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By Michael Beals (auth.), Heinz O. Cordes, Bernhard Gramsch, Harold Widom (eds.)

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Melrose and N. R i t t e r , I n t e r a c t i o n of n o n l i n e a r progressing waves f o r s e m i l i n e a r wave equations, Ann. of Math. 121 (1985), 187-213. [lO] , I n t e r a c t i o n of progressing waves f o r s e m i l i n e a r wave equations I I , p r e p r i n t . [11] J. Rauch and M. Reed, Propagation of s i n g u l a r i t i e s f o r s e m i l i n e a r h y p e r b o l i c equations in one space v a r i a b l e , Ann. of Math. 111 (1980), 531-552. [12] , Nonlinear m i c r o l o c a l a n a l y s i s of s e m i l i n e a r h y p e r b o l i c systems in one space dimension, Duke Math.

10) H(1,~,1,1) of of ~, conoids. intersection. 11) N = the - 3x 2 corresponding meets the s e t triple z = x, Ix = Y = zl 3x 2 by on t h a t w = [2~2-5~±2(~-1)/~(~-3)]x} ~2 - 4~ intersection. In o t h e r words, I(~,i,1,1), a r e o b t a i n e d the new s e t o f (i,i,i,~)~ one o f For e x a m p l e , = x, c o n o i d s g e n e r a t e d by the l i n e s (1,1,1,~) I is the c o r r e s p o n d i n g c o n o i d is 0 = (~2-4~)w2-2(2~2-5~)wx - This y i e l d s the ~ = 4 and examine the f o u r H(1,~,1,1) N H(1,1,6,1) n H(1,1,1,~) symmetry, and from with by the o r d e r e d 2(~2-2~)(wy+wz+yz) + 2a(wx+xy+xz)}.

These kernels have pointwise a s y m p t o t i c expansions as t --* 0 and WD denotes the difference of the fiber traces of the c o n s t a n t terms in this expansion. T h e second contribution turns out to be o(1) as R ---* oo in view of (26) since ( - 0 2 + So)(O:~ + So) = (0~ + So)(-Ox + So). T h e third contribution is c o m p u t e d in [B+S,4]; it is independent of R and equals (28) ~1 ( r / (So ) - dim ker So) where rlso(z) : = ~ sgn s Ist - z sespec so\{o} 46 is the r/-function introduced in [A+P+S].

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