GAUGE

TRANSFORMATION

OF SIMPLE

WAVES

i

M.W. KALINOWSKI*

Institute o f Nuclear Research, 00-681 Warsaw, Ho~a 69, Poland

ABSTRACT. In this paper we find some connections between simple (Riemann) waves and nonlinear representations of local gauge groups originating from orthogonal and pseudo-orthogonal groups. These representations are obtained by introducing fibre bundles for Riemann waves and are symmetries of the original equation.

1.

INTRODUCTION

Let us consider a system of partial differential equations of the form v= l, 2,..., n aj8/) (u 1 , u 2 ..... d ) ,-~;, d ( x I . .x. .2,. ox"

x n) = O,

s = 1 , 2 ..... m, m ) l

/ ' = 1 , 2 ..... l (1.1) X = (XI, x 2 ..... x n ) E e,

LI(X) = (I.ll (X), IA2 (X) ..... bll(x) E ~.

which is a quasilinear homogeneous system of the first order with its coefficients depending only on unknown functions. This system may be undetermined, i.e., m ) 1. Let us suppose that the system is a noneUiptical one. That indicates that there exist some nontrivial solutions of the system of algebraic equations nSV,vj'A -j . ,.v = 0

where rank Ila;VXvl[< l

(1.2)

for vectors 3' ER1 and X E R n. The above system of algebraic equations specifies so-called knotted characteristic vectors in hodograph space (the values of the functions u i) Js = R 1 and in physical space e = R n (independent variables). The pair 3' and )t will be called a knotted pair iff it obeys Equation (1.2). This fact will be marked by 7 ~ X. The matrix Liv = 3,J)tv created by a pair of knotted vectors will be a simple integral, because rank IIL~(uo)II = 1. It is convenient to consider X as an element of space of linear forms, e* 9 X: e -+ R 1. On the *Partially supported by NSF contract INT 73-20002A01. Present address: Institute of Philosophy and Sociology, Polish Academy of Sciences, 00-330, Warsaw, Nowy Swiat 72, Poland.

Letters in Mathematical Physics 7 (1983) 479-486. 0377-9017/83/0076-0479 $01.20. 0 1983 by D. Reidel Publishing Company.

479

other hand, in terminology of tensor calculus, if we consider x E e as a contravariant vector, then )t E e* is a covariant vector. In these terms the element L is an element of tensor space TuJ( | e* of the form L = 7 | X. Now we introduce a simple wave, which will suggest a concept of distinguishing simple integral elements from a set of integral elements. Let the mapping u: D -+ JC, D C e be any solution of the system (1.1). We call u a simple wave for a homogeneous system if the tangent mapping du is a simple element at any point Xo E D. Let us consider the smooth curve: P: u -+f(R) in hodograph space Js parametricized by R, so that the tangent vector dr(R) - 3'if(R)) dR

(1.3)

is a characteristic vector. Thus, there exists a field of characteristic covectors X(u) connected with "),(f(R)) defined on the curve F: X = X(f(R)). Now we may state that if the curve F K Jf obeys condition (1.3) and if ~o(. ) is any differentiable functions of one variable, then the function u = u(x ) specified in an implicit way by u =f(R),

R = ~(Xv(f(R))x u)

(1.4)

where a]VT]Xv = 0 is the solution of basic system (1.1). Such a solution is called a simple wave (or the Pdemann wave) [ 1 - 3 ] . A proof may be obtained by direct differentiation of implicit relations (1.4). The covector X appearing in this expression fixes the velocity and direction of the wave propagation. The curve F fulfilling condition (1.3) is called a characteristic curve in a space of hodograph Js This tells us that if the mapping u: e -+ Js is a simple wave, then a characteristic curve in space JC is a picture of mapping u. Parameter R specified on this curve is called the Riemann invariant. Now let us consider a nonelliptical equation of the second order

i,]=1

~x ~' ""' ~x n

~x' X. x j / = O.

(1.5)

According to [4, 5] 7 is proportional to X and:

Q(Xa, X2, ..., Xn) = ~

aqXiX} = O.

(1.6)

i,]=1

Equation (1.6) is the equation of a cone of the characteristic covector X, specifying the velocity and direction of propagation of a simple wave. We may obtain all possible simple elements for (1.5) and, consequently, all simple waves after integrating (1.4) (see [4, 5] ).

480

2.

GAUGE TRANSFORMATIONS FOR SINGLE WAVES

Now we construct some geometrical structures for simple (Riemann) waves of (1.5). These structures establish relations between Riemann waves of (1.5) and allow us to introduce nonlinear transformations connecting two Riemann waves (exact solutions of (1.5)). The nonlinear transformations are of a gauge type, and are symmetries of the original equation (1.5). Namely, let matrix A = (aij) have K eigenvalues coi, i = 1, 2, ... K, each of order Ii, EK= 1li = n. In such a case we have a natural group operating on simple elements X (see [1, 2] ). FIX=1 | O(li) each of the group O(li) operates on the coordinates i--1

i--I

i

vi,j= Z it, Z l ~ + l .... , Z l ~ rml

r=l

r=l

without destroying the diagonalization of matrix A, where:

X =B

(vl) V2

=By

= B ( a t . . . . . O~m)V ,

(2.1)

\Vn!

where B is a matrix diagonalizing A. It is easy to see that a l , a2, ... am are parameters of the group K

K

Z | oq,), i=1

m = Z 89 q , - 1). i=1

For details see [4, 5]. Simultaneously we can connect the conformal group with the cone /2

zjv~ = O, (zj is equal to one of c~i)

(2.2)

j=l preserving this cone. Let

zj = eiizji

(2.3)

where ej = sgn zj. By transforming the length

v} =

lx/~Tjlvj

(2.4)

481

we transform (1.7) into

= 0,

/'=1

(2.5)

i.e., into a canonical cone. Because of the fact that the length of vector X, and consequently v or v' do not play any role in the construction of simple elements (3' ~ X), we deal, in fact, not with a conformal group connected with the cone (2.5) but with the group O(p, q). This group conserves a quadratic form n

Q(v ' , v') = • eiv i,2 ,

(2.6)

i=1

where p = the number of integers 1 and q -- the number of integers ( - 1 ) in the sum (2.6). Obviously, p + q -- n (we assume there are no zero eigenvalues). Now, let us notice that the classes of simple elements and, in consequence, simple waves which are constructed according to [4, 5], are related to the choice of a concrete chain of subgroups O(p, q). This chain ends on two-element group {e, -e} or trivial group {e}, hence, O(p, q) = O(Pl, ql) D O(P2, q2) D ... D {e, -e} D (e}

(2.6')

where forpi, qi, Pi+l, qi+l we have the following relations, either pi=Pi+l,

qi=qi+l + 1

or

pi=Pi+l + 1 ,

qi=qi+l,

(2.7) po =p, qo =q.

In this way the dimension of space in which the group operates diminishes for 1 according to [4, 5]. The choice of the sequence of subscripts io, ix . . . . iK corresponds to one of the possible chain of subgroups (2.6). Thus, with each simple element we can connect the following group in a natural way,

Li=

I

|

]

O(li) | r=l

qi).

(2.8)

The origin of each factor of simple product is, of course, different. In general, we connect with Equation (1.5) the group

L =

482

I

| 1-I o(I,) r=l

1

| o p, q).

(2.9)

Group L i acts on a submanifold F i C e* (the manifold of simple elements of a chosen class according to the classification from [4], [5] ). Since a simple element is a function of a point of hodograph space 3s (the space of values of the solution of equation), we may connect very natural fibre bundles with the equation. For every class of simple elements we have a fibre bundle Pi over the base space ~ with structural group Li, typical fibre F i and projection 7ri: Pi -+ ~ . It is easy to see that dim(Li) = dim(F/) and for every simple element, X E e* we have

X=g. Xo,

(2.1o)

where g E L i and Xo E F i , X0 = const. Taking a local section of P i, we get X(u) =g(a(u))Xo,

u E D C Js

(2.11)

where a is set all parameters of group L i. But in the case of simple waves, we have Riemann invariant R (parametrization in hodograph space 3s Thus, we obtain a special structure, a bundle Hi over the base space (a straight line of Riemann invariant) with structural group Li, typical fibre Fi and projection ~i: Hi -+~- For every local section of IIi we get

X(R) =g(,~(R))Xo, n ~ v c ~ .

(2.12)

Every local section f gives us a simple wave belonging to a chosen class of simple waves (simple elements). If we have two local sections f and g we have two different simple waves of the same type. If we change the section f r o m f t o g, we change functions or(R) to, e.g., ~(R) and we get g(a(R)) = h(R)gq3(R)),

X(o~(R))= h(R)X(~(R)), h(R) ~ L i .

(2.13)

Thus, we see that an action of the 'gauge group' o f L i over a straight line ~ (Riemann invariant) on a simple wave creates a new simple wave of the same type ('gauge group' means that the parameters o f L i depend on R). In [4, 5] Equation (1.3)was solved by using arbitrary functions a(R). We now throw the weight of arbitrality from a to new, more convenient functions. Now we should do this independently for functions a and/5. For a and/3 (see [4, 5] ) one gets algebraic (or transcendent) equations. These equations will express a (or/5) in terms of new functions and their first derivatives with respect to R. The condition of solvability of algebraic (or transcendent) equations provide us with restrictions on the new functions [4, 5]. Varying a(R) to/3(R), we change these new functions and their first derivatives. Thus, we get the gauge transformation connecting two exact solutions (simple waves of the same type). This transformation is a gauge symmetry of the original Equation (1.5).

3. EXAMPLE Consider an equation of potential flow of compressible, perfect gas in two dimensions [4-6] 483

(c2 - 02x)r

+ (c2 - ~ ) r

+ 2(~xCyCy + (~

+( c2 --r162

+ CyCzCyz)= 0.

(3.1)

An exact solution (simple wave of (3.1)) with some arbitrary functions has been found in [4, 5] ~ox = ~2e c sin K(R),

~o2 =

?'12

eG Cos K(R),

~o3 =

"O1%/e 2 H - -

e 2G

(3.2)

where

K(R) =

f R

Ro

(cos ~ -- sin a)

(e2H _ C2)1/2 ed

dR' +c', c '= const ~7~ = r~ = 1,

R= (97 x + Z y +(G), R
= v,.

where chp=---c dR'

el{eG_~(H_G)(e2H(~=

shp=e2

I)ll2

1-

e2H.dH 2) 1/2,

dH =

e~ =e~ =e~, d

2H

(3.3) 2 1/2/

s i n oe =

leH(dI~)2(e2H_e2C)l/2

.(e2H_c=)I/2

C2

2H dH - +(dH]2(e2H-e2G)(e 2H- c 2) l (~-(~)-1) e 2 H ( ~ ( H - G)) ~ "e \dR/

COS~ = C3t

-ill e2H{dH]4(e2H~!

1-

+

_e2C ) . (e21_I _ c2 )

d 2 d [e2H/dH\ 2 \1/2 e2C(~R(H-G)) +2e a ~ ( H - G ) ~ - T Y [ ~ ) - 1 )

2

2

2) -fi-e2G(d(H-G))11

+

le2H (~]4(e2H _ e2G)(e 2H c 2 C4

\dR!

--

)

(For details see [4, 5] ). In this case we have L i = 0(2) 0(1, 1), ~ is a parameter of 0(2) and p of 0(1, 1) ([4, 5]). We change functions p and a into p + AO and a + As. It is a change of gauge by means of functions AO and Aa. And we look for how G and H vail change. In this way we obtain the explicit action of the gauge group on the manifold of functions H and G and their first derivatives with respect to R. We have eH dH ch p = c(eH ) dR'

e u, dill ch (p + Ap) - c(eH ~) dR

(3.4)

but ch (p + Ap) = ch p ch Ap + sh p sh Ap. 484

(3.s)

Inserting (3.4) and (3.3) into (3.5) we get: e Ht dH 1 e I't dH { e2H { ~ ] 2 ),/2 ~ \dR ] - 1 sh (A~). e(eH, ) dR = e(eH) dR ch (Ap) + e2 \e2(

(3.6)

Equation (3.6) is the nonlinear representation of the gauge group originated from O(1, 1) on a manifold of functionsH and G and their first derivatives with respect toR (H, Hi, G, G1, oL,p, Aa,/xp are functions of R). For acting 0(2) we similarly get

(3.7)

sin (a + A~) = sin a cos Ao~+ sin Aa cos a

le2H,{diit~2

eile~

1/,12_l(d/_/~]2

tt~,

dR, - I

,2s-s,_ 2.

(e2H' -e2G'jle

c2

C )--eZG'(d (H1-G'))

2,/2 /

1_eH' (dI-I~]~(e2H' e2C, ),i2. (e2H, _ c~),12

c~

e2leG d (H

cos (AcO"

I

dS<

\dR l

T M

"tl--, ][e2H2diH" ~ G"\c2\-dR] --1/1[21

j

eH(dH]

t- i

| { ~ ] 2 ( e2H-e2G'(e2H-c2)

i\dR ]

) -2'

-e

2aid . . . . . \:11/2/ t~t-r/- IJ,] | /

+

_ e2G)l/2 . (e TM - c2)li 2

e 2 G ( d ( H _ G ) 2[e2HidH"a_l)+(d~)2 t~i-t~ ) (e2H--e2G)(e2H--c2)~

+sin (Aa)"e3/ 1 -

1 2H[dH]4(eZH_e2G). (e2H

--

d 2 e2G(-~(H-G)) +2e G ~R,(-G)t~-t-~) d H /e2H/dIt\2

~el 2H[dH~

-lJ

~1/2 l / ~ . / \ 2

"|t~}"

C2)

d 2 1/2 1 2G ~t~("-G)) te 2H- e 2G-. )re 2H- c 2.)-~,e ]

e2G)(e2S_1 _ c2 )

where C~ = C2(e ~'

), C 2 = c2(e H) = Cg - (~C - 1/2)e 2H, e~ = e~ = ea2 = I. (For details see [4, 51 .) Restrictions have been imposed upon functions H, Hi, G, Ga and a range of parameter R (see

[4, 51 ). Relations (3.6) and (3.8) connect two solutions of Equation (3.1). At the same time, this action is a certain representation of a gauge group (a local one) on a manifold of arbitrary functions and their first derivatives parametrizing the solution. In this way, a symmetry of Equation (3.1) is a nonlinear representation of a gauge group (a local one) which has originated from group L i.

ACKNOWLEDGEMENTS

I thank Profs R. R~czka and R. Zelazny for continuous support during the preparation of the present version of this paper. 485

1/2

REFERENCES 1. Riemann, G.E.B., 'l]ber die Fortpflanzung ebener Luftwellen yon endlicher. Schwingungsweite', Abhandl. Konigl. Ges. Wiss. Gdttingen, 8 (1869). 2. Burnat, M.,Arch. Mech. Stos. 18, 4 (1966). 3. Peradzyriski, Z., Bull. Acad. Pol. Sec. Sci. Tech. 19, 9 (1971). 4. Kalinowski, M.W., 'On the Old-New Method of Solving Nonlinear Equations', Warsaw University, preprint IFT/13/80. 5. Kalinowski, M.W., Lett. Math. Phys. 6, 17 (1982). 6. Landau, L. and Lifshitz, E., Mechanics of Continuous Media, Mir, Moscow, 1952 (in Russian).

(Received July 19. 1983)

486