{"id":"e83ceaed-994a-45a8-8ce5-d4f43cbe5e98","arxiv_id":"2505.24562","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"This paper proves the existence of gravity-driven traveling bore solutions to the 2D free-boundary Navier-Stokes equations in shallow single-layer flow, both surging and ebbing.","lead":"Stevenson and Tice prove, with a long analysis, that a shallow viscous fluid layer on an inclined plane can support traveling bore waves, where the surface height jumps between two constant levels, driven purely by gravity. It is the first rigorous existence proof of such waves in the free-boundary Navier-Stokes equations, a problem that has been open for over a century.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Epsilon-uniform Stokes invertibility (Thm 3.9) is not established as written: the Korn proof in Prop 3.6 drops a non-negligible ||Dψ|| term, leaving the key estimate chain (3.1.65)-(3.1.67) unjustified.","rationale":"The reader's weakest-assumption analysis correctly identified Theorem 3.9 as the load-bearing premise of the contraction argument. My review locates a more specific defect inside that theorem's proof: Proposition 3.6, the thin-domain Korn inequality that supplies the ε-independent coercivity for the Stokes operator, contains an algebraic omission in the derivation of its main estimate. Specifically, the transition from (3.1.65) to (3.1.66) drops the ε^{-1/2}||Dψ|| term forced by the R1ψ contribution, and the subsequent adsorption step (3.1.67) therefore does not follow. If the omitted term is essential, the Korn constant may grow as ε→0, which would propagate through Proposition 3.7, Proposition 3.8, and Theorem 3.9 to destroy the smallness of the residual forcing terms in Proposition 5.3. This would invalidate the existence theorem. However, I do not assert that the Korn estimate is false; the paper's overall strategy is coherent and the surrounding arguments are careful. The defect is best treated as a condition on the proof: the authors must supply a corrected derivation of (3.1.37). If a corrected interpolation proves the ε-uniform Korn bound, the central argument stands; if not, a counterexample would refute it. Because the reader's verdict was already CONDITIONAL due to truncation and unverified estimates, my finding does not change the verdict category, but it sharpens the condition: Theorem 3.9 should not be accepted as established until Proposition 3.6's omitted-term issue is resolved. The independent support in the paper (consistent ODE analysis, explicit residual computations, clear fixed-point structure) counts in favor of the result, but the load-bearing linear estimate is exactly where the proof needs scrutiny.","tokens_in":82273,"tokens_out":19909,"duration_ms":221388,"concrete_test":"Recompute Proposition 3.6 starting from (3.1.65), retaining the ε^{-1}||Dψ||^2 term when solving for ||Rψ12||. Check whether the resulting bound on ||∇ψ|| in Step 4 is of the form C1||Dψ|| + C2||Dψ||^{1/2}||ψ||^{1/2} + C3ε||ψ|| with C1 < 1 for all small ε; if C1 ≥ 1, the adsorption fails. To test the truth of the Korn estimate itself, attempt to construct a family ψ_ε ∈ H^1(Ω_{εh}) with Tr_{Σ0}(ψ_ε)_2 = 0 and Tr_{Σ0}(ψ_ε)_1 = 0 for which ||∇ψ_ε||_{L^2} / (||Dψ_ε||_{L^2} + ||ψ_ε||_{L^2}) → ∞ as ε → 0. If such a family exists, (3.1.36) fails and Theorem 3.9 collapses; if no such family exists, the gap is purely in the proof presentation and can be repaired by a correct interpolation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader identified Theorem 3.9, estimate (3.2.73), as the load-bearing premise: the fixed-point contraction in Proposition 5.3 needs the inverse of L^ε_h to be bounded independently of ε, with Lipschitz dependence on h. If any inverse norm contained a positive power of ε^{-1}, the residual forcings of order ε^{3/2}|log ε|^9 and 1/|log ε| would not be small enough to close the argument. This concern is correct, and there is a concrete point inside the visible proof where the ε-uniformity is not actually demonstrated. Proposition 3.6 (Korn, II) is the source of the ε-independent constants used in the coercivity of the Stokes bilinear form (3.2.12) and hence in the a priori estimate (3.2.24). In the proof of Proposition 3.6, inequality (3.1.65) contains the term ε^{-1}||Dψ||^2 coming from the R1ψ part of Rψ12 (see (3.1.52)-(3.1.54)). Solving (3.1.65) for ||Rψ12|| therefore yields a term ε^{-1/2}||Dψ||, but this term is absent from the stated bound (3.1.66). If the missing term is restored, the following line (3.1.67) becomes ||∇ψ|| ≲ ||Dψ|| + ε^{1/2}||Rψ12|| ≲ (1+C)||Dψ|| + C||Dψ||^{1/2}||ψ||^{1/2} + Cε||ψ||, with a coefficient of ||Dψ|| that is not small in ε. The advertised adsorption step to (3.1.37) is therefore not justified by the displayed inequalities. This does not prove the Korn estimate false, but it means the epsilon-uniformity of the Stokes inverse in Theorem 3.9 is not established by the text as written, and the central existence theorem currently rests on an unverified estimate.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims the first rigorous construction of two-dimensional traveling bore wave solutions to the free-boundary incompressible Navier-Stokes equations in a shallow single-layer fluid over an inclined plane. The strategy is to derive, via a formal shallow-water scaling, a one-dimensional Liénard-type ODE for the leading-order free surface height, prove existence of heteroclinic orbits connecting two distinct shear-flow equilibria using phase-plane and energy arguments, then build actual Navier-Stokes bore solutions as small perturbations of these orbits. The perturbation argument combines a nonautonomous heteroclinic persistence theorem, a detailed linear theory for a Stokes operator in ε-thin domains with stress and Navier-slip boundary conditions, a careful ansatz separating ODE and residual unknowns, and a contraction-mapping fixed point. The main theorem asserts existence of smooth surging and ebbing bores for explicit parameter regions C_{-1} ∪ C_1, with surface tension optional and with uniform-in-ε estimates down to a small-depth threshold.","tokens_in":82704,"tokens_out":7860,"duration_ms":99972,"significance":"If the proof is completed, the result would be a substantial advance: it would provide the first construction of nontrivial gravity-driven traveling viscous surface waves with different asymptotic heights, answering a longstanding question highlighted by Rayleigh's inviscid impossibility argument. The paper is self-contained in its main architecture: the heteroclinic orbits are produced from explicit ODE coefficients, the parameter tuning (1.4.5) is explicit, the thin-domain Stokes theory is developed from scratch, and the fixed-point scheme gives quantitative control of the residual. The paper also gives concrete falsifiable predictions in the form of parameter regions and leading-order profiles, and it carefully separates the ODE analysis, the PDE linear theory, and the nonlinear fixed-point argument. The main concern is whether one central estimate in the thin-domain Korn inequality is actually proved with the ε-uniform constants required downstream.","major_comments":[{"comment":"The proof of the thin-domain Korn estimate is not complete as written. The displayed line (3.1.65) contains the term ε^{-1}||Dψ||^2_{L^2(Ω_{εh})}, which comes directly from the bound (3.1.54) on R_1ψ. Taking square roots in (3.1.65) therefore produces a term ε^{-1/2}||Dψ||_{L^2(Ω_{εh})} in the estimate for ||Rψ12||_{L^2(R)}. This term is absent from the claimed bound (3.1.66), and if it is restored the following line (3.1.67) becomes ||∇ψ|| ≲ ||Dψ|| + ||Dψ||^{1/2}||ψ||^{1/2} + ε||ψ||, where the first term has a coefficient that is not small in ε. The advertised absorption leading to (3.1.37) is therefore not justified by the displayed inequalities. The estimate (3.1.36) may well be true, but the text as written does not prove it, and the ε-uniformity of the Stokes inverse in Theorem 3.9 rests on this step.","section":"§3.1, Proposition 3.6, Eqs. (3.1.54)–(3.1.67)"},{"comment":"The fixed-point argument makes the ε-uniform invertibility of the Stokes operator load-bearing. The contraction estimates (5.1.22)–(5.1.23) close only because the residuals are of size ε^{3/2}|log ε|^9 and 1/|log ε|; if the inverse operator norms or Lipschitz constants in Theorem 3.9's estimate (3.2.73) contained any positive power of ε^{-1}, these residuals would not be small enough to apply the contraction mapping theorem. Since the proof of that estimate depends directly on the coercivity (3.2.12), which in turn depends on Proposition 3.6, the gap identified in the preceding comment affects the central existence claim of Theorem 1. The authors need either to supply the missing estimate for the ε^{-1/2}||Dψ|| term or to replace the Korn argument with a different proof of the ε-uniform a priori bound.","section":"§5.3, Proposition 5.3 and Theorem 5.4"}],"minor_comments":[{"comment":"There are several typos in these lemmas: 'strictly deceasing' should be 'strictly decreasing', and in Lemma 2.2 'of and only if' should be 'if and only if'.","section":"§2.1, Lemma 2.1 and Lemma 2.2"},{"comment":"The notation for the average of (1/2)Aψ over the domain Ω_{εh} ∩ Q_ε(x_0) is not fully explicit; the displayed formula should indicate clearly that the integral is normalized by the measure of that set, and the domain of integration should be stated consistently in the bounds that follow.","section":"§3.1, Lemma 3.5, Eq. (3.1.31)"},{"comment":"The notation f_j(R,ε;·) is slightly misleading because the maps f_j do not depend on R except through the admissible sets E^R_ε and F^R_ε; renaming the maps or the sets would improve readability.","section":"§4.3, Definition 4.13"},{"comment":"The long identity for ∂_2p should be checked for missing parentheses or a missing factor of 1/h; as typeset it is difficult to verify by eye, and the derivation is important for the normal-derivative estimate (3.2.67).","section":"§3.2, Eq. (3.2.66)"}],"recommendation":"major_revision","confidential_remarks":"The paper is ambitious and the overall architecture is coherent, but the specific gap in Proposition 3.6 is exactly the kind of load-bearing technical point that must be fixed before the main theorem can be accepted. I would encourage the authors to provide a complete proof or a corrected argument for the Korn estimate; if they do so, the paper would be a strong contribution to the mathematical theory of viscous free-boundary waves. The manuscript's length is appropriate for the depth of the construction, and the exposition of the parameter tuning and the heteroclinic analysis is generally clear."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this is the first existence proof of single-layer bore waves for free-boundary Navier-Stokes, and the first construction of gravity-driven traveling viscous surface waves without a localized forcing. Those are big claims, and the paper largely backs them up. Second, the visible argument is coherent and modular, but there is a small error in the thin-domain Korn estimate that the stress-test flagged; I think it is fixable and not fatal.\n\nThe paper works by justifying the shallow-water limit rigorously: it finds heteroclinic orbits in a Liénard equation, then builds a fixed point around a thin-domain Stokes operator. The parameter tuning (1.4.5) that makes the end states ε-independent is clever and necessary. Section 2.2, the nonautonomous perturbation theory for heteroclinic orbits, is self-contained and generally useful. The authors are honest about limitations: small depth, specific parameter sets C_±1, no claim outside them, and open questions about κ=0 and upstream waves.\n\nThe soft spots. One: in Prop 3.6, inequality (3.1.66) drops the ε^{-1/2}||Dψ|| term that follows from (3.1.65). The stress-test claimed this breaks the ε-uniformity of the Stokes inverse. I disagree. With the term restored, (3.1.67) becomes ||∇ψ|| ≲ ||Dψ|| + ||Dψ||^{1/2}||ψ||^{1/2} + ε||ψ||, which is enough: Young's inequality and a small-ε absorption give (3.1.37) with constants independent of ε. So the gap is a typo-level omission, not a load-bearing flaw. It should be fixed before publication.\n\nTwo: the version I saw truncates at the end of Theorem 5.4, so the final verification of the ε-bound on η,u,p and the far-field limits is not fully visible. A referee should read the end of Section 5 carefully. Three: several estimate bundles in Section 4.3 are described as 'straightforward computations'; they're plausible, but a referee should spot-check them.\n\nBottom line: the significance is high and the construction is credible. This deserves a serious referee, with minor-to-moderate revisions. I'd cite it and bring it to a reading group.","headline":"First rigorous viscous bores; mostly sound, with a fixable Korn-estimate omission that isn't fatal.","tokens_in":83246,"tokens_out":5844,"would_cite":true,"duration_ms":62922,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35C07","76D33","35B40","35J66","76A20","76L05","34C37"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper gives the first rigorous construction of traveling bore waves — smooth fronts joining two distinct shear-flow states — for the free-boundary incompressible Navier-Stokes equations in a single shallow layer of fluid, with…","keywords":["bore waves","traveling fronts","free boundary Navier-Stokes","viscous surface waves","shallow water","heteroclinic orbits","thin domains","Stokes problem"],"falsifier":"Compute, numerically or by spectral analysis, the operator norm of the inverse thin-domain Stokes map $(L^\\epsilon_h)^{-1}$ in the norms of estimate (3.2.73) for a flat profile $h\\equiv1$; if the norm grows like $\\epsilon^{-\\alpha}$ for any $\\alpha>0$, the contraction argument cannot apply. A second check: take $g=8$, the one value excluded from $C_{-1}\\cup C_1$ in Lemma 2.2, and numerically integrate the ODE (1.5.15); the appearance of a heteroclinic orbit there would show the parameter sets are not the end of the story, though Theorem 1 itself claims only existence in $C_\\iota$, not non-existence outside.","tokens_in":81957,"feed_emoji":"🌊","tokens_out":12371,"duration_ms":125222,"temperature":0.7,"pith_summary":"The paper proves that a single shallow layer of viscous, incompressible fluid with a free surface can support traveling bore waves — fronts that join two different constant heights and shear flows far upstream and downstream — with gravity as the only driving force. This is the first construction of such fronts for the free-boundary Navier-Stokes equations; previously, all known viscous traveling surface waves required an external moving force or stress, and classical results showed inviscid single layers cannot sustain bores. The construction rigorously justifies the shallow-water limit by reducing the traveling PDE system to an ODE for the surface height, exhibiting heteroclinic orbits of that ODE, and then lifting each orbit to an exact Navier-Stokes solution through a fixed-point argument in thin domains. The result covers both 'surging' bores (subcritical, small Froude number) and 'ebbing' bores (supercritical), does not require surface tension, and yields bores of large or small height jump depending on the parameter choice.","feed_headline":"Gravity alone can sustain viscous bore waves","feed_subtitle":"First rigorous proof that gravity alone drives viscous bores, both surging and ebbing.","key_machinery":"The argument is carried by four coupled mechanisms. First, the parameter-tuning identities (1.4.5) fix the asymptotic heights $0<H_-<H_+<1$ so they are independent of $\\epsilon$ and coincide with the zeros of the relative-flux polynomial (1.4.2). Second, the shallow-water reduction replaces the PDE by the Liénard-type ODE $\\rho''=F(\\rho)-G(\\rho)\\rho'$ with $\\rho=\\log H$ and $F$, $G$ as in (1.5.16); when $(g,A)\\in C_\\iota$ the associated potential $V$ and the sign of $G$ force heteroclinic orbits (solutions connecting the two equilibria $\\rho_-$ and $\\rho_+$) as proved in Theorem 2.5. Third, a general nonautonomous perturbation theorem (Theorem 2.10) shows these orbits persist under small, time-dependent perturbations, producing the 'bore map' that solves the coupled ODEs as a function of the residual unknowns. Fourth, the thin-domain Stokes problem (3.2.1) with stress boundary conditions and Navier slip is shown to be invertible with norms uniform in $\\epsilon$ and Lipschitz in the profile $h$ (Theorem 3.9); with a coupling $r_1,r_3,r_4$ chosen to cancel the adversarial linear terms, the residual PDE becomes a contractive fixed-point problem (Proposition 5.3), using the $\\epsilon^{3/2}|\\log\\epsilon|^9$ and $1/|\\log\\epsilon|$ smallness of the residual sources.","core_discovery":"Fix viscosity $\\mu>0$, slip parameter $a>0$, a sign $\\iota\\in\\{-1,1\\}$, a gravity-height pair $(g,A)$ in the explicit sets $C_\\iota$ defined in (1.5.2), and surface tension $\\sigma\\ge0$. The paper proves there is a small-depth threshold $\\epsilon_*$ such that for every shallowness parameter $\\epsilon\\in(0,\\epsilon_*)$ the nondimensionalized free-boundary Navier-Stokes system (1.4.8) has a smooth classical bore wave in the sense of Definition 0: a free surface $H+\\eta$, a velocity field, and a pressure that solve the system, converge to two distinct shear flows with heights $H_+$ and $H_-$ at opposite infinities, and are surging if $\\iota=-1$ and ebbing if $\\iota=1$. The leading-order free surface $H$ solves the ODE (1.5.3) with limits $H(\\iota x)\\to H_\\pm$ as $x\\to\\pm\\infty$, and the residual $(\\eta,u,p)$ is of size $O(\\epsilon)$ in the relevant norms. Corollary 4 converts this into dimensional Eulerian bores for essentially every leading-order wave speed $\\gamma$, with the surging/ebbing distinction governed by a Froude number $\\mathrm{Fr}^2=\\gamma\\kappa/(2ga)$.","pith_inferences":["The single value $g=8$ omitted by the theorem is exactly the critical-Froude line $\\mathrm{Fr}=1$; the paper makes no claim there, but a natural conjecture is that critical-speed bores, if they exist, need a different construction or break down at the shallow-water level — a concrete next test.","The modular design — heteroclinic germ plus nonautonomous persistence plus thin-domain fixed point — suggests the same machinery could produce other gravity-driven viscous waves the authors flag as open: upstream-traveling bores ($\\gamma<0$), purely vertical gravity ($\\kappa=0$), or other capillary scaling regimes.","The estimate that really carries the proof is the $\\epsilon$-uniform invertibility of the thin-domain Stokes operator (Theorem 3.9); an independent proof or numerical verification of that estimate would determine how far the shallow-water construction can be pushed beyond the stated threshold."],"forward_implications":["Bores exist for arbitrarily shallow layers, with the free surface, velocity, and pressure differing from their asymptotic shear states by $O(\\epsilon)$; the construction yields both 'small' bores ($A$ close to $1$) and 'large' bores with height jump close to $1$ ($A$ close to $0$).","As $\\epsilon\\to0$ the Navier-Stokes bore lies within $O(\\epsilon)$ of a traveling-wave solution of the one-dimensional viscous shallow water equations with laminar drag (Corollary 3), a rigorous confirmation of the formal shallow-water limit.","In physical units, for any viscosity, horizontal gravity, slip parameter, vertical gravity, and surface tension, every leading-order wave speed $\\gamma$ except $\\gamma=2ga/\\kappa$ admits smooth Eulerian bore waves in sufficiently shallow layers; surging bores occur exactly when the Froude number $\\mathrm{Fr}<1$ and ebbing bores when $\\mathrm{Fr}>1$ (Corollary 4).","Surface tension is not needed: the theorem holds with $\\sigma=0$, so a purely gravitational mechanism suffices to sustain nontrivial viscous traveling surface waves."],"supporting_citations":[{"why":"supplies the classical proof that inviscid single-layer bores are impossible, the motivation for including viscosity.","marker":"[73]"},{"why":"gives the first construction of traveling viscous free-boundary waves and the derivation of the traveling system used here.","marker":"[47]"},{"why":"provides the inclined-plane and Navier-slip formulation of the traveling free-boundary Navier-Stokes system.","marker":"[41]"},{"why":"derives the viscous shallow-water limit and supplies the ODE heteroclinic germ at the core of the construction.","marker":"[81]"},{"why":"provides the thin-domain Korn inequality theory adapted into Proposition 3.6.","marker":"[50]"},{"why":"supplies the thin-domain Korn technique that the paper modifies for unbounded domains with stress boundary conditions.","marker":"[51]"},{"why":"provides the elliptic regularity framework used to establish strong solvability of the thin-domain Stokes problem.","marker":"[2]"},{"why":"gives the phase-plane and stable/unstable manifold results used to construct the heteroclinic orbits.","marker":"[70]"},{"why":"supplies the contraction mapping with parameter and hyperbolic diffeomorphism tools used in the nonautonomous perturbation theorem.","marker":"[39]"},{"why":"frames the nonautonomous perturbation theory near hyperbolic equilibria that the authors adapt.","marker":"[7]"}],"fun_headline_variants":["First proof: gravity alone drives viscous bore waves","Surging and ebbing bores from gravity only","Exact bores via shallow water limit","Traveling bore waves: first existence result","Free boundary bores proven in Navier-Stokes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole fixed-point construction rests on the claim that the solution operator for the linearized Stokes problem in the thin layer has norm bounded independently of the layer depth $\\epsilon$ (and depends Lipschitz-continuously on the profile); if that norm grew like any positive power of $1/\\epsilon$, the residual errors, which are only small like $\\epsilon^{3/2}|\\log\\epsilon|^9$ and $1/|\\log\\epsilon|$, would not be small enough to close the contraction.","fun_headline_variants_meta":{"raw":{"variants":["First proof: gravity alone drives viscous bore waves","Surging and ebbing bores from gravity only","Exact bores via shallow water limit","Traveling bore waves: first existence result","Free boundary bores proven in Navier-Stokes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000374,"raw_usage":{"total_tokens":2000,"prompt_tokens":949,"completion_tokens":1051,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":979}},"tokens_in":565,"tokens_out":1051,"duration_ms":13332,"temperature":1.0,"reasoning_tokens":979,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:20:49.272781+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, numerically or by spectral analysis, the operator norm of the inverse thin-domain Stokes map $(L^\\epsilon_h)^{-1}$ in the norms of estimate (3.2.73) for a flat profile $h\\equiv1$; if the norm grows like $\\epsilon^{-\\alpha}$ for any $\\alpha>0$, the contraction argument cannot apply. A second check: take $g=8$, the one value excluded from $C_{-1}\\cup C_1$ in Lemma 2.2, and numerically integrate the ODE (1.5.15); the appearance of a heteroclinic orbit there would show the parameter sets are not the end of the story, though Theorem 1 itself claims only existence in $C_\\iota$, not non-existence outside.","supporting_citations":[{"cited_title":"Rayleigh","cited_arxiv_id":null,"evidence_quote":"supplies the classical proof that inviscid single-layer bores are impossible, the motivation for including viscosity."},{"cited_title":"Leoni and I","cited_arxiv_id":null,"evidence_quote":"gives the first construction of traveling viscous free-boundary waves and the derivation of the traveling system used here."},{"cited_title":"Koganemaru and I","cited_arxiv_id":null,"evidence_quote":"provides the inclined-plane and Navier-slip formulation of the traveling free-boundary Navier-Stokes system."},{"cited_title":"Stationary wave solutions to two dimensional viscous shallow water equations: theory of small and large solutions","cited_arxiv_id":"2502.11899","evidence_quote":"derives the viscous shallow-water limit and supplies the ODE heteroclinic germ at the core of the construction."},{"cited_title":"Lewicka.Calculus of Variations on Thin Prestressed Films—Asymptotic Methods in Elasticity, volume 101 ofProgress in Nonlinear Differential Equations and their Applications","cited_arxiv_id":null,"evidence_quote":"provides the thin-domain Korn inequality theory adapted into Proposition 3.6."},{"cited_title":"Lewicka and S","cited_arxiv_id":null,"evidence_quote":"supplies the thin-domain Korn technique that the paper modifies for unbounded domains with stress boundary conditions."},{"cited_title":"Perko.Differential Equations and Dynamical Systems, volume 7 ofTexts in Applied Mathematics","cited_arxiv_id":null,"evidence_quote":"gives the phase-plane and stable/unstable manifold results used to construct the heteroclinic orbits."}],"review_version":1}