{"id":"5de739f2-f53b-4141-822d-148a2fa760ee","arxiv_id":"2508.07231","paper_version":4,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Stable and unique recovery of stationary coefficients in a nonlinear Schrödinger equation is established from boundary measurements, under analytic nonlinearity and small-data conditions.","lead":"This paper proves that unknown coefficients in a class of nonlinear Schrödinger equations can be stably recovered from measurements on the boundary. The result uses high-order linearization and Carleman estimates, giving a rigorous basis for inverse problems in quantum and wave imaging.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; proof details unavailable, so correctness remains unverifiable from the abstract.","rationale":"The reader's verdict is UNVERDICTED due to the absence of the full text, which is the appropriate disposition for an abstract-only review. My stress-test does not uncover a concrete technical flaw in the central claim as stated. The method described (high-order linearization plus Carleman estimates) is a standard approach for nonlinear inverse problems, and the explicit assumption of local analyticity is a reasonable and commonly used condition for making high-order linearization viable. The reader's identified weakest assumption—that analyticity is needed to obtain infinitely many linearized problems—is indeed a necessary condition for the method, but it is not a hidden assumption; it is stated in the abstract. Therefore, I agree with the reader's assessment that the key uncertainty lies in the correctness of the proof, which cannot be evaluated without the full manuscript. No adjustment to the verdict is needed.","tokens_in":709,"tokens_out":5556,"duration_ms":55568,"concrete_test":"Obtain the full text and independently verify the key Carleman estimate by checking the weight function, the boundary subset condition, and the dependence of the constant on the linearization order. Then verify that the high-order linearization yields a Fredholm map with the claimed stability modulus as the initial-data smallness parameter tends to zero. If either step fails, the central stability claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"This stress-test pass finds no specific technical flaw in the abstract's central claim. The argument relies on high-order linearization, which requires the nonlinearity to be locally analytic (as stated), and Carleman estimates for the linear Schrödinger equation, which are standard in this context. The reader's concern about analyticity is valid but is an explicit hypothesis, not a hidden assumption. The main limitation is that the abstract provides no derivations; the correctness of the Carleman estimate, the well-posedness result, and the stability estimate cannot be assessed. Without the full text, the paper remains unverdictable, but no concrete objection to the argument can be identified from the abstract alone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper (arXiv:2508.07231, abstract only) claims stable and unique determination of stationary coefficients in a class of nonlinear dynamical Schrödinger equations with locally analytic nonlinear terms, given knowledge of the coefficients near the boundary and measured Neumann data on a boundary subset. Two measurement regimes are discussed: a subset satisfying a geometric condition, and an arbitrary subset under stronger regularity and size assumptions on the coefficients. The proposed method combines high-order linearization with Carleman estimates for the linear Schrödinger equation, together with small-data well-posedness. No proofs, theorem statements, or derivations are available in the reviewed material; the full text was not provided.","tokens_in":800,"tokens_out":1765,"duration_ms":20324,"significance":"If the claimed result is correct, it would be a meaningful advance in inverse problems for nonlinear Schrödinger equations, going beyond linear uniqueness/stability by using analyticity of the nonlinearity to generate infinitely many independent linearized problems. The arbitrary-subset result, under stronger assumptions, is particularly notable. The approach is methodologically credible: high-order linearization is a well-established technique, and Carleman estimates are the standard tool for stable coefficient recovery. The explicit mention of locally analytic nonlinear terms and the geometric condition indicates awareness of the technical requirements. However, because the manuscript contains only the abstract, the significance is necessarily conditional on the details of the Carleman estimates, the well-posedness theory, and the stability inequalities, none of which can be inspected here.","major_comments":[{"comment":"The central claim—stable and unique determination of coefficients—is stated but not supported by any proof or precise statement in the available material. The Carleman estimate, the geometric condition on the measurement subset, the function spaces, the smallness assumptions, and the exact stability inequality are all absent. These are load-bearing components of the result. Without them, the correctness of the claim cannot be assessed. This is not an allegation of error, but a fundamental limitation of the reviewable material.","section":"Abstract (entire available text)"},{"comment":"The local analyticity of the nonlinearity is the structural assumption that makes high-order linearization possible. The abstract does not specify the precise class of nonlinearities (e.g., polynomial, entire, or with prescribed growth bounds), nor does it state how the high-order linearized problems are derived and why they are well-posed. Since the recovery procedure depends on extracting arbitrarily many Taylor coefficients of the boundary map, the exact analyticity assumption is central. The manuscript needs to state this assumption precisely and justify the linearization procedure at each order.","section":"Abstract, 'locally analytic nonlinear terms'"}],"minor_comments":[{"comment":"The phrase 'stable and unique determination' is used without defining the measurement map or the notion of stability (e.g., Lipschitz, log-Lipschitz, conditional Hölder). The roles of 'small initial data' and 'trivial boundary data' should be clarified: are they necessary for well-posedness or for the linearization procedure?","section":"Abstract, terminology"}],"recommendation":"uncertain","confidential_remarks":"The review is based solely on the abstract because no full text was available. If the manuscript is under consideration, the editor should obtain the complete paper; the abstract alone is insufficient to judge the validity of the Carleman estimates or the linearization argument. The topic is within the scope of a mathematical analysis journal, and there is no visible internal inconsistency, but the absence of proof details precludes a stronger recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Since we only have the abstract, I'll be honest about what I can and can't judge. The paper's package is clear: treat well-posedness for small data with trivial boundary data, then use high-order linearization to turn the nonlinear inverse problem into a family of linear ones, then apply Carleman estimates for the linear Schrödinger equation to get stable coefficient recovery. That is a natural and coherent strategy, and the abstract suggests the authors actually carried it out for both a large boundary subset satisfying a geometric condition and, under stronger regularity/size assumptions, arbitrary subsets. If the proof goes through, this is a solid contribution to the inverse problems literature, not a paradigm shift but a credible extension of linear results to a nonlinear setting.\n\nThe main soft spot is the one you'd expect: local analyticity of the nonlinear term. That's explicitly stated, not hidden, and it's the standard price for high-order linearization. Still, it limits applicability — finitely smooth nonlinearities won't work with this method, and the stability estimate presumably degrades with the order. I'd want the authors to say something about how the analyticity assumption interacts with the smallness of the data, and whether the geometric condition on the boundary subset is really needed or an artifact of the Carleman proof.\n\nThe proof details are the whole ballgame here. No derivations in the abstract, so I can't verify the Carleman estimates, the well-posedness claim, or the stability inequality. That's not a flaw in the paper; it's a limitation of this review. The stress-test note found no concrete objection, and neither did I. The circularity burden looks low; the argument is genuinely from boundary measurements, not fitted parameters.\n\nCitation-wise, nothing in the abstract raises red flags, but I'd wait to see the full text before citing it. For peer review, yes: this deserves a serious referee. The combination of high-order linearization with Carleman estimates for nonlinear Schrödinger is worthwhile, and the stability claim is sharp enough that a referee should check the technical steps carefully. I would not desk-reject it on the abstract.\n\nAll in all: plausible, well-scoped, likely correct. Send it out.","headline":"Abstract-only, but the high-order linearization plus Carleman approach for nonlinear Schrödinger is a sensible, likely-correct extension; deserves a real referee.","tokens_in":1275,"tokens_out":1418,"would_cite":false,"duration_ms":14924,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35Q55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For nonlinear Schrödinger equations with analytic nonlinearities, stationary coefficients are stably and uniquely recoverable from boundary flux measurements, provided the coefficients are known near the boundary.","keywords":["nonlinear Schrödinger equation","inverse boundary value problem","high-order linearization","Carleman estimates","stability","unique determination","analytic nonlinearity","stationary coefficients"],"falsifier":"Fix the cubic nonlinear Schrödinger equation $i u_t + \\Delta u + q(x) u + |u|^2 u = 0$ on a bounded domain with zero Dirichlet boundary data, and let two potentials $q_1,q_2$ agree near the boundary but differ inside. For a small boundary input of amplitude $\\epsilon$, compute the third-order term in $\\epsilon$ of the Neumann data. If these third-order Neumann data coincide for every such input while $q_1\\ne q_2$ in the interior, the uniqueness claim is false; if they differ, the difference as a function of the potential difference tests the stated stability bound. A one-dimensional numerical","tokens_in":572,"feed_emoji":"📡","tokens_out":9587,"duration_ms":82681,"temperature":0.7,"pith_summary":"The paper establishes that in a dynamical Schrödinger equation with a locally analytic nonlinear term, the stationary coefficients are uniquely and stably determined by measuring the normal derivative of the solution on part of the boundary, assuming the coefficients are known near that boundary. The proof works by high-order linearization: because the nonlinearity is analytic, the solution map can be differentiated arbitrarily many times at zero, turning the nonlinear inverse problem into an infinite sequence of linear inverse problems. Stability for each linearized problem comes from Carleman estimates for the linear Schrödinger equation. This matters because it shows that boundary measurements can pin down internal coefficients in realistic nonlinear quantum and optical models, without needing to know the nonlinearity's precise form in advance.","feed_headline":"Stable recovery of coefficients in nonlinear Schrödinger equations","feed_subtitle":"Locally analytic nonlinearities enable high-order linearization, turning boundary data into many linear inverse problems.","key_machinery":"High-order linearization: repeatedly differentiating the solution-to-boundary-data map with respect to small-amplitude excitations, producing a hierarchy of linear Schrödinger equations whose sources are built from lower-order derivatives; analyticity of the nonlinearity guarantees all derivatives exist and that the $k$-th Taylor coefficient of the boundary response is isolated by the $k$-th linearized problem. Carleman estimates: exponentially weighted $L^2$ estimates for the linear Schrödinger equation that provide quantitative unique continuation and stability, transferring smallness of the boundary data difference to smallness of the coefficient difference.","core_discovery":"The paper treats the inverse problem of recovering time-independent coefficients in a dynamical nonlinear Schrödinger equation $i\\partial_t u + \\Delta u + q(x)u + f(x,u,\\bar u)=0$ on a bounded domain in $\\mathbb{R}^n$, with zero Dirichlet boundary data and small initial data; here $q$ is a stationary potential and $f$ is a locally analytic nonlinearity that may itself depend on stationary coefficients. The main discovery is that these stationary coefficients are uniquely and stably determined by the Neumann data (the normal derivative $\\partial_\\nu u$ on the boundary) measured on a subset of the boundary, assuming the coefficients are already known close to the boundary. This is proved in tw","pith_inferences":["If the nonlinearity is only $C^k$ rather than analytic, the hierarchy stops after $k$ steps, so coefficients beyond the $k$-th Taylor order likely become invisible; the stability modulus should degrade accordingly. This is an inference, not a claim of the paper.","The same strategy of high-order linearization plus Carleman estimates may apply to other nonlinear evolution equations (e.g., nonlinear wave or heat equations) with analytic nonlinearities, so the method may be general beyond Schrödinger equations.","Because coefficients must be known near the measured boundary, the result implicitly describes a two-stage experimental protocol: first calibrate a boundary layer, then recover the interior coefficients from subsequent boundary flux measurements; the interior recovery's stability should weaken with distance from the boundary.","For a cubic nonlinearity, the third-order linearized problem corresponds to a four-wave mixing term; this gives a concrete numerical testbed: simulate two potentials that agree near the boundary and check whether their third-order Neumann data are distinguishable."],"forward_implications":["The same boundary data set used for a linear Schrödinger inverse problem is sufficient for the nonlinear problem, as long as the nonlinearity is analytic and the coefficients are known near the boundary.","High-order linearization is constructive: sending multiple small-amplitude inputs and taking differences of the recorded Neumann data isolates each Taylor order, so the nonlinearity's coefficients can be recovered one order at a time.","The geometric-condition case and the arbitrary-subset case quantify the trade-off between measurement access and required coefficient regularity: larger measurement sets need milder assumptions, while small patches need stronger bounds on the size and smoothness of the coefficients.","The small-data well-posedness result provides the rigorous foundation for the solution map used in the recovery, ensuring the high-order derivatives taken in the linearization are well defined."],"supporting_citations":[],"fun_headline_variants":["Boundary data pin down nonlinear Schrödinger coefficients","Stable inverse method for nonlinear Schrödinger potentials","Recover coefficients in nonlinear Schrödinger from boundary","Nonlinear Schrödinger: stable coefficient recovery from boundary"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The nonlinearity must be locally analytic (representable by a convergent power series around zero), because the entire method relies on taking infinitely many derivatives of the solution map at zero to separate the effects of the coefficients.","fun_headline_variants_meta":{"raw":{"variants":["Boundary data pin down nonlinear Schrödinger coefficients","Stable inverse method for nonlinear Schrödinger potentials","Recover coefficients in nonlinear Schrödinger from boundary","Nonlinear Schrödinger: stable coefficient recovery from boundary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000539,"raw_usage":{"total_tokens":2373,"prompt_tokens":645,"completion_tokens":1728,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":389,"completion_tokens_details":{"reasoning_tokens":1668}},"tokens_in":389,"tokens_out":1728,"duration_ms":10923,"temperature":1.0,"reasoning_tokens":1668,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T22:14:22.675422+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix the cubic nonlinear Schrödinger equation $i u_t + \\Delta u + q(x) u + |u|^2 u = 0$ on a bounded domain with zero Dirichlet boundary data, and let two potentials $q_1,q_2$ agree near the boundary but differ inside. For a small boundary input of amplitude $\\epsilon$, compute the third-order term in $\\epsilon$ of the Neumann data. If these third-order Neumann data coincide for every such input while $q_1\\ne q_2$ in the interior, the uniqueness claim is false; if they differ, the difference as a function of the potential difference tests the stated stability bound. A one-dimensional numerical","supporting_citations":[],"review_version":1}