{"id":"168fe395-5b21-49b8-a713-b6cc055e6fdb","arxiv_id":"1908.06415","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For shifted-step initial data, the long-time solution of the nonlocal focusing NLS splits into 4n+2 sectors: alternating sectors decay to 0 or approach explicit constants.","lead":"This paper derives the long-time behavior of solutions of the nonlocal focusing nonlinear Schrödinger equation starting from a shifted step: the x/t plane splits into many sectors that alternately decay to zero or approach constants. The number of sectors grows with the step's position, a non-translation-invariance effect for an integrable system.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Closeness-to-shifted-step claim is unsupported: Assumptions A(b) (winding/zero-count) are proved only for the exact shifted step, with no perturbation stability argument.","rationale":"The reader's weakest assumption identifies exactly the gap I consider load-bearing: Assumptions A(b) are verified only for the exact shifted step, and no stability argument is provided for perturbations, which the abstract claims to cover. My analysis of the proof confirms that the ordering and winding conditions are not decorative; they are used to select which residue conditions survive the steepest-descent reduction, and Remark 8 explicitly shows the mechanism fails if the ordering is violated. Because this is a missing proof rather than a demonstrated falsehood, the appropriate verdict remains CONDITIONAL (the reader's verdict), so no adjustment is needed. The concrete numerical/analytic test I propose would settle whether the concern actually lands: if the spectral data are stable under small perturbations, the gap is easily filled; if not, the theorem's scope must be narrowed.","tokens_in":24770,"tokens_out":7118,"duration_ms":79549,"concrete_test":"For the one-parameter family q0^ε(x) = χ_R(x) + ε φ(x), with φ a fixed compactly supported C^∞ function and ε ∈ [−ε0, ε0], solve the direct scattering problem (2.3) numerically to spectral accuracy and compute: (i) the zeros of a1 in C+ (e.g., by contour integration or Newton iteration on a1(k)=0), and (ii) the argument integral I(ε, y) = ∫_{−∞}^{y} d arg(a1 a2)(k) for y near the unperturbed value −ω_{n−m+1}. If for every small ε the zero count is 2n+1 with the ordering (2.32) and there exists y(ε) with I(ε, y(ε)) = (2m−1)π, then Assumptions A are stable for this perturbation; if some ε yields a different zero count or no such y, the abstract's 'close to' claim fails. Repeat for several generic φ (e.g., Gaussian bump, one-sided bump) to test robustness.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's advertised result (abstract, Section 1) is for step-like initial data 'close to' the shifted step χ_R, but Theorem 1 is conditional on Assumptions A. The critical part is A(b): the ordering (2.32) and the winding conditions (2.33), which fix Im ν(−ξ) ∈ (−1/2, 1/2) and determine which residues grow. Proposition 2 verifies (2.32)–(2.33) only for the exact shifted-step data (2.7), and only for R in the open interval; at the endpoint R = (2n+1)π/(2A) extra real zeros appear (Proposition 2(iii)). No theorem or lemma shows that a small perturbation of χ_R preserves: (a) the number 2n+1 of zeros of a1 in C+ and their simplicity, (b) the ordering Re p_n < −ω_n < ... < Re p_1 < −ω_1 < 0, and (c) the exact winding values (2.33). The asymptotic reduction in Section 3 depends on these in an essential way: Proposition 3 removes growing residues by multiplying by factors (k+ω_{n−s})/(k−p_{n−s}), and Remark 8 shows that if the ordering fails, two growing residue conditions land on the same column and the method breaks down. Since the winding integral for a contour ending at a spectral-data-dependent point is not automatically locally constant, stability is not a free consequence of continuity. Thus the 'family of step-like initial data' in the title/abstract is not actually covered by the proof; only the exact shifted step and data already satisfying Assumptions A are covered.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper considers the long-time asymptotics of solutions of the integrable nonlocal focusing nonlinear Schrödinger equation iq_t + q_xx + 2 q^2 \\bar q(-x,t)=0 with step-like initial data. The advertised result is that when the initial profile is close to the shifted step A H(x-R), with R in ((2n-1)π/(2A),(2n+1)π/(2A)), the (x,t)-plane is divided into 4n+2 sectors, with alternating decay q=o(1) and nonzero constant limits; the constants and the subleading power-log corrections are given explicitly in terms of the scattering data. The main theorem (Theorem 1) is conditional on Assumptions A on the spectral functions, which include an ordering condition (2.32) and winding conditions (2.33). Proposition 2 verifies Assumptions A for the exact shifted step. The proof is based on the inverse scattering transform, a formulation of the problem as a matrix Riemann-Hilbert problem with residues, and a nonlinear steepest descent analysis with a parabolic cylinder local parametrix taken from the authors' earlier paper [42].","tokens_in":25079,"tokens_out":8893,"duration_ms":90407,"significance":"If the asymptotic formulae are correct, the paper gives a concrete illustration of the lack of translation invariance of the NNLS equation: the sector structure itself changes with the location R of the step. Its strengths are the explicit formulas for the leading constants and the precise power-law rates, which make the predictions readily checkable, and the fact that the abstract sector count is tied to winding properties of the scattering data rather than to adjustable parameters. The paper is nevertheless a reduction to a known steepest-descent scheme rather than a fully self-contained proof, and the advertised 'family of step-like initial data close to the shifted step' is not shown to satisfy the assumptions under which the theorem is proved.","major_comments":[{"comment":"Theorem 1 is stated under Assumptions A, but Assumptions A(b) (the ordering (2.32) and the winding conditions (2.33)) are verified in Proposition 2 only for the exact shifted-step initial data (2.7), and only in the open interval ((2n-1)π/(2A),(2n+1)π/(2A)). The abstract and introduction advertise data 'close to the shifted step', but no perturbation lemma is given showing that small perturbations preserve (a) the number and simplicity of the zeros of a1 in C+, (b) the ordering Re p_n < -ω_n < ... < Re p_1 < -ω_1 < 0 in (2.32), and (c) the exact values of the winding integrals (2.33). This stability is not automatic: the winding integrals end at spectral-data-dependent points, so a small change of data can in principle change the value of the integral. Since Remark 8 and Proposition 3 show that the sector decomposition relies essentially on this ordering and on the residues growing in specific columns, the advertised class of initial data is not covered by the proof. The authors should either prove such a stability result with a quantitative notion of closeness, or restate the main theorem and abstract for initial data satisfying Assumptions A.","section":"Assumptions A / Proposition 2"},{"comment":"Proposition 3 is central because it converts the full Riemann-Hilbert problem to the model problems (3.25) and (3.27), yet its proof 'ignores' exponentially decaying residue conditions and parts of the jump matrix without giving an explicit error bound for (3.23)-(3.24). The sketch of Theorem 1 likewise imports the parabolic cylinder parametrix and the remainder estimates from [42]; however, the present problem contains m singular points -ω_{n-s}, 2n+1 growing/decaying residue conditions, and an additional residue at k=0, and the constants α_j contain factors (ξ+p_{n-s})^{-2} or (p_{n-s}-ξ)^{-2}. It is therefore not immediate that the estimates of [42] remain uniform as ξ approaches the sector boundaries Re p_{n-m} and ω_{n-m}, where those factors degenerate. A rigorous derivation of the remainders R_j(ξ,t) in (3.42)-(3.43), or a precise statement of the conditions under which the estimates of [42] apply verbatim, is needed.","section":"§3.2, Proposition 3 and Sketch of proof of Theorem 1"},{"comment":"The deformation of the Riemann-Hilbert problem from the real axis to the cross Γ assumes that the reflection coefficients r1 and r2 can be analytically continued into the whole complex plane, but this hypothesis does not appear in the statement of Theorem 1. The proof therefore does not cover all data satisfying Assumptions A unless the rational approximation mentioned in the text is carried out with uniform error estimates. The theorem should either include analytic continuation as an explicit hypothesis, or the proof should supply the approximation argument in detail.","section":"§3.2, paragraph before Eq. (3.16)"}],"minor_comments":[{"comment":"The sector inequalities mix ξ and -ξ without indicating which variable is used; for ξ>0, intervals such as -ω_{n-m+1}<ξ<Re p_{n-m} are empty, while the intended interval in the first line is -Re p_{n-m}<ξ<ω_{n-m+1}. Please rewrite the sectors in terms of a single variable, or state explicitly the symmetry that covers ξ<0.","section":"Corollary 1, eq. (3.37)"},{"comment":"'Steepest decent' should read 'steepest descent' in the two places where it appears.","section":"Section 1"},{"comment":"The statement of Theorem 1 from the authors' earlier paper [43] is cited but not summarized; since the present paper compares its results with that theorem, a brief statement of the comparison would help the reader.","section":"References"},{"comment":"The endpoint case R=(2n+1)π/(2A) is excluded from Theorem 1, but the statement of Proposition 2(iii) does not explain how the extra real zeros ±A/2 interact with the ordering (2.32); a sentence making clear that the open interval is essential would prevent confusion.","section":"Proposition 2(iii)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript leans very heavily on the authors' own previous paper [42] for the parabolic cylinder parametrix and the remainder estimates, and the proof of Theorem 1 is only a sketch. If the journal expects a largely self-contained proof for a main theorem, the editor may wish to require a fuller derivation. There is also a mismatch between the abstract's promise of a 'family' of initial data and the conditional nature of the theorem; this is correctable in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things before reading this one. First, the core result is real and worth taking seriously: for the NNLS equation with the exact shifted step q0(x)=A H(x-R), the paper works out the long-time behavior in full, and the sector-counting structure (4n+2 sectors, alternating decay and constants, explicit constants in terms of spectral data) is new for R in the intervals ((2n-1)pi/(2A), (2n+1)pi/(2A)). Second, the advertised 'family of step-like initial data close to the shifted step' is not actually covered by the proof. Assumptions A, especially the winding conditions (2.33), are verified in Proposition 2 only for the exact shifted step. No stability argument shows that a small perturbation preserves the zero count, the ordering Re p_n < -omega_n < ... < 0, or the exact winding values. Since the winding integral is not automatically locally constant, this is not a free continuity consequence. The abstract and introduction overstate the scope. What is genuinely good: the spectral functions for the shifted step are computed explicitly; the zero structure and winding are proved in detail; the partial delta functions (3.3)-(3.5) and the transformation (3.29) that removes growing residues are new and clever; the asymptotic constants are completely explicit, with no fitted parameters. The result for the exact shifted step appears correct and is a solid advance over the authors' earlier n=0 paper. The soft spots are real but proportionate. Theorem 1 is stated with only a 'Sketch of proof' that imports the parabolic cylinder parametrix and remainder control from the companion paper [42]. That is acceptable in this subfield, but it means the paper is not self-contained. More importantly, Proposition 3's reduction ignores decaying residues and jumps without explicit error bounds; a referee should ask for those estimates or a clear statement that standard arguments supply them. The endpoint R=(2n+1)pi/(2A) is excluded, which is fine since extra real zeros appear. Bottom line: this deserves a serious referee. The mathematical core for the exact shifted step is substantial and likely correct. The 'close to' claim needs to be either proved or removed. I would send it to review with a request for major revision: add a perturbation-stability lemma for Assumptions A, or honestly restrict the title/abstract/theorem to the exact shifted step and data already satisfying Assumptions A. If I were working on nonlocal NLS asymptotics, I would cite the exact-step result.","headline":"Solid result for the exact shifted step, with an overreaching 'close to' claim that the proof does not actually cover — referee it, but require a scope fix or a perturbation argument.","tokens_in":740,"tokens_out":977,"would_cite":true,"duration_ms":24594,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","37K15","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the long-time profile of the nonlocal focusing NLS equation with shifted step-like data splits into 4n+2 alternating sectors of decay and nonzero constants, with constants set by the spectral data.","keywords":["nonlocal NLS","long-time asymptotics","Riemann-Hilbert problem","nonlinear steepest descent","step-like initial data","discrete spectrum","parabolic cylinder functions","PT symmetry"],"falsifier":"Compute the scattering coefficient $a_1(k)$ for $q_0(x)=A H(x-R)+\\varepsilon e^{-x^2}$ with a small $\\varepsilon$; if the zero count in $\\mathbb{C}^+$ or the winding integrals (2.33) change, the predicted $4n+2$ sector structure fails. A complementary check is to simulate the NNLS equation numerically from the exact shifted step and compare the plateau constants on each ray with the formula (3.37).","tokens_in":24558,"feed_emoji":"🌊","tokens_out":8935,"duration_ms":82641,"temperature":0.7,"pith_summary":"The paper studies the long-time behavior of solutions of the integrable nonlocal focusing nonlinear Schrödinger equation $iq_t+q_{xx}+2q^2\\bar q(-x,t)=0$ when the initial datum tends to $0$ at $-\\infty$ and to $A>0$ at $+\\infty$ and is, in a precise sense, close to a step of height $A$ located at $x=R$. Its central claim is that for $R$ in the interval $((2n-1)\\pi/(2A), (2n+1)\\pi/(2A))$, the half-plane splits, as $t\\to\\infty$, into $4n+2$ ray sectors: in $2n+1$ alternating sectors the solution decays to $0$, and in the other $2n+1$ sectors it approaches explicit nonzero constants that are computed from the spectral functions of the initial profile. This matters because the nonlocal equation is not translation invariant, so the location $R$ of the step genuinely changes the asymptotic picture; the previously known zero-step analysis only covered $R<\\pi/(2A)$. The method is the matrix Riemann–Hilbert formulation of inverse scattering together with the nonlinear steepest descent analysis, and the paper derives both the leading constants and the algebraic-in-time corrections to them.","feed_headline":"Nonlocal NLS: decay sectors alternate with plateaus","feed_subtitle":"As t grows, a shifted step yields 2n+1 decaying wedges and 2n+1 constant wedges.","key_machinery":"The central object is the solution $M(x,t,k)$ of a $2\\times2$ matrix Riemann–Hilbert problem on the real $k$-axis, whose $12$-entry at large $k$ gives $q(x,t)$. Two mechanisms carry the argument. First, a scalar multiplier $\\delta(k,\\xi)$ is assembled from partial Cauchy integrals over the intervals between consecutive winding points $-\\omega_{n-s}$, chosen so that its jump absorbs $1+r_1(k)r_2(k)$ on $(-\\infty,-\\xi)$ and, crucially, so that $\\operatorname{Im}\\nu(-\\xi)\\in(-1/2,1/2)$; this makes the deformed jump matrices converge to the identity as $t\\to\\infty$. Second, rational factors $\\prod_{s=0}^{m-1}((k+\\omega_{n-s})/(k-p_{n-s}))^{\\sigma_3}$ are inserted to remove the singularities at $k=-\\omega_{n-s}$ and to turn the exponentially growing residue conditions at the zeros $p_{n-s}$ into decaying ones. At the stationary phase point $k=-\\xi$, a local model problem is solved explicitly with parabolic cylinder functions, which supplies the constants $\\alpha_j(\\xi)$ and the power corrections.","core_discovery":"On the paper's own terms, the discovery is Corollary 1 / eq. (3.37): for each $m=0,\\dots,n$, the sectors $-\\operatorname{Re} p_{n-m}<\\xi<\\omega_{n-m+1}$ have $q(x,t)=A\\delta^2(0,\\xi)\\prod_{s=0}^{m-1}(\\omega_{n-s}/p_{n-s})^2+o(1)$, the sectors $-\\omega_{n-m+1}<\\xi<-\\operatorname{Re} p_{n-m}$ and $\\omega_{n-m}<\\xi<-\\operatorname{Re}p_{n-m}$ have $q(x,t)=o(1)$, and the sectors $\\operatorname{Re}p_{n-m}<\\xi<-\\omega_{n-m}$ have $q(x,t)=-4p_{n-m}^2/(A\\delta^2(0,-\\xi))\\prod_{s=0}^{m-1}(p_{n-s}/\\omega_{n-s})^2+o(1)$. The refined Theorem 1 replaces each $o(1)$ by the leading algebraic terms of order $t^{-1/2\\pm\\operatorname{Im}\\nu}$ with constants expressed through parabolic cylinder functions. Here $p_j$ are the zeros of the spectral function $a_1(k)$ in the upper half-plane, $\\omega_j$ are the points where the accumulated argument of $a_1 a_2$ crosses odd multiples of $\\pi$, and $\\delta$ is the scalar factor created by deforming the Riemann–Hilbert problem.","pith_inferences":["A natural extension, not pursued in the paper, is to nonlocal mKdV or other PT-symmetric integrable equations: the same residue-absorption mechanism should produce analogous alternating plateau sectors once the step is shifted.","Because the paper verifies Assumptions A only for the exact shifted step, the statement 'close to' carries an implicit stability conjecture; proving a Rouché-type perturbation theorem would either confirm the sector count for small perturbations or reveal that the plateau structure is nongeneric.","The dependence of the plateaus on $\\xi$ suggests a clean experimental signature: in a PT-symmetric optical setting, a scan of $|q|$ along a fixed line at large $t$ should show step-like jumps at the sector boundaries, with heights predicted by (3.37)."],"forward_implications":["For each fixed $A$ and $R$, the asymptotic landscape is completely determined by the spectral data of the initial profile: no further PDE evolution is needed to predict the leading behavior along any ray.","Increasing $R$ across $(2n+1)\\pi/(2A)$ adds two new sectors, so the number of plateaus grows one step at a time as the step is moved farther from the origin.","In the constant sectors the limit depends on $\\xi=x/(4t)$, so two observers moving at different speeds in the same sector see different limiting amplitudes, not a single global constant.","The algebraic corrections $t^{-1/2\\pm\\operatorname{Im}\\nu}$ are slow enough that the constants are approached like weak powers of time, which makes the plateau visible but extremely slowly settling when $\\operatorname{Im}\\nu\\approx 0$."],"supporting_citations":[{"why":"Provides the Riemann–Hilbert formalism for the NNLS step problem and the asymptotics for the unshifted step, which this paper extends to shifted steps.","marker":"[43]"},{"why":"The companion asymptotic analysis of the same equation, whose parabolic-cylinder model and deformation steps are reused here.","marker":"[42]"},{"why":"Supplies the nonlinear steepest descent method used to deform the oscillatory Riemann–Hilbert problem.","marker":"[20]"},{"why":"Introduced the nonlocal NLS equation and its Lax pair, the starting model.","marker":"[5]"},{"why":"Gives the inverse scattering transform for the nonlocal NLS equation on the line, the scattering framework used here.","marker":"[6]"},{"why":"Long-time asymptotics of integrable wave equations via Riemann–Hilbert problems, the source of the partial-delta and contour-deformation ideas.","marker":"[16]"}],"fun_headline_variants":["Nonlocal NLS: step-like data create alternating decay and plateaus","Decay and constant sectors alternate in nonlocal NLS asymptotics","Nonlocal NLS: 4n+2 sectors, alternating decay and constants","Step-like nonlocal NLS: decay wedges alternate with plateau wedges","Nonlocal NLS long-time: alternating decay and constant regions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the assumption that small perturbations of the exact shifted step keep the same number of discrete-spectrum points and the same winding of the spectral argument, a property that is proved only for the exact step itself.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal NLS: step-like data create alternating decay and plateaus","Decay and constant sectors alternate in nonlocal NLS asymptotics","Nonlocal NLS: 4n+2 sectors, alternating decay and constants","Step-like nonlocal NLS: decay wedges alternate with plateau wedges","Nonlocal NLS long-time: alternating decay and constant regions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000698,"raw_usage":{"total_tokens":3257,"prompt_tokens":1154,"completion_tokens":2103,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":770,"completion_tokens_details":{"reasoning_tokens":2007}},"tokens_in":770,"tokens_out":2103,"duration_ms":15134,"temperature":1.0,"reasoning_tokens":2007,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:46:41.607932+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the scattering coefficient $a_1(k)$ for $q_0(x)=A H(x-R)+\\varepsilon e^{-x^2}$ with a small $\\varepsilon$; if the zero count in $\\mathbb{C}^+$ or the winding integrals (2.33) change, the predicted $4n+2$ sector structure fails. A complementary check is to simulate the NNLS equation numerically from the exact shifted step and compare the plateau constants on each ray with the formula (3.37).","supporting_citations":[{"cited_title":"Long-time asymptotics for the integrable nonlocal nonlinear Schr\\\"odinger equation with step-like initial data","cited_arxiv_id":"1906.08489","evidence_quote":"Provides the Riemann–Hilbert formalism for the NNLS step problem and the asymptotics for the unshifted step, which this paper extends to shifted steps."},{"cited_title":"Rybalko, D","cited_arxiv_id":null,"evidence_quote":"The companion asymptotic analysis of the same equation, whose parabolic-cylinder model and deformation steps are reused here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the nonlinear steepest descent method used to deform the oscillatory Riemann–Hilbert problem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the nonlocal NLS equation and its Lax pair, the starting model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the inverse scattering transform for the nonlocal NLS equation on the line, the scattering framework used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Long-time asymptotics of integrable wave equations via Riemann–Hilbert problems, the source of the partial-delta and contour-deformation ideas."}],"review_version":1}