Download PDF by Sydney R. Parker, Armen H. Zemanian (auth.), Adhemar: Rational Approximation in Systems Engineering

By Sydney R. Parker, Armen H. Zemanian (auth.), Adhemar Bultheel, Patrick Dewilde (eds.)

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2 , which has a partial fraction representation: (24) each term of which can be associated with a soliton. ), in which A. 2 , and zi into z/, so that the above asymptotic analysis can be repeated. 's, I ~ n ~ N, a little further, and to see how the successive approximations behave with respect to the emerging solitons. According to the above analogy, we shall limit ourselves to the KdV-case. z, with [1. 1 and t, > 0. ,zt' = E JLjZ/, r = 0,1, ... 2n- 1, (27) j= I so that their asymptotic behaviour can be investigated.

Asymptotic behavior For large n we believe that the behavior of the polynomials p 1 (z) is given by (4) Moreover, for zES except near branch points we expect P;(Z) - x;(z) + X/(Z) . n-oo (5) Here { x1 ( z) } is a set of functions analytic on sheet m of CR or, equivalently, analytic in the complex plane outsideS. They are the solution of the equations m Ei) x1 (z) = 0, k= l, ... ,m-1, zES (6) J~I m E/;(Z) XJ(z) = Q(z), zEs, (7) }~I where Q ( z) is a function analytic on sheets 1, ... , m- 1 of CR.

Supported in part by Natural Sciences & Engineering Research Council, Canada. Department of Physics, University of Western Ontario, London, Ontario N6A 3K7, Canada. 306 NUTTALL 2. The Riemann surface ffi The description of the proposed asymptotic behavior involves a Riemann surface ffi that we believe is associated with each set of functions {jj(z)). We have in general no proof of the existence and uniqueness of this surface but in this section we present characteristics which we believe might determine ffi.

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