{"id":"c69dc4ef-5f0e-427c-8037-daa34f95299e","arxiv_id":"2608.05540","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A splitting method that composes the exact Benjamin-Ono flow with a linear perturbation step converges at first order in H^s for initial data in H^{s+1}, and its ILW variant converges to the Benjamin-Ono solution in the deep-water limit.","lead":"A new time-stepping scheme for a family of quasilinear water-wave equations, including the Intermediate Long Wave equation, solves the Benjamin-Ono part exactly and treats the rest as a small linear step. The authors prove first-order convergence in time with only one extra derivative of the initial data, and show in simulations that the method nearly preserves energy over long times without the small time steps other schemes need.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Birkhoff-map derivative bounds are used uniformly on balls, but Proposition 3.1 states only local estimates; Theorem 1.2's stability induction depends on this unproved uniformity.","rationale":"The central claim is Theorem 1.2, and its proof hinges on Lemma 3.3: the splitting map S=phi^tau_BO e^{tau A} is 1+O(tau)-Lipschitz in Birkhoff coordinates. That Lipschitz constant is the only reason the global Gronwall sum over T/tau steps does not degrade. Lemma 3.3's proof uses (10)--(11) as though the derivative bounds were uniform on bounded balls, but Proposition 3.1 as written gives only pointwise neighborhoods. Without uniformity there is no fixed exponent c for all steps, and the induction in Section 3.4 cannot close. This matches exactly the reader's weakest assumption. The condition (2) typo is real and must be corrected, but it does not affect the internal proof structure as deeply as the uniform Birkhoff-bound question does. The proposed check determines whether the missing uniformity is already in the cited literature; if it is, the proof is likely repairable and the verdict remains CONDITIONAL. If it is not, the theorem as stated is unsupported and the verdict should move to UNVERDICTED or REJECT. For now, I keep the reader's CONDITIONAL verdict unchanged.","tokens_in":19124,"tokens_out":28717,"duration_ms":254993,"concrete_test":"Check [24] for a statement that Phi is real-analytic on all H^s_0 and that dPhi and dPhi^{-1} are bounded, and dPhi is Lipschitz, on every bounded subset with constants depending only on the radius. If present, add that uniform version to Proposition 3.1 and verify that Lemma 3.3's constants become independent of the pair (f,g). If absent, test Lemma 3.3 directly: for fixed M>0, attempt to construct pairs (f_r,g_r) in the H^s-ball with ||f_r-g_r|| tending to 0 but with the pairs never lying in a common neighborhood V where (11) holds with a uniform C. If such a sequence exists, the Gronwall argument in Lemma 3.3 cannot be closed as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.1 gives, for each v, a neighborhood V and a constant C such that (10)--(11) hold pointwise. Lemma 3.3 then bounds the Birkhoff-coordinate difference of e^{tA}f and e^{tA}g by a Gronwall integral whose integrand applies (10)--(11) for all r in [0,tau] and all f,g in a fixed H^s-ball. The proof silently converts these local estimates into a constant C(M) valid on the entire ball, and Section 3.4 repeats this on the R_Phi-ball across all T/tau steps. This conversion does not follow from the stated Proposition 3.1: a bounded analytic map on an infinite-dimensional ball need not have uniformly bounded derivative unless a global bound is supplied. The sentence 'Thanks to (10), for any ball B...' is therefore an unsupported step. If [24] actually proves real-analyticity and boundedness on bounded subsets, the uniform version can be recovered via Cauchy estimates; but the manuscript neither states nor proves this. This is the hinge of Lemma 3.3 and the induction in Section 3.4. Separately, condition (2) should be corrected to a(-k)=overline{a(k)} for the examples to fit the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a first-order time-splitting method for a class of perturbed Benjamin--Ono equations ∂t u = ∂x|D|u − 2u ∂xu + A u, where A is a bounded Fourier multiplier. The scheme composes the exact Benjamin--Ono flow φ^τ_BO (evaluated via Gérard's explicit formula) with the exact linear flow e^{τA}. The main analytic result (Theorem 1.2) is first-order convergence in H^s for initial data u0 ∈ H^{s+1}_0, with a constant linear in ||a||_{ℓ∞}. Theorem 1.3 provides an O(e^{-2δ}) convergence rate for the ILW scheme to the BO flow in the deep-water limit. The paper also reports numerical experiments on energy preservation and long-time dynamics, including a numerical study of soliton resolution for ILW and KdV–BO. The proof strategy is to lift the discrete flow to Birkhoff coordinates for BO and to perform the whole stability and induction argument on the h^{s+1/2} side.","tokens_in":19286,"tokens_out":18411,"duration_ms":150194,"significance":"If the gaps identified below are fixed, the paper would be a significant contribution: it would give a first-order H^s convergence theorem for a quasilinear class of dispersive PDEs with only one additional derivative of regularity, substantially improving on classical Lie splitting analyses for KdV and BO. The use of Birkhoff coordinates to obtain a stability bound with Lipschitz constant 1 + O(τ) is conceptually novel and potentially influential. The deep-water limit result and the numerical evidence for long-time energy preservation without a quadratic time-step restriction are valuable, although the fully discrete analysis is explicitly deferred. The paper clearly credits its reliance on the authors' earlier explicit-formula implementations [3,4], which are used only in the numerics and do not enter the semi-discrete convergence proof.","major_comments":[{"comment":"Condition (2) states a(−k) = a(k), but for A to map real-valued functions to real-valued functions the correct condition is a(−k) = \\overline{a(k)}. All three examples satisfy only the latter: for ILW, a(−k) = −a(k) = \\overline{a(k)} since a(k) is purely imaginary; for Smith and KdVBO the same identity holds. As written, Theorem 1.2 does not apply to the examples in the introduction, and Theorem 1.3 inherits the problem. The condition and all subsequent uses must be corrected.","section":"Section 1, Eq. (2)"},{"comment":"The local estimates (10)–(11) are pointwise in v: for every v there is a neighborhood V and a constant C. Lemma 3.3 applies these estimates to all w(r) and z(r) along trajectories starting from arbitrary f,g in a ball of radius M, and silently uses a single constant C(M) uniformly over the entire ball. The sentence 'Thanks to (10), for any ball B of H^s_0 there exist two positive constants c, C ...' asserts a uniform norm equivalence that is not a consequence of the stated local derivative bounds. The same uniformity is needed in (15) and throughout the induction in Section 3.4, and in the conversion from h^{s+1/2} estimates back to H^s in Lemma 3.7. Without a global or uniform-on-bounded-sets version of these estimates, the Gronwall argument in Lemma 3.3 and the induction do not close. The authors should either state and prove (or precisely quote from [24]) that Φ and Φ^{-1} have derivatives uniformly bounded on bounded subsets of H^s_0, or restructure the stability argument to avoid this uniformity requirement.","section":"Section 3.1, Proposition 3.1 and Lemma 3.3"},{"comment":"The proof applies Theorem 1.2 to the ILW flow uδ, which requires a bound on sup_{t∈[0,T]} ||uδ(t)||_{H^{s+1}} that is uniform in δ for all δ ≥ δ0. However, Proposition 3.10 only provides uniform boundedness in H^s. If the H^{s+1} norm of uδ can grow with δ (or if no uniform bound is known), the constant C in Theorem 1.2 depends on δ, and the stated e^{-2δ} rate does not follow. The authors should supply a uniform H^{s+1} bound — either by proof or by precise citation — or else state Theorem 1.3 with a constant that may depend on the depth parameter in a controlled way.","section":"Section 3.5, proof of Theorem 1.3"}],"minor_comments":[{"comment":"The reference 'By Lemma 3.1' should be 'By Proposition 3.1'; there is no Lemma 3.1 in the manuscript.","section":"Section 3.4, after Eq. (15)"},{"comment":"The expression u^0_K = (Π_K + Π_K) u0 is a typo; it should involve the conjugate projector, e.g., Π_K + \\overline{Π}_K, to define a real-valued initial condition.","section":"Section 4, Eq. (20)"},{"comment":"The symmetry condition for real initial data should be \\hat{U}_{−k} = \\overline{\\hat{U}_k}, not \\hat{U}_{−k} = \\hat{U}_k as stated.","section":"Section 4.1, initial data construction"},{"comment":"The sentence 'global well-posedness for u0 ∈ H^s_0, s ≥ 0, has recently been obtained in Gassot–Laurens [20]' attributes to [20] a statement about the whole class (1), but [20] treats the ILW equation. Theorem 1.2 is conditional on an existing solution in C([0,T], H^{s+1}_0), so this does not invalidate the theorem, but the presentation should be corrected to avoid overstating the well-posedness theory for the non-integrable examples.","section":"Introduction, after Eq. (1)"},{"comment":"The sentence 'which can we written in the form (1)' contains a typo ('can we written' should be 'can be written').","section":"Section 1, KdV–BO equation"},{"comment":"The Galilean transformation formula contains the term −1/a(0), which is undefined when a(0)=0; the formula should be stated in a way that covers the examples.","section":"Remark 1.5"}],"recommendation":"major_revision","confidential_remarks":"The paper has a clear and elegant proof strategy, and the main convergence theorem is likely correct if the uniform Birkhoff-map bounds can be supplied. The two most serious issues are (i) the incorrect condition a(−k)=a(k) in (2), which is easy to fix but essential, and (ii) the unsupported passage from local to uniform derivative bounds for Φ, which is load-bearing for Lemma 3.3 and the whole induction. The uniform H^{s+1} bound needed in Theorem 1.3 is a third gap that should not be overlooked. If these are resolved, the paper would be a strong fit for the journal; in its current form the proof as written does not close. The numerical claims are heuristic but the experiments are clearly described."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious paper and deserves a good referee, but it needs a revision before I would trust Theorem 1.2 as stated.\n\nWhat is genuinely new: splitting the equation into the exactly solvable BO flow and a bounded linear perturbation is a natural and original idea. The analysis in Birkhoff coordinates, extracting a 1+O(τ) Lipschitz constant for the numerical flow, is a real technical contribution. It buys a substantially better regularity requirement — first-order H^s convergence with only one extra derivative — versus s+3 for KdV Lie splitting or H^{5/2} for the prior BO splitting. Theorem 1.3, the discrete deep-water limit for ILW, is also new and cleanly proved given the ingredients.\n\nSoft spots, in proportion to their seriousness:\n\nFirst, condition (2) writes a(−k)=a(k). All three examples satisfy a(−k)=overline{a(k)}, which is the real-operator condition. The theorem as stated therefore does not apply to ILW, Smith, or KdV–BO. The proofs only use boundedness of a, so I read this as a typo, but it has to be corrected before the main theorem matches the paper's own examples.\n\nSecond, and more load-bearing: the stability estimate Lemma 3.3 converts the local derivative bounds (10)–(11) from Proposition 3.1 into a uniform bound on a ball. Proposition 3.1 as stated gives, for each v, a neighborhood V and a constant C. The sentence \"Thanks to (10), for any ball B...\" does not follow from the stated proposition. The induction in Section 3.4 iterates the step T/τ times and needs a single constant on the whole R_Φ-ball. If the Birkhoff map in [24] is actually analytic with bounded derivative on bounded subsets, then a uniform version can be recovered by Cauchy estimates, but the paper should say so explicitly or prove the needed uniformity. This is the hinge of the convergence proof, so it deserves direct attention.\n\nThird, the fully discrete scheme used in the experiments is not covered by the convergence analysis; Remark 3.9 says this honestly, so it is a limitation, not a hidden flaw.\n\nThe numerical work is presented fairly: energy preservation without the quadratic time-step restriction is shown in convincing experiments, and the soliton resolution numerics are exploratory and labeled as such.\n\nWho is this for? Numerical analysts working on quasilinear dispersive PDEs, and researchers in integrable systems who care about turning explicit formulas into practical schemes. The paper deserves a serious referee, not a desk rejection, and with the condition (2) correction and a strengthened treatment of the Birkhoff-map uniformity, I would expect it to become a solid publication.\n\nRecommendation: send to peer review, and ask the authors to address the two issues above head-on.","headline":"Novel BO-based splitting with first-order H^s convergence at s+1 regularity; the stated theorem has a sign condition typo and the key stability hinge uses a uniformity assumption borrowed from cited work rather than proved.","tokens_in":19919,"tokens_out":1523,"would_cite":true,"duration_ms":16045,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37K10","65M70","65M15","35Q55","35Q35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves first-order $H^s$ convergence for a perturbative splitting scheme for the Benjamin–Ono class, requiring only one additional derivative on the initial data.","keywords":["Benjamin–Ono equation","Intermediate Long Wave equation","splitting methods","Birkhoff coordinates","quasilinear dispersive PDEs","low-regularity convergence","soliton resolution","energy preservation"],"falsifier":"For the Smith equation (or KdV–BO) at fixed high spatial resolution, take $u_0\\in H^{s+1}_0$ with a bounded $H^{s+1}$ norm but provably not in $H^{s+2}$, and measure $\\sup_{n\\tau\\le T}\\|u(n\\tau)-u^n\\|_{H^s}$ for $\\tau=2^{-m}$; if the log-log slope is strictly below 1, or the prefactor grows without bound as the $H^{s+1}$ norm stays bounded, Theorem 1.2 is false. A second check: with a multiplier $A$ whose symbol $a(k)$ is bounded but non-decaying, the threshold $\\tau_0$ in (18) shrinks like $1/(C_{\\mathrm{loc}}\\|a\\|_{\\ell^\\infty} e^{\\tilde c T}T)$; a counterexample would be a choice for which no positive $\\tau_0$ yields the stated bound.","tokens_in":18848,"feed_emoji":"🌊","tokens_out":10932,"duration_ms":88502,"temperature":0.7,"pith_summary":"This paper tries to establish that a new splitting scheme solves a broad class of perturbed Benjamin–Ono equations—including the integrable Intermediate Long Wave (ILW) equation and non-integrable models such as the Smith and KdV–BO equations—with first-order accuracy in $H^s$ and with only one additional Sobolev derivative on the initial data. This matters because the equations are quasilinear: standard Sobolev or Bourgain-space arguments cannot control the derivative loss, and classical Lie splitting for the same equations either demands more regularity or a restrictive quadratic time-step condition for stability. The proposed scheme separates the Benjamin–Ono part, which is integrated exactly in time using an explicit formula, from a zero-order Fourier multiplier perturbation, integrated linearly. In Birkhoff coordinates the one-step map is Lipschitz with constant $1+O(\\tau)$, which makes a Gronwall iteration close with no loss of derivatives. If the proof is right, rigorous low-regularity and long-time numerics for ILW and its non-integrable relatives become feasible, including deep-water limits and soliton-resolution studies.","feed_headline":"First-order splitting scheme proven for ILW and perturbed BO","feed_subtitle":"It requires only one extra derivative on the initial data, not three or more.","key_machinery":"The machine is the Birkhoff map $\\Phi: H^s_0 \\to h^{s+1/2}$, a nonlinear Fourier transform in which the unperturbed BO flow is diagonal: $\\frac{d}{dt}\\zeta_n(v(t))=i\\omega_n(v(t))\\zeta_n(v(t))$ with $\\omega_n(v)=n^2-2\\sum_{k\\ge 0}\\min\\{k,n\\}|\\zeta_k(v)|^2$. Here $h^{s+1/2}$ is the weighted $\\ell^2$ sequence space with norm $(\\sum_{n\\ge1} n^{2s+1}|z_n|^2)^{1/2}$. In these coordinates the splitting map $w\\mapsto \\varphi^\\tau_{\\mathrm{BO}} e^{\\tau A}w$ is shown, via the analytic derivative bounds (10)–(11), to be Lipschitz with the precise constant $1+c\\tau$ on bounded sets; this exact leading constant $1$, rather than some $C>1$, is what prevents the $n=T/\\tau$ iterates from compounding an $O(1)$ factor into an exponential-in-$1/\\tau$ blow-up. The explicit solution formula (4) supplies $\\varphi^\\tau_{\\mathrm{BO}}$ exactly in time, so the only time error is the split between BO and $A$. Local error is then $O(\\|a\\|_{\\ell^\\infty}\\tau^2)$ in $H^s$, global stability is carried in $h^{s+1/2}$, and an induction keeps all numerical iterates inside a fixed bounded ball $R_\\Phi$.","core_discovery":"The central claim is Theorem 1.2: for $s\\ge 0$, $T>0$, initial data $u_0\\in H^{s+1}_0$, and a Fourier multiplier $A$ with bounded, real, even symbol $a$, if $u$ is the exact solution of (1) and $u^n$ is the splitting scheme $u^{n+1}=\\varphi^\\tau_{\\mathrm{BO}} e^{\\tau A}u^n$, then for all sufficiently small $\\tau$, $\\sup_{n\\tau\\le T}\\|u(n\\tau)-u^n\\|_{H^s} \\le C\\|a\\|_{\\ell^\\infty}\\tau$, with $C$ depending on $s$, $\\|u_0\\|_{H^{s+1}}$, $\\|a\\|_{\\ell^\\infty}$, and $T$. This is a first-order error bound that asks for only one additional derivative, rather than the three or more required by Lie-splitting analyses for KdV or BO. The paper also proves Theorem 1.3: for the ILW equation with depth $\\delta$, the splitting scheme converges to the BO solution with error $(C_0 T e^{C_0 T}+C\\tau)e^{-2\\delta}$, giving a discrete deep-water limit. Numerically, the scheme keeps the ILW energy nearly conserved up to $T=5000$ under a linear time-step condition and produces soliton-resolution dynamics for ILW and a KdV–BO model.","pith_inferences":["The paper leaves a fully discrete convergence analysis in Birkhoff coordinates open (Remark 3.9); a natural next step would be to prove that truncation to $K$ Fourier modes perturbs the first $K$ Birkhoff coordinates in a controllable way, turning the semi-discrete theorem into a fully discrete one.","The numerical energy near-preservation without a quadratic CFL suggests a modified-energy or adiabatic-invariant explanation in Birkhoff coordinates; the authors note the rigorous justification is ongoing, so this is an inference from their numerics rather than a proved claim.","The conjectured soliton count $\\lceil 2c/3\\rceil$ for KdV–BO, if confirmed, would mean a non-integrable perturbation of BO still has a robust mass-based soliton counting law; testing it for other perturbations, such as the Smith equation, would reveal how much of BO's integrable rigidity survives.","On the full line, the exponential-in-$T$ constant could perhaps be improved to a uniform-in-time bound using dispersion and scattering, as has been done for NLS splitting methods; the paper only remarks on this possibility, so such an extension remains open."],"forward_implications":["For initial data in $H^{s+1}_0$, the bound $\\sup_{n\\tau\\le T}\\|u(n\\tau)-u^n\\|_{H^s}\\le C\\|a\\|_{\\ell^\\infty}\\tau$ gives rigorous first-order time convergence with no loss of derivatives beyond the single extra derivative, for ILW, Smith, and KdV–BO equations.","Because both subflows are evaluated exactly in time, the time-discretization error is purely from splitting; the remaining spatial error is controlled by the spectral schemes built on formula (4), giving stable low-regularity computations.","For ILW with large depth $\\delta$, the scheme tracks the BO solution with exponentially small error $e^{-2\\delta}$ plus the $O(\\tau)$ splitting error, so a single code covers the ILW-to-BO transition.","The experiments show energy near-preservation up to $T=5000$ under a linear $\\tau\\sim K^{-1}$ step, avoiding the quadratic $\\tau\\sim K^{-2}$ restriction imposed on Lie splitting and RK4.","The observed dynamics for ILW and KdV–BO are consistent with soliton resolution; for KdV–BO with rational initial data of mass $c$, the observed number of asymptotic solitons is $\\lceil 2c/3\\rceil$."],"supporting_citations":[{"why":"Supplies the Birkhoff map analyticity and the derivative bounds (10)–(11) that the stability and induction arguments rely on.","marker":"[24]"},{"why":"Introduces Birkhoff coordinates for BO and the diagonal frequency dynamics (8)–(9) used to get the leading constant $1$ in the stability bound.","marker":"[23]"},{"why":"Provides the explicit solution formula (4) that makes the BO subflow computable exactly in time and therefore free of time-discretization error.","marker":"[22]"},{"why":"Establishes global well-posedness and a priori bounds for ILW and derives the Duhamel formula in Birkhoff coordinates used in the local error analysis.","marker":"[20]"},{"why":"Gives the deep-water limit from ILW to BO and uniform bounds needed for the exponential-in-$\\delta$ convergence of Theorem 1.3.","marker":"[9]"},{"why":"Constructs the spectrally accurate fully discrete scheme based on the explicit BO formula, used for the numerical experiments.","marker":"[3]"},{"why":"Generalizes the explicit-formula schemes and provides the $L^2$-mass-preserving variants used to guarantee discrete mass conservation.","marker":"[4]"},{"why":"Provides the classical Lie-splitting error analysis for BO, the comparison baseline showing the new method's reduced regularity requirement.","marker":"[13]"}],"fun_headline_variants":["First-order splitting scheme needs only one extra derivative","Splitting method for ILW and BO: one extra derivative, first-order","Deep-water limit and long-time energy: new splitting scheme","BO formula splitting: first-order, fewer regularity requirements","Splitting method for ILW: first-order with one extra derivative"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported Proposition 3.1—that the Birkhoff map and its inverse are analytic on bounded subsets of $H^s_0$ with derivative bounds (10)–(11)—together with the assumed existence of a global $H^{s+1}$ solution; if either fails uniformly on a fixed bounded ball, the induction that keeps all iterates in a ball does not close.","fun_headline_variants_meta":{"raw":{"variants":["First-order splitting scheme needs only one extra derivative","Splitting method for ILW and BO: one extra derivative, first-order","Deep-water limit and long-time energy: new splitting scheme","BO formula splitting: first-order, fewer regularity requirements","Splitting method for ILW: first-order with one extra derivative"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000901,"raw_usage":{"total_tokens":3942,"prompt_tokens":1069,"completion_tokens":2873,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":2787}},"tokens_in":685,"tokens_out":2873,"duration_ms":17426,"temperature":1.0,"reasoning_tokens":2787,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T11:14:31.699783+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the Smith equation (or KdV–BO) at fixed high spatial resolution, take $u_0\\in H^{s+1}_0$ with a bounded $H^{s+1}$ norm but provably not in $H^{s+2}$, and measure $\\sup_{n\\tau\\le T}\\|u(n\\tau)-u^n\\|_{H^s}$ for $\\tau=2^{-m}$; if the log-log slope is strictly below 1, or the prefactor grows without bound as the $H^{s+1}$ norm stays bounded, Theorem 1.2 is false. A second check: with a multiplier $A$ whose symbol $a(k)$ is bounded but non-decaying, the threshold $\\tau_0$ in (18) shrinks like $1/(C_{\\mathrm{loc}}\\|a\\|_{\\ell^\\infty} e^{\\tilde c T}T)$; a counterexample would be a choice for which no positive $\\tau_0$ yields the stated bound.","supporting_citations":[{"cited_title":"Gérard, T","cited_arxiv_id":null,"evidence_quote":"Supplies the Birkhoff map analyticity and the derivative bounds (10)–(11) that the stability and induction arguments rely on."},{"cited_title":"Gérard and T","cited_arxiv_id":null,"evidence_quote":"Introduces Birkhoff coordinates for BO and the diagonal frequency dynamics (8)–(9) used to get the leading constant $1$ in the stability bound."},{"cited_title":"Gérard,An explicit formula for the Benjamin–Ono equation, Tunisian J","cited_arxiv_id":null,"evidence_quote":"Provides the explicit solution formula (4) that makes the BO subflow computable exactly in time and therefore free of time-discretization error."},{"cited_title":"Chapouto, G","cited_arxiv_id":null,"evidence_quote":"Gives the deep-water limit from ILW to BO and uniform bounds needed for the exponential-in-$\\delta$ convergence of Theorem 1.3."},{"cited_title":"Alama Bronsard, X","cited_arxiv_id":null,"evidence_quote":"Constructs the spectrally accurate fully discrete scheme based on the explicit BO formula, used for the numerical experiments."},{"cited_title":"Dutta, H","cited_arxiv_id":null,"evidence_quote":"Provides the classical Lie-splitting error analysis for BO, the comparison baseline showing the new method's reduced regularity requirement."}],"review_version":1}