{"id":"3648bb65-84c4-431a-8c04-21400bbad5bb","arxiv_id":"2501.03485","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"For the modified Camassa-Holm equation, densely packed N-soliton solutions aggregate into a one-soliton, n-soliton, or line-distribution state as N→∞, depending on the region containing the discrete spectral points.","lead":"This paper analyzes the large-N limit of N-soliton solutions of a shallow-water wave equation. It shows that, depending on the arrangement of the soliton parameters, the crowd can aggregate into a single soliton, an n-soliton state, or a state described by a new Riemann-Hilbert problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.4's sign inequalities on the lens contours are explicitly admitted as unproved; the line-domain N∞-soliton asymptotics (Theorem 5.1) therefore depends on a numerical check, not a rigorous estimate, leaving a principal result conditional.","rationale":"I read the paper in good faith and checked the quadrature-domain arguments in detail. Propositions 4.1 and 4.2 are internally consistent: Lemma 4.1 uses a uniform Riemann-sum approximation, Lemma 4.2 applies Green's theorem to the quadrature-domain boundary, and the resulting jump matrices exactly match the one- and n-soliton RH problems with the stated discrete spectra and norming constants. The quadrant symmetries of the mCH scattering data are handled correctly in the residue conditions. Thus the aggregation claims (i) and (ii) in the abstract appear sound. The line-domain section, however, contains an explicit admission that the key sign inequalities of Lemma 5.4 'can not be rigorously shown' and are only numerically checked. This inequality is not peripheral: it is the mechanism that makes the lens jumps exponentially small in y, which in turn justifies the reduction to the outer parametrix and the small-norm estimate leading to Theorem 5.1. Without a proof of Lemma 5.4, the line-domain asymptotics is not established. This is precisely the weakest assumption identified by the reader, and I agree with that assessment. I also note the outer parametrix is imported from Ref. [42] without a self-contained verification, but that is a secondary gap compared to the unproved sign condition. The paper is plausibly correct and the quadrature results are a real contribution, but the line-domain theorem is conditional on a missing analytic estimate. Therefore the reader's CONDITIONAL verdict is appropriate; my stress test does not move that verdict.","tokens_in":43043,"tokens_out":18969,"duration_ms":171549,"concrete_test":"For a representative range of parameter pairs (a,b) with 1<a<b, e.g., (1.5,2), (2,3), (1.1,10), compute g(z) from Eq. (5.84) with m4,m5 given by Eqs. (5.86)–(5.88), then evaluate Re(2g(z) − (z+1/z)/2) on a fine grid along the four lens arcs O1±,O2±,O3±,O4± defined in §5.1, using high-precision arithmetic. If any point violates the signs claimed in Lemma 5.4, the lemma is false and Theorem 5.1 collapses; if the signs hold for the full parameter grid, the gap is a missing proof rather than a counterexample, and the paper's line-domain result should remain conditional pending a rigorous sign estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The line-domain asymptotics (Section 5) is one of the paper's four stated contributions, and Theorem 5.1 gives the O(1/|y|) behavior of u(x) as y→−∞. The derivation hinges on the lens-opening step, which requires Lemma 5.4: Re(2g(z) − (z+1/z)/2) < 0 on O1±∪O2± and > 0 on O3±∪O4±. The authors state verbatim that 'these two inequalities can not be rigorously shown' and are only checked numerically (Section 5.1). If this sign condition fails anywhere on the lens arcs, the off-diagonal entries of the jump matrix in Eq. (5.118) are not exponentially small, the error estimate VE = I + O(e^{−c|y|}) in Eq. (5.139) is unjustified, the small-norm RH problem for E does not have the claimed contraction, and Theorem 5.1's conclusion does not follow. This is a genuine missing proof, not a stylistic gap: the sign of Re(2g(z) − (z+1/z)/2) is a quantitative property of the explicit g-function in Eq. (5.84) that must be established for all parameters 1<a<b. The remainder of the line-domain argument—including the imported outer parametrix and endpoint local parametrices—is standard conditional on this inequality, so Lemma 5.4 is the single most load-bearing unproved step for the line-domain claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the large-N limit of N-soliton solutions for the modified Camassa-Holm equation with linear dispersion and vanishing boundary conditions, using the Riemann-Hilbert representation of the N-soliton solution. The authors consider discrete spectra distributed in three types of regions: quadrature domains, line segments, and elliptic domains. For quadrature domains, they prove that the limiting modified RH problem collapses to the RH problem of a one-soliton solution when ℓ=n=1 (Proposition 4.1) and to an n-soliton solution when ℓ=n (Proposition 4.2). For the line domain, they carry out a Deift-Zhou steepest descent analysis in the limit y→−∞, obtaining an O(1/|y|) asymptotic formula in Theorem 5.1 in terms of an outer parametrix expressed by theta functions and local Bessel parametrices. For the elliptic domain, they assert equivalence to the line case. The paper explicitly acknowledges that the key sign inequalities in Lemma 5.4, which justify the lens-opening step in the line-domain analysis, cannot be rigorously proved and are only checked numerically.","tokens_in":43409,"tokens_out":5293,"duration_ms":49893,"significance":"If the results hold, this would be the first N∞-soliton asymptotics for the mCH equation with linear dispersion, extending the soliton-shielding phenomenon of Bertola-Grava-Orsatti from NLS to a Camassa-Holm-type equation. The quadrature-domain part is a genuine strength: Lemmas 4.1 and 4.2 give transparent Riemann-sum limits and Green-theorem reductions, and Propositions 4.1 and 4.2 provide explicit limiting one- and n-soliton states. The line-domain part is technically ambitious but rests on an unproved sign estimate that the authors explicitly flag. The paper is also honest in stating what is and what is not proved, which is helpful for assessing the conditional nature of the line-domain theorem.","major_comments":[{"comment":"The sign inequalities Re(2g(z) - (z+1/z)/2) < 0 on O1±∪O2± and > 0 on O3±∪O4± are explicitly admitted to be unproved and are only checked numerically. These inequalities are the mechanism that makes the off-diagonal jump entries in Eq. (5.118) exponentially small as y→−∞, which in turn justifies the estimate V_E = I + O(e^{-c|y|}) in Eq. (5.139), the small-norm RH problem, and finally Theorem 5.1. Without a rigorous proof valid for all 1<a<b and all admissible r2, the line-domain asymptotic formula (5.142) is conditional. This is a load-bearing gap in a principal contribution of the paper, not a stylistic issue.","section":"5.1, Lemma 5.4, Eq. (5.120)"},{"comment":"The conclusion that the elliptic-domain N∞-soliton problem is 'equivalent to the case of the line region' is not demonstrated. Lemma 6.1 only rewrites the boundary integrals over ∂Ω2 as integrals over [ia1, ia2] with a weight ΔF(ζ); it does not show that the RH problem 16, whose jumps are on the small circles Γ5± and Γ6±, can be deformed or mapped to the line-domain RH problem 4 or 5 with jump matrices of the form (5.62). Since the equivalence is stated as one of the paper's four contributions, a precise deformation argument or a construction of the mapping between the two RH problems is needed.","section":"6, Eqs. (6.146)-(6.149)"},{"comment":"The outer parametrix N^o(y;z) is imported from Ref. [42] by referring to 'RH Problem 4.2.3' and 'Theorem 4.3.1' of that reference. However, the jump conditions of RH Problem 13 contain diagonal jumps on Σ2, Σ4, Σ6 with exponential factors e^{xm_i+n_i}, which do not appear in the semiclassical NLS model treated in Ref. [42]. The applicability of the theta-function formula to this modified jump structure is not established; the authors should either provide a derivation of the outer parametrix for the present jump matrices or give a precise reference to a theorem that covers this case.","section":"5.2, Eq. (5.124)"}],"minor_comments":[{"comment":"There are typographical errors in the abstract: 'Riemann-Hilbert problem,;' should be cleaned up, and the keyword 'Riemann-Hiblert problem' should be 'Riemann-Hilbert problem'.","section":"Abstract"},{"comment":"The formula for the norming constants in the statement of Proposition 4.2 appears to omit the denominator: the proof and the jump matrices in Eqs. (4.48)-(4.52) use c_j = s_3^2 r_1(ζ_j) / ∏_{k≠j}(ζ_j−ζ_k), while the proposition statement as printed shows the product without a denominator. Please correct the statement to match the proof.","section":"Proposition 4.2"},{"comment":"The equation labels (5.86) and (5.88) are duplicated in the display for m4 and m5; renumber to avoid confusion.","section":"Eqs. (5.86)-(5.88)"},{"comment":"The definition r3±(z) := ±r2(z) on Σ1∪Σ3∪Σ5∪Σ7 is not fully explained; the sign convention should be tied to the side of the branch cut and to the direction of the contour so that the factorizations in Eqs. (5.114)-(5.115) are unambiguous.","section":"Section 5.1, around Eq. (5.113)"},{"comment":"The theorem states an asymptotic formula for u(x) as y→−∞, but x and y are related by Eq. (5.143). The statement would be clearer if it specified the uniformity class of the error O(1/|y|) with respect to the parameter dependence and the relation between x and y.","section":"Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a clear and honest structure: the quadrature-domain results are self-contained and appear correct, while the line-domain theorem hinges on an openly unproved sign inequality. I would encourage the editors to treat this as a conditional result: the paper could be acceptable if Lemma 5.4 is either proved or explicitly reformulated as a conjecture with the line-domain theorem stated conditionally. The elliptic-domain equivalence also needs a real argument, not just a statement. The paper is within scope for the journal, but the current form is not ready for acceptance as a rigorous asymptotic-analysis paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper does two very different things. The quadrature-domain part (Section 4) is a careful adaptation of Bertola-Grava-Orsatti's soliton-shielding argument to the mCH equation with linear dispersion, and it works. The Riemann-sum to area-integral lemmas are standard but checked, the Green-theorem reductions hold, and Propositions 4.1 and 4.2 deliver exactly what they claim: a dense soliton ensemble collapsing to a one-soliton or n-soliton state, with the discrete spectrum sitting at the center of the quadrature domain. That is a genuine new result for this equation and worth having.\n\nThe line-domain part (Section 5) is much weaker. The modified RH problem is constructed fine, but Theorem 5.1 depends on Lemma 5.4, a sign inequality on the lens contours. The authors state plainly that it 'can not be rigorously shown' and offer only numerics. That is not a stylistic gap. The sign of Re(2g(z) - (z+1/z)/2) is a quantitative property of an explicit g-function and has to be established for all 1<a<b. Without it, the lens-opening step does not give exponentially small jumps, the small-norm RH problem for E has no justification, and the O(1/|y|) formula does not follow. The rest of the line-domain machinery, including the outer parametrix imported from Kamvissis-McLaughlin-Miller, is standard conditional on that inequality, but as written the theorem is one numerical check away from a proof.\n\nThe elliptic-domain section is also sketchy. Lemma 6.1 reduces the elliptic boundary integrals to line integrals and then declares equivalence to the line case. That may be correct, but it is not demonstrated at the level of Section 4, and it inherits the line-domain gap.\n\nOn citations: the dependence on [2] is direct and acknowledged, and that is legitimate here; the framework is being adapted, not repackaged. There is no circularity or fitting-to-data anywhere.\n\nBottom line: the quadrature-domain aggregation results deserve a serious referee and can likely be published as a solid contribution. The line-domain theorem should be either proved or explicitly labeled as conditional. A serious editor should send this to peer review, but the referee should demand a proof of Lemma 5.4 or a revised, weakened claim. For people working on soliton gases and mCH, this is a useful paper despite the gap.","headline":"Quadrature-domain soliton shielding for mCH is real and correct, but the line-domain theorem hinges on an inequality the authors admit they cannot prove.","tokens_in":43974,"tokens_out":2419,"would_cite":true,"duration_ms":23968,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P25","35Q15","37K40","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"An infinite-soliton ensemble of the modified Camassa–Holm equation collapses to an exact one-soliton or n-soliton state when its discrete spectra fill a quadrature domain.","keywords":["modified Camassa-Holm equation","linear dispersion","N-infinity soliton asymptotics","Riemann-Hilbert problem","quadrature domain","soliton shielding","Riemann theta function","steepest descent method"],"falsifier":"Fix a concrete density $r(z)$ (for instance a constant) and values $a,b$ with $1<a<b$, compute $g(z)$ from equation (5.84), and evaluate $\\operatorname{Re}(2g(z)-\\frac{1}{2}(z+1/z))$ at sample points on the four lens boundaries; Lemma 5.4 predicts a strict sign on each lens, so a single point with the opposite sign would refute the lens-opening step and Theorem 5.1. A complementary check is to simulate the $N$-soliton solution of (1.1) for large $N$ with discrete spectra sampled from the density on $(ia,ib)$ and compare it, for large negative $y$, with the formula (5.142)–(5.143).","tokens_in":42778,"feed_emoji":"🌊","tokens_out":12154,"duration_ms":93312,"temperature":0.7,"pith_summary":"The authors set out to establish what happens to the N-soliton solutions of the modified Camassa–Holm equation with linear dispersion and vanishing boundary values, $m_t+(m(u^2-u_x^2)^2)_x+\\kappa u_x=0$ with $m=u-u_{xx}$, when $N$ tends to infinity and the discrete spectral points uniformly fill a prescribed region of the complex plane. Their central claim is that the limiting $N_\\infty$-soliton is controlled entirely by the geometry of that region. When the region is a quadrature domain of the form $\\{z: |(z-s_1)^\\ell-s_2|<s_3\\}$, the dense limit is exactly a finite-soliton solution: with $\\ell=n=1$ it is the one-soliton solution whose discrete spectral point is the center $\\zeta_0=s_1+s_2$, and with $\\ell=n$ it is the $n$-soliton solution with explicit spectra and norming constants. When the discrete spectra lie along a line segment, the paper derives a solvable Riemann–Hilbert problem and, as $y\\to-\\infty$, an explicit leading-order asymptotic formula with an $O(1/|y|)$ error; an elliptic region is shown to reduce to the line case. A sympathetic reader cares because this predicts soliton shielding for the mCH equation: a dense soliton gas can organize into a single smooth soliton or a finite soliton train without leftover radiation.","feed_headline":"Dense Camassa-Holm soliton gas collapses to finite soliton states","feed_subtitle":"Quadrature spectral domains give exact one- or n-soliton limits; line and elliptic regions give theta-function asymptotics.","key_machinery":"The carrying object is the modified Riemann–Hilbert problem (RH Problem 2): each discrete pole of the standard mCH RH problem is encircled by a small closed contour, and the pole residues are converted into jump matrices on those contours. The large-$N$ limit is taken by choosing norming constants proportional to the local spectral density, $c_j=|\\Omega|r(z_j,z_j^*)/(N\\pi)$, so that the residue sums become quadrature sums over the spectral region. Lemma 4.1 turns those sums into area integrals over $\\Omega$; for quadrature domains, Lemma 4.2 uses Green's theorem to write each area integral as a boundary integral over $\\partial\\Omega$, and the boundary integrals are then evaluated by the residue theorem, producing exactly the jump matrices of a one-soliton or $n$-soliton RH problem. For the line domain, the modified RH problem is deformed into a model problem on the four intervals $(a,b)$, $(1/b,1/a)$, $(-b,-a)$, $(-1/a,-1/b)$; two new scalar functions are constructed — $g(z)$, solving a scalar RH problem with undetermined constants $m_1,m_2,m_3$, and $f(z)$, solving a multiplicative scalar RH problem with constants $n_1,n_2$ — and these reduce the model problem to standard form. Lens opening, Bessel local parametrices at the endpoints, and a small-norm RH problem then give the leading asymptotic behavior.","core_discovery":"The central claim, on the paper's own terms, is a collapse mechanism in the modified Riemann–Hilbert representation of $N$-soliton solutions of the mCH equation with linear dispersion and zero boundaries. Replacing the discrete pole residues by jump matrices on small contours around the poles, and taking the $N\\to\\infty$ limit with norming constants scaling as $c_j=|\\Omega|r(z_j,z_j^*)/(N\\pi)$, the discrete data become integrals: Riemann sums over the spectra converge to area integrals (Lemma 4.1). For quadrature domains these area integrals collapse under Green's theorem to boundary integrals (Lemma 4.2), and for $\\ell=n=1$ the boundary integrals evaluate to the jump matrices of the one-soliton solution with discrete spectrum $\\zeta_0=s_1+s_2$ and norming constant $c_1=s_3^2r_1(\\zeta_0)$ (Proposition 4.1); for $\\ell=n$ they evaluate to the jump matrices of the $n$-soliton solution with spectra $\\zeta_j$ solving $(z-s_1)^n=s_2$ and norming constants $s_3^2r_1(\\zeta_j)/\\prod_{k\\neq j}(\\zeta_j-\\zeta_k)$ (Proposition 4.2). For spectra on a line, the same construction leads to a model Riemann–Hilbert problem on four intervals; the paper introduces scalar functions $g(z)$ and $f(z)$ to reduce it to standard form, solves the outer parametrix with a Riemann $\\theta$ function, constructs Bessel-type local parametrices at the endpoints, and obtains the $y\\to-\\infty$ formula in which $u(x)$ is expressed through the entries of the outer parametrix at $z=1$ plus $O(1/|y|)$, with $x(y)=y-\\ln(f^2(1)e^{2yg(1)})+O(1/|y|)$. The elliptic-domain case is reduced to the line case via Lemma 6.1, which converts boundary integrals over the ellipse into line integrals over the focal segment.","pith_inferences":["Inference beyond the paper: because the quadrature-domain collapse uses only the structure of the spectral measure and the residue-to-integral conversion, the same mechanism should apply to other members of the mCH/FORQ family and to the short-pulse limit, provided the Lax-pair symmetries are preserved.","Inference beyond the paper: the line-region construction suggests that the density function $\\rho(z)$ controls the macroscopic profile through the scalar functions $g$ and $f$; a numerical test of the predicted $y\\to-\\infty$ profile against direct $N$-soliton dynamics for finite $N$ would sharpen the practical range of validity of the $O(1/|y|)$ formula.","Inference beyond the paper: the paper fixes $t=0$ for the line-region asymptotics; extending Theorem 5.1 to joint $(y,t)$ sectors would connect the $N_\\infty$-soliton asymptotics with the known long-time asymptotics of the mCH equation.","Inference beyond the paper: since Lemma 5.4 is only numerical, a rigorous proof of the $g$-function sign — or a counterexample — is the natural next step and would settle the line-region result independently of the quadrature-domain claims."],"forward_implications":["For a quadrature domain with $\\ell=n=1$, the $N_\\infty$-soliton is exactly the one-soliton solution with discrete spectrum $\\zeta_0=s_1+s_2$ and norming constant $s_3^2r_1(\\zeta_0)$, so the dense ensemble leaves no radiation.","For a quadrature domain with $\\ell=n$, the $N_\\infty$-soliton is exactly the $n$-soliton solution with discrete spectra $\\zeta_j$ solving $(z-s_1)^n=s_2$ and norming constants $s_3^2r_1(\\zeta_j)/\\prod_{k\\neq j}(\\zeta_j-\\zeta_k)$, a finite interacting soliton train.","For discrete spectra on a line segment, as $y\\to-\\infty$ the solution is given by the outer parametrix built from the Riemann theta function, with an $O(1/|y|)$ correction made explicit by the small-norm RH analysis.","For discrete spectra in an elliptic region, the limiting RH problem reduces to the line case, so the same theta-function asymptotic formula applies.","Taken together, the results establish soliton shielding for the mCH equation with linear dispersion: depending on the spectral region, a dense $N$-soliton gas collapses to a finite soliton state or to a theta-function modulated profile."],"supporting_citations":[{"why":"supplies the N∞-soliton shielding phenomenon and the Riemann-sum-to-integral strategy that the paper adapts to the mCH equation.","marker":"[2]"},{"why":"provides the starting Riemann–Hilbert problem and Lax-pair treatment for the mCH equation with linear dispersion.","marker":"[12]"},{"why":"complements [12] with the large-time RH analysis of the mCH equation that the paper's framework builds on.","marker":"[13]"},{"why":"supplies the nonlinear steepest descent method used for the line-domain asymptotic analysis.","marker":"[23]"},{"why":"provides the outer model problem and its Riemann theta function solution used to write the leading-order parametrix.","marker":"[42]"},{"why":"provides the Bessel model used to construct the local parametrices at the endpoints.","marker":"[45]"},{"why":"gives the mCH Riemann–Hilbert problem and long-time asymptotics for Schwartz initial data under zero boundary.","marker":"[66]"},{"why":"gives the weighted-Sobolev long-time asymptotics for the mCH equation that the new N∞-soliton analysis extends.","marker":"[68]"}],"fun_headline_variants":["Soliton gas collapses to one or n solitons","Quadrature domains turn N-infinity solitons into n solitons","mCH equation: dense soliton spectra condense to finite states","Riemann-Hilbert method reveals soliton collapse in mCH"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The line-region asymptotic formula rests on the sign condition in Lemma 5.4 — the real part of $2g(z)-\\frac{1}{2}(z+1/z)$ is negative on the lenses around $(a,b)$ and positive on the lenses around the negative intervals — which the authors state cannot be rigorously proved and is checked only numerically; if that sign failed anywhere on the lens contours, the lens-opening step would no longer produce exponentially small errors and the $O(1/|y|)$ result would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Soliton gas collapses to one or n solitons","Quadrature domains turn N-infinity solitons into n solitons","mCH equation: dense soliton spectra condense to finite states","Riemann-Hilbert method reveals soliton collapse in mCH"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000794,"raw_usage":{"total_tokens":3648,"prompt_tokens":1249,"completion_tokens":2399,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":865,"completion_tokens_details":{"reasoning_tokens":2322}},"tokens_in":865,"tokens_out":2399,"duration_ms":52572,"temperature":1.0,"reasoning_tokens":2322,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:54:18.720586+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a concrete density $r(z)$ (for instance a constant) and values $a,b$ with $1<a<b$, compute $g(z)$ from equation (5.84), and evaluate $\\operatorname{Re}(2g(z)-\\frac{1}{2}(z+1/z))$ at sample points on the four lens boundaries; Lemma 5.4 predicts a strict sign on each lens, so a single point with the opposite sign would refute the lens-opening step and Theorem 5.1. A complementary check is to simulate the $N$-soliton solution of (1.1) for large $N$ with discrete spectra sampled from the density on $(ia,ib)$ and compare it, for large negative $y$, with the formula (5.142)–(5.143).","supporting_citations":[{"cited_title":"Bertola, T","cited_arxiv_id":null,"evidence_quote":"supplies the N∞-soliton shielding phenomenon and the Riemann-sum-to-integral strategy that the paper adapts to the mCH equation."},{"cited_title":"Boutet de Monvel, I","cited_arxiv_id":null,"evidence_quote":"provides the starting Riemann–Hilbert problem and Lax-pair treatment for the mCH equation with linear dispersion."},{"cited_title":"The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem","cited_arxiv_id":"2011.13235","evidence_quote":"complements [12] with the large-time RH analysis of the mCH equation that the paper's framework builds on."},{"cited_title":"Deift, X","cited_arxiv_id":null,"evidence_quote":"supplies the nonlinear steepest descent method used for the line-domain asymptotic analysis."},{"cited_title":"Kamvissis, K","cited_arxiv_id":null,"evidence_quote":"provides the outer model problem and its Riemann theta function solution used to write the leading-order parametrix."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the Bessel model used to construct the local parametrices at the endpoints."},{"cited_title":"Long-time asyptotics behavior for the integrable modified Camassa-Holm equation with cubic nonlinearity","cited_arxiv_id":"1911.12554","evidence_quote":"gives the mCH Riemann–Hilbert problem and long-time asymptotics for Schwartz initial data under zero boundary."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the weighted-Sobolev long-time asymptotics for the mCH equation that the new N∞-soliton analysis extends."}],"review_version":1}