Abstract
(--co, &>, (4 9 n;>, (A, 3 A>, t&T WY (3) are called instability intervals. All but the first interval in (3) are finite and may shrink to a point under special conditions. Erdelyi [3] discovered situations where all but a finite number of finite instability intervals vanish when q(t) is a suitable elliptic function. Lax [9] showed that a function q(t) which satisfies the nth order Korteweg-de Vries equation requires (1) to have no more than n non-vanishing finite instability intervals. Conversely, if all finite instability intervals vanish, then q(t) in (1) is necessarily a constant. This fact was proved by Borg [ 11, Hochstadt [ 71 and Ungar [ 121. When all but n finite instability intervals vanish q(t) satisfies a differential equation of the form
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