ON UNIQUENESS PROPERTIES OF SOLUTIONS OF THE k-GENERALIZED KDV EQUATIONS

(1.1) ∂tu+ ∂ 3 xu+ u ∂xu = 0, (x, t) ∈ R, k ∈ Z. Our goal is to obtain sufficient conditions on the behavior of the difference u1 − u2 of two solutions u1, u2 of (1.1) at two different times t0 = 0 and t1 = 1 which guarantee that u1 ≡ u2. This kind of uniqueness results has been deduced under the assumption that the solutions coincide in a large sub-domain of R at two different times. In [17] B. Zhang proved that if u1(x, t) is a solution of the KdV, i.e. k = 1 in (1.1), such that u1(x, t) = 0, (x, t) ∈ (b,∞)× {t0, t1} (or (−∞, b)× {t0, t1}), b ∈ R, then u1 ≡ 0, (notice that u2 ≡ 0 is a solution of (1.1)). His proof was based on the inverse scattering method (IST). In [10] this result was extended to any pair of solutions u1, u2 to the generalized KdV equation, which includes non-integrable models. In particular, if u1, u2 are solutions of (1.1) in an appropriate class with u1(x, t) = u2(x, t), for (x, t) ∈ (b,∞)× {t0, t1} (or (−∞, b)× {t0, t1}), then u1 ≡ u2. In [13] L. Robbiano proved the following uniqueness result : Let u be a solution of the equation