2405.01321
Kahan–Hirota–Kimura maps preserving original cubic Hamiltonians
Víctor Mañosa, Chara Pantazi
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s logic is internally consistent and well-supported. It derives the preservation criterion from the modified Hamiltonian of Celledoni–McLachlan–Owren–Quispel (Eq. (2)) and reduces it to an explicit algebraic condition (3) for all cases considered. In R2, Lemma 6 gives the exact condition 2HxHyHxy − HxxHy^2 − HyyHx^2 = 0, and Theorem 4 lists precisely five families for which the KHK map preserves the original Hamiltonian; each of these maps is symplectic (Corollary 15) and the corresponding vector field is a Lie symmetry . In R4, Lemma 8 reduces preservation to two Poisson-bracket identities (13), and Theorem 9 exhibits 54 families satisfying them; all corresponding maps are symplectic (Proposition 16) and the vector fields are Lie symmetries . In R6, Proposition 10 provides an explicit family preserving H for which X is not a Lie symmetry, with X|Φh − DΦh X computed explicitly, and the associated map is not symplectic (Corollary 17; Remark 18) . The model’s core reductions in R2 and R4 align with the paper, but it introduces two substantive inaccuracies: (i) it asserts that the R4 classification yields exactly 54 families, whereas the paper explicitly does not claim completeness (it only provides 54 families found by computation) ; and (ii) it claims “most but not all” R4 families are symplectic, contradicting Proposition 16 which shows all those listed are symplectic .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions \textbf{Journal Tier:} specialist/solid \textbf{Justification:} This paper makes a careful and concrete contribution to the exact preservation problem for Kahan–Hirota–Kimura discretizations with cubic Hamiltonians. The R2 classification is complete and transparent; the R4 case provides a substantial catalogue (54 families) together with structural properties (Lie symmetry, symplecticity, additional integrals); the R6 counterexample is insightful, clarifying that preservation need not imply Lie symmetry nor symplecticity. The presentation is clear overall. Minor revisions could further improve clarity (e.g., providing computational details or code for the Maple-based searches and large expressions).