Taylor Series Expansion for Solutions of the Korteweg- de Vries Equation with respect to Their Initial Values

The initial value problem for the KdV equation @tu+ u@xu + @ 3 xu = 0; u(x; 0) = (x) establishes a nonlinear map K from H(R) to C([ T; T ];H(R)). It has been known for many years that this map K is continuous [2] , [17] and is proved recently being Lipschitz continuous [23]. In this paper it is shown that the nonlinear map K is in nitely many times Frechet di erentiable from H(R) to C([ T; T ];H(R)) . Furthermore, it is proved that K has a Taylor series expansion at any given 2 H(R), i.e.