Taylor Series Expansion for Solutions of the Korteweg-deVries Equation with Respect to Their Initial Values

Abstract The initial value problem (IVP) for the Korteweg-deVries (KdV) equation ∂ t u+u∂ x u+∂ 3 x u=0, u(x, 0) = φ(x) for x ∈ R, t ∈ R establishes a nonlinear map K from the space Hs(R) to the space C([−T, T]; Hs(R)). It has been known for many years that this map K is continuous [Bona and Smith (1975), Kato (1983)] and it was proved recently that this map is Lipschitz continuous [Kenig et al. (1993)]. It is shown in this paper that the nonlinear map K is infinitely many times Frechet differentiable from Hs(R) to C([−T, T]; Hs(R)). Furthermore, it is proved that K has a Taylor series expansion at any given φ ∈ Hs(R), i.e., [formula] where Kn(φ), the nth derivative of K at φ, is an n-linear map from the n-fold space of Hs(R) to C([−T, T]; Hs(R)) and the series converges in the space C([−T, T]; Hs(R)) uniformly for ||h|| ≤ δ for some δ > 0. Each term yn= Kn(φ)[hn] in the series solves a linearized KdV equation. Thus any solutions of the IVP (∗) can be obtained by solving a series of linear problems. By contrast, the corresponding map Kp established by the initial value problem for the periodic KdV equation ∂ t u+u∂ x u+ ∂ 3 x u = 0, u(x, 0) = φ(x) for x ∈ S, t ∈ R, where S is the unit length circle in the plane, is known to be continuous only from Hs(S) to C([−T, T]; Hs(S)). This is due to the lack of smoothing effects for solutions of the periodic KdV equation. In this paper, it is shown that Kp is Lipschitz continuous from Hs+1(S) to C([−T, T]; Hs(S)) and is n times Frechet differentiable from Hs+n+1(S) to C([−T, T]; Hs(S)) for any n ≥ 1. The method developed in this paper applies to other nonlinear dispersive wave equations.