Quantitative constructive representation of the KdV flow in $H^3$ via $L^2$-energy-preserving centered-difference Chernoff approximations and Picard iteration

Abstract

We give a constructive representation of the solution of the KdV equation $u_t=-u_{xxx}+6uu_x$ on $\mathbb R$ for real initial data in $H^3(\mathbb R)$. The approximations couple an exact, unitary centered-difference Chernoff product for the Airy group with a skew-symmetric discretization of the quadratic term, and are computed by Picard iteration; they preserve the $L^2$ norm exactly, as does the limit. A two-mesh Cauchy argument shows that the whole sequence converges in $C([0,T];H^2)$, at rate $O(n^{-\frac{2(s-2)}5})$ for $H^s$ data ($3\leq s\leq 7$), and in $L^2$ at rate $O(n^{-6/5})$ for $H^3$ data. This scheme, to the best of our knowledge, is novel and is of independent interest. As a by-product we obtain existence and uniqueness of strong solutions, Lipschitz dependence on the data in the $H^2$ norm, uniformly on bounded subsets of $H^3$ and compact time intervals, and joint locally Lipschitz dependence of $(t,f)\mapsto S(t)f$ into $L^2$; the classical conservation laws extend the flow globally in time.