By Wahlen E.

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Extra info for A blow-up result for the periodic Camassa-Holm equation

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For the solutions 6 and the C o r o l l a r y theorems of i s o l a t e d ding c o n s i d e r a t i o n s reader (56) which is d e f i n e d in the n e i g h b o u r h o o d singularities The results of T h e o r e m deduce general and w of in each point of D and r e p r e s e n t functions expansion solution and ~(z) the p o s s i b i l i t y for the solutions singularities. are sketched offer briefly of (56) In the sequel (for further to in the the c o r r e s p o n details the to [18]). be h o l o m o r p h i c in the disk u(%) ={zl,z-z o' < P} Let w be a solution of (56) d e f i n e d and u n i q u e in the p u n c t u r e d open 26 disk o) ={z Io< Then, from the Corollary and by indefinite it follows

K'-n+k-1)h. 16 Let y(z) and 6(z) be holomorphic or meromorphic functions in D which satisfy the conditions (114). = - o in D with Then, w = ~IK* n + (n-1)@I ( 2n-k ... r(n-k),r (~) (K'-n)kh with (K'-n) k = (K*-n)(K*-n+l) ... (K*-n+k-1) for (K*-n)oh = h, represents a solution of the differential w + [B- z~ in D. equation n(n+l)7'~r] w = 0 2 k £ IN , 49 The above results suggest to ask for differential operators of the form (134) L = alr which map a solution of the + a2s differential w (135) + a3 equation + Bw = O zE onto a solution v = Lw (136) of the differential equation (137) V + B'v = O.

E = -1) 40 Theorem 13 a) For every solution w of the d i f f e r e n t i a l (1+ezz)2w equation + en(n+l)w = O, £ = +1, (92) n 6 IN, zz defined in D, there exist (101) w = Eng + Enf two f u n c t i o n s g(z), cI z n I°n E (l+ez~) z n+l 1 %z n b) C o n v e r s e l y , sents for a r b i t r a r y a solution of c) For every given f(2n+l)(z) functions f(z) £ H(D), %~n g(z), such that (l+~z~)n+l f(z) E H(D) " (101) repre- (92) in D. solution are u n i q u e l y w of (92) determined by g(2n+l)(z) = the f u n c t i o n s g(2n+l)(z) and D~+lw (102) (1+£z~)2n+2 ' Dn+l~ (103) f(2n+l)(z) In this case polynomial the g e n e r a t o r s p(z) of d e g r e e rators ~(z) (IO4) and ~(z) ~(z) g(z) = (l+~z~)2n+2 " and f(z) 2n.