PHYSICAL REVIEW D VOLUME 35, NUMBER 6 15 MARCH 1987

Inverse scattering method and n-step superpotential for axially symmetric, static solutions of a models

Tibor Kiss-Toth Faculty of Education, University of Osijek, 54000 Osijek, Yugoslavia

(Received 23 September 1986)

The superpotential for the n-step soliton solution is derived for an axially symmetric, static solu- tion of a models by using the appropriately modified inverse scattering method of Belinsky and Zakharov. Finite-energy solutions are constructed by the method.

I. INTRODUCTION

Over the past few years there has been significant pro- with the well-known constraint gress in the understanding of nonlinear o models in two dimensions.' Higher-dimensional models have also been $i$i= 1 . used in different fields of theoretical physics. Hirayama, This action yields the field equations Chia Tze, Ishida, and Kawabe pointed out the connecting links among the axially symmetric versions of o models, a,,a@di=hdi , - - - ,- the Ernst equation of general relativity describing station- ary and axially symmetric vacua, and the SU(2) vacuum where h is the Lagrange multiplier field.

Yang-Mills fields.' The self-dual solutions of classical A convenient form of the same action is

gauge theories can be derived from an equation which resembles a four-dimensional nonlinear o model.3

For the Ernst equation various solution-generating techniques have been found, e.g., Backlund transforma- tions and an inverse scattering method.4s5 Using these methods one can construct the asymptotically flat, sta- tionary, axially symmetric vacua in general relativity and as a completely different application the static, axially symmetric self-dual monopoles for Yang-Mills fields with a Higgs scalar in the adjoint representation.6

It was shown that this method can be applied for SU( N ) principal a models also.7 The process we follow is the generalization of the standard method of Belinsky and ~akharov. ' Recently, we have succeeded in constructing an important function which we call a superpotential in these model^.^ We have hinted previously that with the help of this superpotential it would be possible to calcu- late the energy of static, axially symmetric a models. In the present paper we deduce an exact finite-energy solu- tion starting from a special seed solution.

Let us now explain the structure of our present paper. The formal side of the method considered here has been described in a detailed manner in our earlier papers.799 Therefore, we shall present here only the basic points which are necessary for a complete exposition.

Section I1 deals with the inverse scattering method for static, axially symmetric SU(N) principal o models. In Sec. I11 we generalize the superpotential for these models. In Sec. IV we calculate the energy of a single soliton solu- tion by this method.

11. AXIALLY SYMMETRIC, STATIC FOUR-DIMENSIONAL u MODELS

First, we consider the usual form of the action for the nonlinear chiral models:

with g = 4 0 + i ~ i + i E S U ( 2 ) and T['S are the familiar Pauli matrices.

The generalization of the model is trivial by considering g E SU( N ) giving the so-called principal SU( N ) a models. The field equations of the theory (4) are

If we impose on Eq. (5) the condition of the axial sym- metry the system in the static case reduces to

a r ~ r a r g g - 1 ~ + a , s ~ r a x 3 g g - ' ) = ~ , (6)

where r = ( x ~ ~ + x ~ ~ ) " ~ . These equations are similar to the well-known equations

of Belinsky and Zakharov. The only difference is that here the matrix g is unitary.

The key to our solution-generating method is to find a pair of linear operators for Eq. (61,

with the following notation:

where h is an arbitrary complex ( r and x3 independent) parameter. The condition of unitarity, imposes anti- Hermiticity on the matrices U and V.

We could say that (7) represents a linearization of (6) because the integrability condition of Eqs. (7) are just (6).

Our procedure generates new solutions of (7) if we know some particular solution of the problem (6) which

35 1855 - @ 1987 The American Physical Society

1856 TIBOR KISS-TOTH 3 5 -

we denoted by go, Uo, and Vo. We also assume that for go we have a solution \Vo of (7 ) with a natural boundary condition for \V( h,r,x3 ) at h=O:

We look for a new solution of (7 ) in the form

which implies that the new solution of (6 ) is given as

g =X(0 )go . (10)

For X we obtain, substituting ( 9 ) into (7 ) ,

The properties of X in the complex plane h are limited by the unitarity of g. Namely, [ X t ( X ) ] - I satisfies the same equation as X and the simplest way to guarantee the unitarity of X is

x ( ~ ) x + ( x ) = I , (12)

where the overbar denotes complex conjugation and dagger stands for the adjoint. Furthermore, the compati- bility of (12) with (8 ) demands X ( co , r , x , ) = I .

The soliton solutions of our framework correspond to the existence of simple poles in the matrix X ( h ) . There- fore, we write this matrix in the form

and in a similar way

where n denotes the number of poles. The residues Rk and Sk and the pole positions p k and v k are functions of r and x 3 only.

Substituting ( 13) into ( 1 1 ) and comparing the various residues of the two sides we can completely determine the pole trajectories p k , while for the construction of matrices Rk we need also the supplementary condition (12). In this way we find that pk satisfies

whose solutions are the roots of a surprisingly simple quadratic algebraic equation

where mk are arbitrary complex constants. Taking the residua at the poles pk = h in the relation

X X - ' = I we can determine the components of the degen- erate matrices Rk in the form

( k . i k 1 ( k , j k I with mb qb = 0 , where 1 < s k ( N - 1 are the dimen- sions of the subspaces associated with each pole.

From the unitarity constraint (12) it follows

( k , i k I The vectors ma defined in (16) can be obtained

directly requiring that Eq. ( 1 1 ) be satisfied at the poles ( k , i k )

h = p k . The solutions for ma are given in terms of \Vo as

( k , i k i where mo, form a set of s k arbitrary N-dimensional constant vectors.

( k , ik ) The remaining task is to get nu from condition (12) .

They are the solutions of a system of 2:=, si linear equa- tions

where l- is a matrix with elements

with

y ( l , j I ) ( k , i k i ( 1 , ~ ~ ) - i k , f k l =mb m b .

Finally according to (81, (91, and (13) the new solution of the matrix g is

For a new solution of our Eq. (6) it must not be forgot- ten that beside the unitarity of g we have to satisfy the supplementary condition detg = 1 . We can calculate the determinant of g as given in (21). This form is not the most convenient to do the calculation and we notice that the same result can be obtained also iteratively, i.e., apply- ing a similar procedure successively n times, using in every step X with a single pole only. This is formally equivalent to the introduction of the n solitons one at a time successively. Therefore, it is enough to calculate the determinant of X containing only a single pole. In this case X 1 is expressed as

where P is a Hermitian projector and accordingly has the

3 5 - INVERSE SCATTERING METHOD AND n-STEP . . . 1857

properties

The determinant of (22) at h=O is

It is easy to show that the "renormalized" gPh has the form

and also solves Eq. (6). We shall call such a solution a physical one and denote it by gph.

111. THE CONSTRUCTION OF THE SUPERPOTENTIAL

From now on we shall consider the axially symmetric case only. It is easy to show that using Eq. (6) one can construct a function in the following way:

We shall call r the superpotential. Justifying this name we note that this quantity as given in (24), is identical to the superpotential introduced in Ref. 10 and further stud- ied in Ref. 11, when g is a symmetric 2X 2 matrix.

With the aid of the previous inverse scattering method one can construct a new r from go and the corresponding r0 in the form

It is also convenient to do this calculation in two stages. First we calculate the value of r when we substitute in (24) the nonphysical value of g, and then we use a simple pro- cedure to find the physical value of the superpotential.

This transformation can be iterated giving the possibili- ty of generating infinitely many new solutions from an in- itial one.

If we rewrite (24) in a more convenient form

we are able to integrate (241, and we obtain the new super- potential in the form9

(27) with s the dimension of the projector P, y an s x s matrix, and c l an arbitrary constant.

The great virtue of (27) is that it gives the possibility of iteration. After the first step in which we went from go and ro to r 1 containing one soliton, corresponding to the presence of only one pole, we take r l and g l as the start- ing solution; and add one more pole at h=p2 with an as- sociated projector of dimension s2. After the repeated calculation with these new ingredients we get r2:

Here r 1 is given by (27) and y2 is an s2 Xs2 matrix the analogue of y l . The new 7 2 also satisfies (24). Repeating this procedure, i.e., introducing further poles, we get a su- perposition law, which emphasizes the solitonic nature of our solutions, because of similarity to the properties of some solitons in two-dimensional (2D) theories.

Even more remarkable we are able in a special case to express these results of iterations in a single formula writ- ten directly in terms of original ingredients.

Equations (27) and (28) suggest that this formula in the general n-soliton case, when all projectors are one dimen- sional, has the form

We have seen in (27) that this expression is valid for n = 1 and now we have to prove that (29) holds for n + 1, by using the method of mathematical induction.

We suppose that we have some solution r n , Y n of our problem. We shall construct a solution 7, +, , Y n +, from it by introducing m solitons, corresponding to poles h=pn h=pn +2 , . . . , h = y n +,. We assume that the superpotential in this case takes the form

where c, +, is an arbitrary constant, and r is given by h = p n . This procedure leads to the expressions for the ( n + i ) - ( n + j )

mb m b new Y n and * , - I :

rn+i,n+j= (3 1) ~n +i-Pn +j (33a)

The vectors mAn+" are constructed according to (17):

In+i)- ( n +i) ma -mot [ * n - l ( ~ n +i , r ,~s) Ica . (32) (33b)

Now, consider a known solution Y n - l , r n - l and let us add to it one more soliton, corresponding to the pole where projector Pn is constructed from Y n - 1:

1858 TIBOR KISS-TOTH 3 5 -

The vectors 1:"' are given by the expression

Beside the vector (35) we shall need also the vector [An + ; I :

where m c + " represent the same arbitrary constant vec- tors as in (32). We introduce a new matrix En +,,, + B as

[Ln + a ) ~ ( n +B) b

En +a,, +@= (37) ~n +a-pn +B

where we adopt a new convention about indices; namely, the running indices i and j go through the values 1,2, . . . , m while a and f l denote indices that go through the m + 1 values O,1,2, . . . , m.

Combining (32), (33b1, and (34) and using (37) we obtain an ex ression for the vectors min+" in terms of the vec- tors l!") and lin +jl:

Now from (38) and (3 1) we find the connection between the matrices Tn +i ,n + j and En +,,, +B:

From this equation it follows that the determinant of the matrix En+a,n+B is expressed in term of the de t rn +i.n + j as

On the basis of (27) we write the formula for 7,:

Substituting (41) in (30) and using (40) we get

r n +m = ~ n - I +In cn +m + 1detEn +a,n +B I

This result together with (35)-(37) is nothing but the expression (29) itself except that it is for the case m + 1. The solution T, +, is obtained from rn by adding n + 1 solitons to the latter.

Now in a simple way one can determine the physical value of the superpotential, 7ph, which is obtained when gph is substituted in (24) instead of g. The key point is to observe that replacing g by g p h = g ~ (which satisfies the supplementary condition detg = I), where M =(detg)-"'v~2v, in (26), we get the following additional terms only:

Therefore, we can look for r P h in the form Now if we put together (441, (47) and (27), (28) and compare this expression to (25) we see that T in (25) is

7 P h = 7 + F . (44) given by the sum of several logarithms, coming from both From (26), (43), and (44) we get for the normalization fac- the factor and from the nonphysical tor F: value of 7. We note that the resulting formula is entirely

I r explicit; and once go and \Yo(h) are determined, it con- (a, + i a X 3 ) F = - - -[(lndetg),,2-(lndetg),x32] tains as free parameters the number of associated dimen-

N 4 sions sk and the ok parameters of the poles p k as well as the r and x 3 independent vectors, m;,, etc.

+iL( ln detg),,(ln detglZx3 2 IV. THE ENERGY OF THE SU(2)

where detg is PRINCIPAL u MODEL 'i

detg = n 1% 1 detg. . (46) With the help of the superpotential it is possible to cal- culate the energy of the static, axially symmetric case

I

The integration based on repeated application of (14) without knowing the g matrix explicitly.

after a painful amount of mathematics, gives Indeed using (24) and the property that

AT= - + ~ r ( g , , ~ g - )2 , (48)

one can express the energy of this configuration as

E = ~ o m r d r ~ ~ m ( q , + ~ , x , x , ) d ~ 3 - 00 . (49) (47)

Thus once we constructed the new superpotential with our

35 - INVERSE SCATTERING METHOD AND n-STEP . .

algorithm we can determine the energy of the resulting solution immediately.

In this section we shall consider a soliton solution for a simple case of SU(2) nonlinear a models. The soliton will be situated at x 3 = 0 All the results will be obtained easi- ly from the preceding general analysis. In this case w takes the value w = i a . Then it is convenient to introduce oblate spheroidal coordinates, defined as

r =a[ ( 1 +a2)( 1 - P 2 ) ] 1 / 2 , x 3 = a P a ,

In these coordinates p has the form

We suppose a background solution in the form

where K, m, and 6 are arbitrary real constants. We shall now construct the superpotential. To determine r for our case, first we have to calculate r o from (24) with the aid of our background solution (52). r0 is given by the expres- sion

Furthermore, we need To with go on the right-hand side of (7):

For this special case the projector is

Now we can calculate dety using (20) as

Now the only remaining task is the calculation of the normalization factor F because we have to determine the physical value of r . According to (47) for s = 1 we can write the expression

Combining all these relations we finally get, for r P h in our case,

Taking the arbitrary coefficients m and K equal to zero, fied inverse scattering method which was worked out by we obtain, for r p h , Belinsky and Zakharov for the Ernst equation. The

I method gives us the possibility to generate infinitely many

,rph = -!-In (a2+ 1 )'I2 (59) new superpotentials, containing some free parameters, 4 i a ( ~ ~ + / ? ~ ) ' / ~ from a background solution. The construction of these -

We write Eq. (49) in oblate spheroidal coordinates: new solutions requires pure algebra only, if the starting solutions are once fixed.

E = lom ~ - + ~ l d o d ~ [ ( l+u2)T,uu+( l - ~ ~ ) ~ , ~ ~ With the aid of this superpotential it is possible to cal- culate the energy for the static, axially symmetric case,

+ar,u-Pr,~ l . (60) without knowing the matrix g explicitly.

Finally integrating (60) with (59) we get 'lT E = - . 2

(61)

Applying our n-step formulas on the same seed solution we shall obtain the corresponding finite-energy n-soliton solution.

V. CONCLUSIONS

In this paper we have given a general framework for constructing the n-step superpotential for the axially sym- metric, static SU( N ) a models. This was done by a modi-

Indeed in the last part of the paper using r in the one soliton case we have obtained a finite energy. Thus once we constructed the new superpotential with our algorithm we can determine the energy of resulting solution immedi- ately for any number of steps.

ACKNOWLEDGMENTS

The author would like to thank Dr. Stana Kiss-Toth, Dr. Z. Horvath, and Dr. L. Palla, who have offered a number of helpful suggestions in the preparation of this paper.

1860 TIBOR KISS-TOTH - 35

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