Quantitative constructive representation of the KdV flow in H3 via L2-energy-preserving centered-difference Chernoff approximations and Picard iteration
Abstract
We give a constructive representation of the solution of the KdV equation ut=−uxxx+6uux on ℝ for real initial data in H3(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 L2 norm exactly, as does the limit. A two-mesh Cauchy argument shows that the whole sequence converges in C([0,T];H2), at rate O(n−(2(s−2))/5) for Hs data (3≤ s≤ 7), and in L2 at rate O(n−6/5) for H3 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 H2 norm, uniformly on bounded subsets of H3 and compact time intervals, and joint locally Lipschitz dependence of (t,f)↦ S(t)f into L2; the classical conservation laws extend the flow globally in time.
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BibTeX
@misc{koudjinan2026quantitative,
title = {{Quantitative constructive representation of the KdV flow in $H^3$ via $L^2$-energy-preserving centered-difference Chernoff approximations and Picard iteration}},
author = {Comlan Edmond Koudjinan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31634},
eprint = {2610.31634},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31634},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantitative constructive representation of the KdV flow in H^3 via L^2-energy-preserving centered-difference Chernoff approximations and Picard iteration
AU - Comlan Edmond Koudjinan
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31634
UR - https://arxiv.org/abs/2610.31634
AB - We give a constructive representation of the solution of the KdV equation u_t=−u_xxx+6uu_x on ℝ for real initial data in H^3(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^−(2(s−2))/5) for H^s data (3≤ s≤ 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)↦ S(t)f into L^2; the classical conservation laws extend the flow globally in time.
ER -