Construction of solutions to the L2-critical KdV equation with a given asymptotic behaviour

We consider the critical Korteweg-de Vries (KdV) equation: [GRAPHICS] Let R-j(t, x) = Q(cj)(x - x(j) - c(j)t) (j = 1, ..., N) be N soliton solutions to this equation. Denote U(t) the KdV linear group, and let V is an element of H-1 be with sufficient decay on the right; that is, let (1 + x(+)(2+delta 0)) V is an element of H-1 be for some delta(0) > 0. We construct a solution u(t) to the critical KdV equation such that [GRAPHICS]