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By N.N. Bogoliubov

On the finish of 1960 we made plans to write down a monograph concerning the normal ideas of quantum box conception and their experimental implications. We meant essentially to provide an account of the growth of the speculation of dispersion kinfolk because the visual appeal of the booklet of Bogolubov, Medvedev and Polivanov ([BMP]. As an advent we would have liked to incorporate a overview of some of the ways to axiomatic box idea. This creation needed to hide not just the formula of Bogolubov, Medvedev and Polivanov, in accordance with the equipment of practical derivatives of the 5-matrix and the situation of microcausality, but in addition the sphere formula linked to the names of Wightman, Haag, Lehmann, Symanzik, Zimmermann, and others. through the paintings the initiatives (and with them the dimensions) of the creation grew better and bigger, till it constructed into this e-book.

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Q | j , and we again use Lemma A2 to complete the induction. 31b) are absolutely and uniformly (in n) summable if Q 1, R 1 < ∞, where · 1 is the 1 -norm. 26) converges uniformly for all |z| ≥ 1.

83) and this also gives the evolution of the norming constants: C j (t) = C j (0)e−4ik j t , 2 ¯2 C¯ j (t) = C¯ j (0)e4i k j t . 3 Soliton solutions 39 The expressions for the evolution of the scattering data allow one to solve the initial-value problem for the NLS. 58b). 85a) ¯ C¯ m e−2i km x N¯ m (x) . 85b) j=1 J¯ The one-soliton solution is obtained for J = J¯ = 1. 41), we get ∗ N1(1) (x) |C1 |2 e2i(k1 −k1 )x C1∗ −2ik1∗ x =− e 1 − 2 k1 − k1∗ k1 − k1∗ ∗ N1(2) (x) = 1− |C1 |2 e2i(k1 −k1 )x k1 − k1∗ −1 −1 2 , where, as usual, N1 (x) = (N1(1) (x), N1(2) (x))T .

K−1 =−∞ k=−∞ k−1 =−∞ j |R | ; j! 31a). 30) yields for |z| ≥ 1 n−1 |Q k | Ck2 j+1,(2) Cn2 j+2,(1) (z) ≤ k=−∞ n−1 |Q k | ≤ k=−∞ ≤ n−1 k=−∞ |Rk | ( j + 1)! k−1 =−∞ |Q | j k−1 =−∞ |R | j+1 ( j + 1)! j! j+1 n−1 |Q k | k=−∞ k−1 =−∞ j! |Q | j , and we again use Lemma A2 to complete the induction. 31b) are absolutely and uniformly (in n) summable if Q 1, R 1 < ∞, where · 1 is the 1 -norm. 26) converges uniformly for all |z| ≥ 1.

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