A blow-up result for the periodic Camassa-Holm equation - download pdf or read online

By Wahlen E.

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An-l,2 a],n_ 1 a n~n-| al'n-1 .. a|2 ...... atn I t~ ........... a I ] n-l,2 n-l,nl 4'alli a12 "" a l n ~. . . . . . . an_l,n_l:anl an2 . ann 44 Nous pouvons maintenant gtablir Shanks et l'E-algorithme Th~or~me le rfisultat fondamental [223] qui relie la transformation : 36 : (n) = ek(Sn) e2k d~monstration s ~n) = (n) = i/ek (&Sn) ~2k+l : il est facile de voir imm6diatement I/eo ( A Sn) = I/AS n _ n) que : (Sn) = Sn+ 2 S n = el 42 S S2 n+l n supposons avoir d6montr6 la proprift6 jusqu'~ la colonne encore vraie pour la colonne 2k+].

La m~thode des trapezes au calcul de I avec les pas h h 2 = hl/~ avec ~ > x diff~rentiable le proc~d6 de Richardson o = H, h approch~es 1 = ho/~, de I ~ l'aide ... pour acc~l~rer cette suite en prenant l'erreur est un polynSme pair en h. On a alors : 31 = T(n) k I + a2 ( k) h 2k+2 + ... k n d'o~ (n+|) I Tk ~(n) ik - I lim n+~ h 2k+2 ( n+ I. ~ ) = n = lim n -~° l 2k+2 d'autre part on a : hn+k+ 1 2n = h n _ ~ 1 _ 2n+2k+2 2k+2 ce qui d ~ m o n t r e que ~/k {T~+I} fix~(k ~ converge plus vite que {Tkn)} f k fix~ car la condition du th~or~me 26 est satisfaite.

Sn+2k - S Rempla~ons la seconde difference avec la seconde Sn . . . . Yn > N ligne par sa difference avec la premiere, et ainsi de suite. On obtient 1 Sn+ k * ......... ainsi la troisi~me par sa : | kS n ......... kSn+ k . . . . . . . . . . . IkSn+k- 1 '" &Sn+2k-l/ D'o~ la transformation gSn+k- I .... kSn+2k- 1 ek(S n) de Shanks qui, en faisant de nouveau des comb inaisons de lignes et de colonnes, s'~crit S n ........ Sn+ k Sn+ 1 ...... Sn+k+ 1 Sn+ k . . . Sn+2k &2Sn . . .

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A blow-up result for the periodic Camassa-Holm equation by Wahlen E.

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