Analyticity of Solutions of the Generalized Korteweg–de Vries Equation with Respect to Their Initial Values

Abstract

The initial value problem (IVP) of the generalized Korteweg–de Vries (KdV) equation \[ \partial _t u + \partial _x (a(u)) + \partial _x^3 u = 0,\quad u(x,0) = \phi (x)\] is well posed in the classical Sobolev space $H^s (R)$ with $s > {3 / 4}$, which establishes a nonlinear map K from $H^s (R)$ to $C ( [ - T,T];H^s (R)$. It is shown that (i) if $a = a(x)$ is a $C^\infty $ function on R to R, then K is infinitely many times Frechet differentiable; (ii) if $a = a(x)$ is a polynomial, then K is analytic, i.e., for any $\phi \in H^s (R)$, K has a Taylor series expansion \[ K(\phi + h) = \sum_{n = 0}^\infty {\frac{1}{{n!}}} K^{(n)} (\phi )[h^n ].\] Each term $y_n = K^{(n)} (\phi )[h^n ]$ in the series solves a linearized KdV equation. Consequently, any “small” perturbation $K(\phi + h)$ of $K(\phi )$ can be obtained by solving a series of linear problems. The proof of these results relies on various smoothing properties of the associated linear KdV equation.