 

. The present work can be regarded as the first step of it. By applying the results of the present paper, we can obtain the various transformation formulas concerned with the algebro-geometric elliptic potential. These results will be reported in the forthcoming paper.

The contents of the present paper are as follows. In Sec- tion 2, we explain the fundamental materials which are ne- cessary for the present work. In Section 3, calculation of the commutator of the Dirac operator and the differential operator constructed from the semi-commutative operator of the KdV hierarchy is carried out, and the main theorem of the present paper is stated. Section 4 is devoted to the proof of the main theorem. In Section 5, we construct the recursion operator associated with the mKdV hierarchy.

2. Preliminaries

The KdV polynomials Zn u are differential

polyno-mials defined by the recurrence relation

 

 

 

u ,n1, 2, ,

Z u

 

u

1

n n

Z u   u Z (11)

with the condition 0 , where is the for- mal pseudo-differential operator defined by

 

1 

 

1 1

   

1 3

2 4

u D  u xu x D D

   

 

. (12)

Then Zn u

 

u x

turn out to be the differential polynomi- als in . For examples, we have

 

1 ,

 

3 2 1 .

2 8 8

1 2

Z uu Z uuu

 

, uu x t

If depends also on the time variable t, then the evolution equation

 

33

3 1

4 8

u u u

u x 2

Z u

t x x

 

 

is nothing but the KdV equation. Hence, we call them the KdV polynomials. See [3] for more details of the KdV polynomials.

In what follows, we will often use the higher order

derivatives of the differential polynomials u

 

 

. So, for the brevity, we will use the following notations of derivatives of the KdV polynomials defined by

 

d

, 0,1, 2, , d

k k

j k j

X u Z u k

x

  

 0

 

 

j j

X uZ u

   

 1

 

1

,

n n

. Thus we have where

A u H uX u

 

 

 

vv x

From now on, we restrict ourselves to the stationary problem, i.e., the function under consideration depends only on the space variable x.

One immediately verifies the identities

 

   

.

H u L v L v 

 

S v

 

(13) Define the multi-component operator by

 

 

 

 

2

0

0

1 0 0

.

0 1 0

H u S v

H u

u x

D

u x

 

 

 

 

 

 

    

   

 

2

   

   

0

 

0 L v L v

P v S v

L v L v

 

 

 



 

(14)

By (13), one can show immediately the operator iden-tity

. (15)

On the other hand, define the multi-component differ- ential operator A vn

 

 

by

 

 

0

0

n n

n

A u A v

A u

 

 

 

 

   

   

   

 

 

 

 

1

1

1 1 ,

, 0

0 ,

0 . 0

n

n

n

n

n A v S v

A u H u

A u H u

X u

X u

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

n

. (16)

Then, by (5), (14), and (16), we have immediately

(17)

A

Therefore the operator  v are semi-commutative with the operator S v

 

.

   

,

3. Calculation of

A vn P v 

 

n

v by K

Define the scalar differential operators

 

 

 

 

 

n n n

(3)

 

J

   

   

 

,

0

n n

n

   

 

.

0 n

n

A v P v A v P v

Kv

   

 

  P v A v

Kv

 

 

 

 

 

n

(19)

By the definition (6) of the operator A u , we have immediately

 

 

 

 

 

 

 

 

 

 

 

0 1

0

0 1

1 1

2 2

1 2 n

n j j

j

j j

K v X u D X u

L v X u D X u

 





 

 

 

 

n j.

L v

H u  

  

 

 

   

 

. n j

n j

n j

(20) By (15), we have

 

 

 

 

 

n j

H u L v L v

L v L

L v H

  

L v L v

v L v

u

 

 

Furthermore, by the definition (2) of L v

 

, one verifies

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0

0

1 1

0 2

1 2 1 2

1 2

n

n j

j

j j

j j

K v X u X

v X u X u

v X u X u

 

 

 

 

 

 

2

,

j

n j

u D

H u

  

 

 

,

j B Zj u

 

2 . B  D v

 

(21)

where we used the identity

 

B Zu (22)

where

The identity (22) is derived in [4], and is called the fundamental identity of the Darboux transformation in it, By the fundamental identity (22), we have immediately

 

 

 

 

 

 

0 0

1 1

1 2

j j

j j

v X u X

X u

 

 

 

0.

u

X u D

 

   

 

Put

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0 2

1 1

0 2 ,

n

n n

n n

X u D

X u

X u

 

 

 

(23)

then we can express the identity (21) in terms of

1 2 1 2

1 2

n n

J v X u

v X u

v X u

 

 

 

 

n v

as

 

 

1

 

 

, 0,1, 2,

n n n

KvJvK v H un 

For n 0,1

. (24) , one verifies easily

 

0 1 2 Kvv,

 

1 3 2

1 8 4

K v   v v v. Thus, for

 

 

0,1, n n

nKvKv hold, and t

no erentia e multipricative

op-erator. Thus, the ope n

hey are t diff l operators, but are th

rator A u

 

and the Dirac operator

 

P v turns out to be semi-commu tive for ta n0,1. For general n, we have the following theorem which is the main result of the present paper.

Theorem 1. The multi-component differential opera-tors A vn

 

and P v

are semi-commutative, i.e.,

 

n

Kv are the multiplicative operators, and they coin-cide with each other, i.e., the e uality q

 

 

n n

KvKv 5) (2 hold for all n. Moreover, if we denote them as Kn

 

v ,

then the identities

 

1

 

1

 

2

n n n

1

K vZu Zu (26) hold for all nZN

 

0 .

If the potential v depends also i.e.,

on the time variable t,

 

,

vv x t , the evolution equation

 

3

v

K1 1 3 3 2

8 4

v v

v v

t x x

  

   

  

is nothing but the mKdV (−) equation. Hence polynomials n

, we call the differential K v

, nZN

0

the

4. The Proof of Theorem 1

We prove the theorem by induction. Firstl for n 0 mKdV (−) polynomials.

y,  ,

one verifies

 

0 1

. 2

K vv

On the other hand, by (10), we have

   

 

 

 

 

0

2vK v

 

0 0

1   1 1 

1 1

2 X uX u 2 2 u 2u 

2 2

1

1 1 1

2 2 2

1 .

v v v v

   

 

(4)

 

 

 

1 1 2 n n  

 

0 0

n

K vX u X u

holds. Then, for n, we have

 

 

1  0

 

 0

2

n n

n n

 

 

.

KvJvX u X uH u

From (21) and (23), we have

 

 

 

 

 

 

 

 

 

 

 

 

 

0 2

0 2 0

2

2 2 2

1

.

n n n n

n n n

1 1

1 1 1

2

Kv   vX u  vX uv X u

v X u X u v X u

          

This implies that n

 

Kv are the multiplicative

op-erators, i.e., A vn

 

and P v

 

a

Similarly, by induction, we can show that

re semi-commutative.

 

n

Kv

coincide with each other.

Next we show the identity (26). By straightforward calculation, one can show

 

 

 

 

 

 

 

 

 

 1

 

d 2 2

1 1 3

n  1

 

 1

 

 0

 

2 2

2 2 2

2

n

 0

 

 0

 

2 n

3 2

2 1

d 1 1 1

2

.

n n n

n n

n n

n

K v X u vX u

X u X

u          Put x

v   v u  v Xu

(27)

v X uv X uvv Xu

  

vv Xuv X  

 

 

 

n

 

,

1 2

n n

K vKvKv

then, by (27), we have

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 3 1

1 2 1 2 1

0

0 0

d 1 d d

d 2 d d

1 1 1 1

4 2

1 2

1

2 2

2 2 ,

n n n

n n n

n n n

n

n n

K v K v K v

x x x

0

1

n

2 4X u X u v X u

v X u v X u v X u

v X u

vv X u vv X u

                           

where we used the relation

v X u

     

 

 

 

 

 

1 1 0 0 1 1

2 n 2 n

n n

v X u

vv X u

 

 

 

 

which are derived immediately from (22). Thus, we have

 

 

v X u

vv X u

  

 

 

 

 

 

 

 

 

 

 

 

 

 

0 1 3 2 0 1 3 2

d 1 1

2

d 2 2

1 4 1 2 2 1 . 4 n n n n n n n

K v v vv X u

x

v v X u X u

v vv X u

v v X u X u

                            (28)

By (7), we have

 

 

 

 

 

 

 

 

 

0 3 0 3 1

d 2 2 4

1 1

2 4

n

x n

d 1 1

n x

K v u x u x D D X u

x

u x u x D D X u

                   

, we have





By (12) which is the definition of 

 

u

 

 

 

   

 

 

 

 

0 0 1 1 2 1 . 2

n n n

n n

1

K v   u X u u X u

Z u Z u

   

 

 

 

 

This completes the proof of Theorem 1.

5. The Recursion Operator

In the preceding section, we have shown the relation

 

1

 

1

 

1

2 n n

Kn vZ u Z u

hold for all n. Therefore, by (28), we have

 

 

 

 

 

 

2

1 1 1

0 0

1 2 2 1

2 2

1 2

n n n n

n n

K v vv K v K K

v X u X u

u

  

 

 

 

1 1

.

n n

v X uX

 

 

 

 

Then, by (22), we have

  

1 1 n

v D  vK

1 2

1 1 1

1

2 4

n n n n

KD Kv Kvv K

  (29)

cursion operators of The relation (29) defines the re

 

n

K v . Therefore, we have the following theorem. Theorem 2. The formal pseudo-differential operator

 

v

defined by

 

1

 

v D  vf

 

is the recursion operator associated with the mKdV (−) polynomials

1 1 3 2 2

4

v f D  D f v Df vv f 

    

 

n v , nZ N

 

0 , i.e.,

(5)

 

,

n Kn v nN

hold.

Using this rec rator 

 

v , one can calculate

easily the hierarchy KdV (−) polynomials.

REFERENCES

[1] e Vries Equation and Gener-alizations. I. A Remarkable Explicit Nonlinear Transfor-mation,” Journal of Mathematical Physics, Vo

1968, pp. 1202-1204.doi:10.1063/1.1664700

 

v K1

 

v

 

ursion ope of the m

R. M. Miura, “Korteweg—D

l. 9, No. 8, [2] M. Ohmiya, “Spectrum of Darboux Transformation of

ferential Operator,” Osaka Journal of Mathematics,

80.

Osaka

Journal of Mathematics, Vol. 32, No. 2, 1995, pp. 409-

430.

[4] M. Ohmiya an ux Transformation

y KdV Hi-Dif

Vol. 36, No. 4, 1999, pp. 949-9

[3] M. Ohmiya, “KdV Polynomials and Λ-Operator,”

d Y. P. Mishev, “Darbo

and Λ-Operator,” Journal of Mathematics/Tokushima Uni- versity, Vol. 27, 1993, pp. 1-15.

[5] M. Matsushima and M. Ohmiya, “An Algebraic Con-struction of the First Integrals of the Stationar

References

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