{"id":"26f2c34c-025e-4e64-b7e1-e842dc9001b0","arxiv_id":"2607.27118","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Planar monotone and oscillatory dispersive shocks of dissipative KP and multi-D KdV–Burgers are L2-contractive under large multi-D perturbations up to Lipschitz shifts, under explicit viscosity–dispersion–strength bounds.","lead":"The paper proves that planar dispersive shocks of the dissipative KP and multi-D KdV–Burgers equations remain L2-stable under arbitrarily large multi-dimensional perturbations, up to a Lipschitz time shift. This extends recent one-dimensional contraction theory to water-wave and plasma models with transverse directions.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The non-monotone half of the claim rests on numerically-specified profile constants imported from the authors' own unpublished preprint [8]; the Appendix A induction margins are thin, and hypothesis (1.18) is ~200x stricter than the positivity condition the proof actually uses — a sign the constant","rationale":"Read in good faith, the paper does what it says: it extends the authors' 1D KdV–Burgers L²-contraction machinery to dissipative KP and multi-D ZKB, and the genuinely new analytic content (the transverse Poincaré control in (4.15)/(5.9), the vanishing of the KP nonlocal term via the intrinsic zero-mean condition, the global existence argument in §6) is written out and, on my spot checks, internally consistent. I verified the monotone-case algebra end to end: the change of variables (4.10), the weighted Poincaré application of Lemma 3.1, the conversion (4.15) using sup|ũ′| ≤ s²/2ε1 and ∫|ũ′| = 2s, and the positivity of C*, C* under (1.10) all check out. I also re-ran the Proposition 5.3 arithmetic (2.034s is correct given the imported inputs) and the Appendix A recursion (A.17); the indexing of an is off by a factor from the base case a1 = 1/30, but the budget inequalities hold under either reading, so that slip is cosmetic. The reader's chosen weakest assumption — the s = O(√ε2) restriction — is real but is a scope limitation the paper itself states plainly in Remark 1.8; it does not threaten the truth of the stated theorems. What does threaten them is the layer the reader mentioned only in passing: every quantitative input to the oscillatory argument (Theorem 2.1, Propositions 5.1–5.2) is imported from the group's own 2026 preprint [8], the margins in the induction are thin (C0 feasibility window ~6%; M = 4/3 saturated; diffusion budget 0.83–0.64 against a 0.9 cap), and the unexplained ~230x gap between hypothesis (1.18) and the condition C** ≥ 0 actually used in (5.9) suggests the numerical chain has not been independently reconciled even within this paper. Theorem 7.2's proof being \"left to the reader\" adds a smaller gap of the same kind. Credit due: the constants are explicit and checkable, which is exactly what makes the proposed test runnable, and the paper is candid about its regime. Verdict stays CONDITIONAL — same level as the reader — but the condition should be re-pointed: acceptance of the non-monotone results should wait on independent verification of [8]'s profile constants and a correction or derivation of the 7.86 in (1.18)/(7.8).","tokens_in":36320,"tokens_out":15355,"duration_ms":479315,"concrete_test":"Numerically integrate the profile ODE (2.1) for ε1=1, s=1, δ=0.49 (δs ∈ (1/4,1/2), A ≈ 1/2) by shooting along the unstable manifold of (s,0) using the eigenvalue (2.2). Recompute independently: u0/s vs 1.0601; the overshoot ratios in (2.5)/(2.6) vs ρ∗ = 4.64/4.77; λ̄0, λ̄1 in (5.2)–(5.3); the L² ratios in (5.5); and ∫_{-∞}^{ξs}(ũ−s)²dξ vs 0.001s. Substitute the measured values into the Appendix A bookkeeping: check C0 = 13/10 satisfies C0 ≥ 14s/(222s − 198u0) with M = 4/3 ≥ 40C0/39, and that total diffusion use per sub-interval stays below 9/10. Also re-derive (1.18) from C** ≥ 0; if 7.86 cannot be reproduced, the theorem statement needs correction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorems 1.6 and 7.2 reduce to Theorem 5.4, whose Appendix A induction consumes constants imported from [8]: the overshoot u0 ≤ 1.0601s, decay rates ρ∗ = 4.64/4.77, λ̄0 = 0.355, λ̄1 = 9.60, the L² bounds 0.178/0.81/0.82 in (5.5), and the tail bound 0.001s in (5.6). These are not decorative: (i) Step 0 needs C0 ≥ 14s/(222s − 198u0), which requires u0 < 1.1212s — the imported bound 1.0601s leaves only ~6% room, and M = 4/3 is exactly the saturated minimum 40C0/39 at C0 = 13/10; (ii) the diffusion budget per sub-interval (0.83 on (ξ0,∞); 0.63 + 0.01 overlapping on Ji+1) sits close to the 9/10 cap that yields the 1/10 dissipation remainder in (5.8); (iii) the 2.034s L¹-bound on ũ′ feeds directly into C** and hence the admissible shock size. A few-percent error in any one imported constant breaks the chain, and [8] is an unrefereed companion preprint by the same group. Corroborating symptom: (1.18) states s² < 7.86ε2/(M²π²) ≈ 0.448ε2, but the proof only needs C** = 2ε2 − 2.034s²/(8π²M) > 0, i.e. s² < 103.5ε2 — a ~230x discrepancy with no derivation of the 7.86. By contrast the monotone case is essentially self-contained: I checked Lemma 3.2's cancellation (3.6) (valid because ∫(uX − ũ)dξ is y-independent via (1.8)), the (4.15) chain with (4.8), and positivity of C*, C* under (1.10); all consistent. The reader's flagged weakest assumption (s = O(√ε2)) is a candidly stated scope restriction (Remark 1.8), not a correctness risk to the stated theorems; the real soft spot is the imported numerical chain underlying the oscillatory case.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper studies stability of planar viscous-dispersive shocks (both monotone and oscillatory) of the 2D dissipative Kadomtsev–Petviashvili equation (1.1), for both signs λ=±1, and of the multi-dimensional Zakharov–Kuznetsov–Burgers equation (1.4), under arbitrarily large L² perturbations periodic in the transverse variable(s). The main results are: global existence in a class XT respecting the KP zero-mean constraint (Theorem 1.1); L²-contraction up to a Lipschitz shift X(t) with explicit ODE in the monotone regime δ(u−−u+)/2ε1²≤1/4 under s<4π√(ε1ε2) (Theorem 1.3); the analogous contraction in the oscillatory regime 1/4<δs<1/2 under s<√(7.86ε2)/(Mπ), M=4/3 (Theorem 1.6); ZKB analogues (Theorems 7.1–7.2); plus L^p (p>2) time-asymptotic stability and sublinear shift growth. The method is the Kang–Vasseur-type shifted energy identity (Lemma 3.2), a change of variables by the shock profile, the weighted Poincaré inequality of [13], and transverse Poincaré on the torus; the oscillatory case closes via an induction on monotonicity intervals (Appendix A) that imports sharp numerical profile estimates from the authors' companion preprint [8].","tokens_in":36875,"tokens_out":14696,"duration_ms":856738,"significance":"If correct, this is the first multi-dimensional L²-contraction theory with large perturbations for oscillatory dispersive shocks, and a genuine extension of the recent 1D KdV–Burgers theory [2,7,8] to the KP and ZKB settings. Strengths worth naming: the monotone case (Sections 3–4) is self-contained and checkable line by line — the energy identity (3.3), the key cancellation (3.6) exploiting the intrinsic zero-mean condition (1.8) (which treats KP-I and KP-II simultaneously), the profile inequality (4.1), and the closing chain (4.15) are all internally consistent; the shift ODE is explicit; the smallness thresholds are stated with all constants; and a global existence theorem adapted to unequal end states is included. The results come with falsifiable, parameter-explicit hypotheses. The quantitative content of the oscillatory case, however, rests entirely on numerically specified profile constants quoted without proof from the unrefereed companion preprint [8], which tempers the current verifiability of Theorems 1.6 and 7.2.","major_comments":[{"comment":"The constant C** printed in Theorem 1.6, C** = 2ε2 − 2.034 s²/(8π² M), does not match the proof. The chain (5.9) ends with the transverse coefficient (2.034 M s²/(16π²) − ε2), i.e. M in the numerator (the M comes from the shift weight M/(2s) in (3.7) and is untouched by Cauchy–Schwarz). Hence (5.10) is established only with C**/2 = ε2 − 2.034 M s²/(16π²), i.e. C** = 2ε2 − 2.034 M s²/(8π²). Since M = 4/3 > 1, the printed C** is larger than the one the proof delivers, so inequality (1.19) as stated is formally stronger than what is proved. The same mismatch appears in Theorem 7.2. Because hypothesis (1.18) is far stricter than either positivity condition, the theorems survive, but the statement, the proof, and the admissibility criterion C** ≥ 0 must be reconciled.","section":"§5.2, Eqs. (5.9)–(5.10) vs. Theorem 1.6 (and Theorem 7.2)"},{"comment":"The constant 7.86 in the hypothesis s < √(7.86 ε2)/(Mπ) is never derived. The proof only needs positivity of the transverse coefficient in (5.9), i.e. s² < 16π²ε2/(2.034 M) ≈ 58.2 ε2 for M = 4/3 (or ≈103.5 ε2 under the printed C**), whereas (1.18) imposes s² < 7.86 ε2/(M²π²) ≈ 0.448 ε2 — roughly two orders of magnitude stricter, with no explanation. Since Remark 1.8 builds the paper's physical interpretation (admissible shock strength relative to transverse dissipation) on this condition, the authors should either derive 7.86 or replace (1.18)/(7.8) by the sharp condition the argument actually yields. As printed, a reader cannot tell whether 7.86 is a transcription error or a remnant of a different scaling.","section":"Eq. (1.18) (and Eq. (7.8)), Remark 1.8"},{"comment":"Theorems 1.6 and 7.2 reduce to Theorem 5.4, whose induction consumes numerically specified constants imported from the unrefereed companion preprint [8]: u0 ≤ 1.0601s, ρ* = 4.64/4.77, λ̄0 = 0.355, λ̄1 = 9.60, the L² bounds 0.178/0.81/0.82 in (5.5), and the tail bound 0.001s in (5.6). The margins are thin: Step 0 requires C0 ≥ 14s/(222s − 198u0), which needs u0 < 1.1212s — the imported 1.0601s leaves ~6% slack, and M = 4/3 is exactly the saturated minimum 40C0/39 at C0 = 13/10; the diffusion budget on (ξ0,ξ̄1) reaches 0.83 + 0.06 = 0.89 against the 9/10 cap. A few-percent error in any imported constant breaks the chain. The manuscript should (i) state precisely how the constants of [8] are obtained (purely analytic ODE estimates, or computer-assisted, and if so with what rigor), (ii) clarify the publication/refereeing status of [8], and (iii) add a short sensitivity discussion showing whi","section":"§5.1 (Props. 5.1–5.2, Thm 2.1) and Appendix A"}],"minor_comments":[{"comment":"The definition Ji := (ξi, ξi) uses ξi for two distinct objects (the extremum location and the other point where ũ attains the value ui); as printed the interval reads as empty. Please distinguish the two (e.g. bar notation as presumably in [8]).","section":"§5.1, after Prop. 5.1"},{"comment":"\"Proof of the Theorem 5.8\" should read Theorem 5.4.","section":"Appendix A, first line"},{"comment":"The hypothesis \"u0 ∈ H^s(Ω)\" should be u0 − f ∈ H^s(Ω) as in Theorem 1.1; u0 itself does not decay.","section":"Theorems 1.3 and 1.6"},{"comment":"The proof of Theorem 7.2 — one of the four main theorems — is \"left to the reader.\" Given that the paper advertises self-containment and that the constant issues of Major Comments 1–2 propagate to (7.8)–(7.9), the short adaptation of §5 should be included.","section":"§7, Theorem 7.2"},{"comment":"The proof is omitted as similar to [7]. Please state exactly which conclusions of Theorem 1.4 are used later (e.g. boundedness of ∥ũ′∥_{L^{4/3}} in the shift asymptotics) so the omission is checkable.","section":"Theorem 1.4"},{"comment":"The x-derivative coefficient is 1/10 in (5.10) but 1/5 in (1.19); this is consistent only after multiplying the integrated inequality by 2 (likewise C*ε1/2 → C*ε1 in (4.16) vs. (1.11)). One line of explanation would prevent confusion.","section":"Eq. (1.19) vs. (5.10)"},{"comment":"In the estimate 0.1202s/(ρ* − 1) + 2s ≤ 2.034s, state explicitly that ρ* = 4.64 (the smaller of the two rates in (5.1)) is used.","section":"Prop. 5.3"},{"comment":"Typos/notation: \"priori estimates\" (§6, missing \"a\"), \"inequalilty\" (§6, Step 3), \"ξ ↦→ξ\" (§3 and §7, double arrow), and spacing artifacts in the title (\"multip le\", \"sh ock\").","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The entire quantitative content of the oscillatory case (Theorems 1.6 and 7.2) traces to constants quoted from reference [8], an unrefereed arXiv preprint with substantially overlapping authorship; reference [7] is similarly unrefereed. I do not regard self-citation of prior work as problematic per se — the monotone half of the paper is self-contained and I verified its key steps — but the load-bearing inputs u0 ≤ 1.0601s, ρ*, λ̄i, and the L²/tail bounds in (5.5)–(5.6) cannot be checked from this manuscript, and the Appendix A induction is calibrated within a few percent of those constants. The editor may wish to ask for the derivation details of [8]'s estimates (or expedited refereeing of [8] in tandem) before final acceptance. The unexplained factor ~230 between hypothesis (1.18) and the condition the proof uses, together with the M-placement mismatch in C**, suggests the constants in the non-monotone statements have not had a final careful pass."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real news is that they get genuine multi-D L2-contraction (up to a Lipschitz shift) for planar viscous-dispersive shocks of dissipative KP and multi-D ZKB, under large L2 perturbations. That is new relative to their 1D papers [7,8]. The transverse dissipation, the Poincaré control on the torus, and the cancellation of the nonlocal KP term (via the zero-mean condition) are handled cleanly; Lemma 3.2 and the energy identity are self-contained and check out.\n\nWhat they do well: global existence for KP with distinct end-states (Thm 1.1) is standard but useful and carefully written. The monotone case (Thm 1.3 / 7.1) is essentially self-contained once you accept the elementary profile inequality (4.1); the change-of-variable + Poincaré argument closes with explicit constants and the small-shock condition s = O(√ε2) is stated honestly in Remark 1.8. The asymptotic stability and sub-linear shift statements follow in the usual way.\n\nThe soft spot is the non-monotone half. Theorems 1.6 and 7.2 reduce to the induction in Appendix A / Thm 5.4, which imports the whole numerical package from the unrefeered companion [8]: overshoot u0 ≤ 1.0601s, decay rates ρ*, λ̄i, the L2 bounds (5.5)–(5.6), and the L1 bound 2.034s. Those numbers sit close to the edge of the diffusion budget that produces the 1/10 remainder; a few-percent slip breaks the chain. There is also an unexplained ~200× gap between the stated hypothesis (1.18) and the positivity condition actually used for C**. The monotone theory does not have this problem. Dependence on [7,8] is real but not circular—the profile estimates are independent analytic work—yet until [8] is vetted the oscillatory claim remains conditional.\n\nThis is for specialists already working on multi-D stability of viscous-dispersive shocks or KP-type models. They will cite the monotone statements and the existence theory immediately; the oscillatory statements need the companion paper to settle. It deserves a serious referee, with instructions to check the imported constants and the 7.86 discrepancy. I would engage.","headline":"Solid multi-D extension of the authors’ own 1D L2-contraction theory for dispersive shocks; the monotone half is clean, the oscillatory half rides on thin numerical margins imported from an unrefeered companion preprint.","tokens_in":36137,"tokens_out":587,"would_cite":true,"duration_ms":10304,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B35","35Q53","35L67","76L05"],"pacs":[],"model":"grok-4.5","headline":"Planar dispersive shocks of dissipative KP and multi-D KdV-Burgers remain L2-stable under arbitrarily large multi-dimensional perturbations, up to a Lipschitz time shift.","keywords":["dissipative Kadomtsev-Petviashvili","multi-dimensional KdV-Burgers","oscillatory shock","L2 contraction","uniform stability","planar dispersive shock","time-dependent shift"],"falsifier":"Numerically integrate the dissipative KP equation with a fixed oscillatory planar shock of strength s and successively smaller transverse viscosity ε2 until s exceeds the explicit threshold √(7.86 ε2)/(Mπ); if the L2 distance to every Lipschitz translate of the shock grows rather than contracts, the claimed inequality is false.","tokens_in":35623,"feed_emoji":"🌊","tokens_out":996,"duration_ms":24387,"temperature":0.7,"pith_summary":"The paper proves that planar monotone or oscillatory viscous-dispersive shock waves for the dissipative Kadomtsev-Petviashvili equation and the multi-dimensional Zakharov-Kuznetsov-Burgers equation are L2-contracting under large perturbations in the transverse directions. Solutions starting from any large L2 perturbation of the planar profile stay close to a time-shifted copy of that profile, with the shift itself remaining Lipschitz and asymptotically sublinear. The result covers both the monotone regime (viscosity dominates dispersion) and a range of the oscillatory regime (dispersion dominates), and it yields time-asymptotic stability in Lp for p>2. A sympathetic reader cares because multi-dimensional water-wave and plasma models routinely produce such shocks, yet previous L2-contraction theory existed only in one space dimension; the work supplies the first uniform multi-D control that does not require smallness of the initial perturbation.","feed_headline":"Large multi-D perturbations of planar shocks still contract in L2","feed_subtitle":"Dissipative KP and ZKB shocks stay stable under arbitrary transverse noise once the jump is weak enough","key_machinery":"The relative entropy identity obtained after shifting the planar profile by a Lipschitz ODE that cancels the leading linear term, combined with a change of variables along the shock and a Poincaré inequality in the periodic transverse variables that absorbs the residual quadratic terms provided the shock is not too strong relative to transverse viscosity.","core_discovery":"Under explicit parameter restrictions linking shock strength to transverse dissipation, any global solution of dissipative KP (or multi-D ZKB) that is a large L2 perturbation of a planar dispersive shock remains L2-contracting toward a Lipschitz time-dependent translate of that shock, and the L2 distance plus integrated dissipation controls the initial distance. The same contraction implies asymptotic stability in every Lp, p>2, and the shift velocity tends to zero.","pith_inferences":["The method suggests that any viscous-dispersive system whose planar shock satisfies a uniform spectral gap and whose transverse dissipation supplies a Poincaré constant can inherit multi-D L2 contraction by the same relative-entropy-plus-shift argument.","Because the singular limit of vanishing transverse viscosity lies outside the present estimates, the result leaves open whether truly anisotropic physical shocks remain stable; a matched-asymptotics or weighted-energy refinement would be a natural next test.","The explicit decay rates on the oscillatory tails could be fed into a numerical continuation scheme to push the upper bound on the dispersion-to-viscosity ratio beyond 1/2 while retaining multi-D contraction."],"forward_implications":["Global L2 stability of planar dispersive shocks holds for both KP-I and KP-II under large multi-D perturbations once the shock is sufficiently weak relative to transverse viscosity.","The same contraction and asymptotic stability statements transfer verbatim to the multi-dimensional Zakharov-Kuznetsov-Burgers equation with arbitrary transverse dispersion coefficient.","The Lipschitz shift remains sublinear in time, so the long-time shape of the multi-D solution is still that of the original planar profile.","Existence of global solutions with unequal far-field states is obtained as a byproduct for the dissipative KP equation."],"fun_headline_variants":["L2 contraction for large multi-D perturbations of planar dispersive shocks","Planar KP and KdV-Burgers shocks stay L2-contracting under huge 2D noise","Large L2 perturbations of oscillatory planar shocks still contract","Multi-D dispersive shocks remain L2 stable via Lipschitz time shifts","Arbitrary transverse perturbations leave planar shock L2 contraction intact"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The shock jump must be small enough compared with the square root of the transverse dissipation coefficient; otherwise the transverse Poincaré estimate fails to close and the multi-dimensional energy balance no longer yields contraction.","fun_headline_variants_meta":{"raw":{"variants":["L2 contraction for large multi-D perturbations of planar dispersive shocks","Planar KP and KdV-Burgers shocks stay L2-contracting under huge 2D noise","Large L2 perturbations of oscillatory planar shocks still contract","Multi-D dispersive shocks remain L2 stable via Lipschitz time shifts","Arbitrary transverse perturbations leave planar shock L2 contraction intact"]},"model":"grok-4.5","effort":"low","cost_usd":0.004322,"raw_usage":{"total_tokens":1200,"prompt_tokens":665,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":43224000,"prompt_tokens_details":{"text_tokens":665,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":456,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":665,"tokens_out":79,"duration_ms":8427,"temperature":1.0,"reasoning_tokens":456,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T12:29:34.058169+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Numerically integrate the dissipative KP equation with a fixed oscillatory planar shock of strength s and successively smaller transverse viscosity ε2 until s exceeds the explicit threshold √(7.86 ε2)/(Mπ); if the L2 distance to every Lipschitz translate of the shock grows rather than contracts, the claimed inequality is false.","supporting_citations":[],"review_version":1}