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Remark. In the family described above and constructed in [PT3], all minima of the solutions lie below the line U = -1. In fact, as was shown in [KKV], the SBS equation possesses chaotic solutions of a different type for every q E (-,J8, 0). 4. Methods 27 jumps between u = -1 and u = + 1 and small oscillations around these uniform solutions. 17) for a negative constant) that has received considerable attention: u iv + quI! + u - u2 = O. 10) It arises in problems of water waves, and also in the description of localised buckling of elastic beams [HBT, HW].

12) (O,~], so q < --. 12) to eliminate ul/ in this expression, we arrive at the required lower bound for u(~). 3. Jl - r(E, q), where r(E, q) is the unique positive root of the cubic equation X 3 + (q2 8-1 ) 1 2 x 2 -"2Eq =0 In (q<--v8). We conclude this section with estimates for the period 2L. ). 5. 1) that is odd with respect to its zeros and even with respect to its critical points. Suppose that it has energy E E (0, and that lIulioo < 1. ) (a) For any q E R, (b) For any q ~ -,J8, we have L(E) ~ 00 as E '\t 0.

1: The curve of first local maxima C+ in the (a, u)-plane. Because; is continuous, so is ¢, and it suffices to study the sign of ¢. 4. Methods 31 properties of the problem and some hard analysis. 8a) for 0: large. 1. Thus, as 0: increases, the curve C+ crosses the line u = 1, say at 0:0. At the point of intersection ;(0:0), we then have u = 1, u' = 0, u" = 0, where the last equality holds because the energy E is zero. We claim that u'" f= 0, for if u lll = 0, then by uniqueness, u == 1, which is impossible because u(o) = 0.