 By V. A. Marchenko, A. Boutet de Monvel, H. McKean (Editors)

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Extra resources for Mathematical Physics, Analysis and Geometry - Volume 3

Sample text

17) Qk Proof. First, we remark that, for any k, (17) is an extention of the case ε = ±1 of Ik (u) = 0 stated in Section 2. Therefore, we concentrate on (15) and (16) only. 1, , work equally well for general nonlinearities. Therefore, we look at periodic waves (k < ∞). Fixed κ ∈ (0, 1), let ϕT ∈ C0∞ (R) be a nonnegative function such that ϕT = 1 (T + T κ )/2, and ϕ (j ) (x) Cj /|x|j , j = on [−T /2, T /2], ϕT (x) = 0 if |x| 1, 2, . , if T /2 |x| (T + T κ )/2 (the construction of such a function will be given later on).

2), with all xi (t) distinct for almost all times (see below). 2) except at isolated instants of time. 2). This is trivially true, since it follows from the work of Thickstun  that it is true in the soliton limit, which is a special case of the elliptic case. 4b) is solvable for any positive integer N = 2. 4a–b). 4b) in Section 5. This deformation concept does not lead to all elliptic solutions of the KdV equation, but only to those that have nonsingular soliton limits. 8) in this way; for these solutions it is necessary to find an initial guess by some other means.

8) in this way; for these solutions it is necessary to find an initial guess by some other means. 2), a solution corresponding to one particular Treibich–Verdier potential is discussed. It is the only solution discussed in this paper that does not have a nonsingular soliton limit. Its inclusion allows us to point out some differences with the other examples in a very concrete way. 1, then for a given N > 4, nonequivalent configurations satis• If |ω1 /ω2| fying the constraint exist that do not flow into each other under the KdV flow and which cannot be translated into each other.