Instanton solutions in the anisotropic σ model

Volume 126B, number 1,2

PHYSICS LETTERS

23 June 1983

INSTANTON SOLUTIONS IN THE ANISOTROPIC ¢r MODEL Susmita G A N G U L Y Physics Department, Calcutta University, 92, A.P. C. Road, Calcutta-700009, India and Dipankar R A Y Applied Mathematics Department, Calcutta University, 92, A.P.C. Road, Calcutta-700009, India Received 7 June 1982

In a recent paper a set of equations have been obtained for a nonlinear anisotropic o model in two-dimensional euclidean space. The present note presents the complete set of solutions of these equations.

1. Introduction

_ 1 f lx2 + y 2 - (X - y ) 2 ] 2d2 x . s=-~

In a recent paper K u n d u [1] has obtained the following equations for a nonlinear anisotropic o m o d e l in two-dimensional euclidean space.

+/322 = sin2/3

+ 3'22>,

(/3, 7) are the polar angles of the isovector'n a E S 2. K u n d u [1 ] has obtained a particular set of solutions of eqs. (1) as follows: /3 = 2 tan -1 [ep+x/~ e~) n/z] , 3' = na

/3~ +/32 = 2 sin/3 (/313'1 + /32T2 ) ,

(2b)

(3)

where p2 =x'2 + y 22 ,

tana=x2/x 1 ,

2(/311 +/322 ) -- sin/3 (711 + 3'22) n and c are constants. In the present note we seek to obtain the c o m p l e t e set of solutions of eqs. (1).

-- sin 2/3 (7~ + 3'22) = 0 , sin/3[2 sin/3 (3'11 + 3'22) -- (/311 +/322)]

2. Solutions + 2 sin 2/3 (/3171 +/323'2) -- cos/3 (/312 +/3~) = 0 , (1) Case 1.

where /31 = ~t3(X1 'x2)/~lxl '

/3 = constant.

/32 ~ c)/3(xl,x2) [~x2 ,

This leads to the trivial solution

/311 -= ~2/3(xl ,x2)/~xl 2,

X=0=Y.

and so on. Defining X~V/3,

Case 2. Y =sin/3V3,,

(2a)

Using some elementary algebra one can eliminate (711 + T22), (72 + ,),2) and (/313'1 +/3272) from the

the action integral reads

0 031-9163/83/0000-0000/$

/3 v~ constant.

03.00 © 1983 North-Holland

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Volume 126B, n u m b e r 1,2

PHYSICS LETTERS

four equations o f ( l ) to obtain an equation for/3 as sin/3 (/311 + t322) - cos/3 (/32 +/32) = 0 ,

(6d)

From eqs. (4), (6a) and (6b) _1

Ull +u22 =0,

3`u--/,

where

= fcosec/3

sin/3[2 sin/3 (3`uu + 3`uo) -/3uu] + 2(sin 2/3)/3u"lu -- (cos/3)/32 = 0 .

which is equivalent to

U

23 June 1983

3`0

+~/3

(7)

Putting eqs. (4) and (7) into (6) it is easy to see that all the eqs. (6) and hence all eqs. (1) are satisfied provided u and v satisfy (5).

d/3,

i.e.

3. Conclusion

13= 2 t a n - l ( e u ) .

(4)

Let v be the solution of the Laplace equation conjugate to u i.e.

Therefore if one leaves out the trivial solutions that lead to X = 0, Y= 0, the complete set of solutions of the set of eqs. (1) and (2) are given by

U1 =l) 2 ,

/3=2tan-l(eu),

U2 = - v

1 .

(5)

Making a transformation of variables (x 1 , x 2) ~ (u, v) by using (5) and noting that by virtue of(4)/3 is a function o f u only eqs. (1) reduce to

3`=~u+~vx/~,

(8)

/32 = sin2/3 (3`2 + 3`2),

(6a)

where u and v are mutually conjugate solutions of the Laplace equation and X, Y and S are given by eqs. (2). It is easy to check that the solution given by Kundu [1] [eqs. (3)] is a special case of(8).

/32 = 2 sin/3/3uTu ,

(6b)

References

2/3uu - sin/3 (3`uu + 3`~) - sin 2/3 (3`2 + 3`2) = 0 ,

(6c)

[1] A. Kundu, Phys. Lett. ll0B (1982) 61.

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