{"id":"e038ff60-c8bd-44a5-be6e-b801dbf50d76","arxiv_id":"1908.04745","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A nonintegrable discrete nonlocal NLS model is shown numerically to have linearly unstable stationary waves and Cauchy solutions whose large-amplitude growth time and location depend sensitively on the initial phase.","lead":"This paper numerically studies a discrete, non-integrable version of the nonlocal nonlinear Schrödinger equation, a model from parity-time-symmetric physics. It reports stationary waves that are linearly unstable, and numerical evidence that solutions can grow to large amplitude at initial-phase-dependent locations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Blow-up claim not supported: threshold '|u|=100' in scheme (34) is never validated against Cauchy problem (9), and the threshold time grows from 11.8 to 163.5 as N goes 32 to 128.","rationale":"The reader's identified weakest assumption focuses on the stationary-solution fixed-point residual, which is a legitimate concern for the linear-stability part of the paper. However, the paper's headline contribution is the claimed blow-up phenomenon and phase-sensitive dynamics of the continuum nonlocal NLS Cauchy problem (9). That claim is more load-bearing because it motivates the entire paper and appears in the abstract, introduction, and conclusions. The numerical evidence in Section 4 does not connect the discrete simulations to (9): no convergence check is reported for scheme (34), and the threshold time for |u|=100 grows substantially with refinement, which is the opposite of what one expects for a resolved finite-time singularity. Therefore the central claim is currently unsupported. The stationary-solution residual issue remains relevant to the secondary stability result, but it does not by itself invalidate the main numerical observation of phase-dependent large-amplitude growth in the discrete model. The appropriate verdict is still conditional: the paper can be made publishable if the authors reframe the results as properties of the discrete nonintegrable model or provide convincing evidence of convergence to continuum blow-up, and if they verify the fixed-point residual for the stationary solutions. My recommendation of CONDITIONAL matches the reader's verdict, but the required condition should explicitly include the convergence/reframing of the blow-up claim, not only the residual check.","tokens_in":12777,"tokens_out":5350,"duration_ms":61256,"concrete_test":"Solve Cauchy problem (9) directly with an independent, validated high-order method, e.g., a Fourier pseudospectral discretization with N=512 and N=1024 and a fourth-order time integrator, using the same parameters L=2*sqrt(2)*pi, a=1, epsilon=0.1, mu=2*pi/L, phi0=pi/256, and integrate until t=200 while tracking max_x |u(x,t)|. If the maximum does not exceed a large converged threshold, or if the threshold-crossing time does not converge to a finite value under refinement, then the reported blow-up is an artifact of the nonintegrable scheme (34) and the central claim that Cauchy problem (9) blows up is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that the numerical solution of Cauchy problem (9) yields blow-up and that initial data strongly affects the blow-up time and position (Section 1, p.3; Section 4). This is not supported by the evidence. The simulations solve the nonintegrable spatial discretization (34), not the continuum equation (9), and no convergence study of (34) to (9) is provided. The 'blow-up' criterion is simply max_n |u_n| reaching 100 (Figs. 7-9), and the runs are stopped at that threshold, so no finite-time singularity is demonstrated. Worse, for fixed phi0=pi/256 the threshold time grows dramatically with mesh refinement: t=11.8075 for N=32, t=81.5007 for N=64, and t=163.4591 for N=128 (Figs. 7-9). A genuine finite-time blow-up of (9) should exhibit a threshold time that converges as h->0; instead the time increases by an order of magnitude, suggesting that the observed large-amplitude behavior is a property of the discrete model, a numerical artifact, or a slow quasiperiodic excursion rather than a validated blow-up of the PDE. The paper's own statement that the solutions 'become quasiperiodic before the solution blows up with the mesh refinement increasing' (Section 4, p.11) strengthens this concern. Thus the headline claim as stated is unverified, and the paper should either prove or reframe it as a property of the nonintegrable discrete equation, not of the continuum Cauchy problem (9).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the nonintegrable spatial discrete nonlocal NLS equation (6), a PT-symmetric discretization of the reverse-space nonlocal NLS (4). It constructs stationary solitary-wave solutions by a discrete Fourier transform and a modified Neumann/Petviashvili-type iteration (18), examines their linear stability via an eigenvalue problem (28), and numerically investigates the Cauchy problem with periodic boundary conditions using the nonintegrable scheme (34) and the integrable scheme (35). The reported findings are that the stationary waves are linearly unstable, that the nonintegrable scheme (34) can produce large-amplitude growth that the standard discrete NLS (36) does not, and that the blow-up time and location depend strongly on the initial phase.","tokens_in":13136,"tokens_out":4883,"duration_ms":50190,"significance":"If the results are read as properties of the discrete nonlocal models, the paper is a useful numerical case study: the stationary-solution iteration is parameter-free, the stability comparison between the nonintegrable and integrable discrete nonlocal equations is suggestive, and the demonstrated sensitivity to the initial phase in (34)/(35) contrasts with the standard discrete NLS (36). The paper also ships concrete parameter sets and figures that allow the main computations to be reproduced. The central weakness is that the headline claim is phrased for the continuum Cauchy problem (9), whereas all evidence is obtained for the discrete schemes (34)/(35), and the 'blow-up' diagnostic is an unvalidated amplitude threshold. With a corrected interpretation or an added convergence study, the work would be a reasonable contribution.","major_comments":[{"comment":"The claim that 'the numerical solution of Cauchy problem (9) yields the blow-up phenomenon' is not supported by the evidence, because the simulations solve the discrete scheme (34), not the continuum equation (9), and no convergence or residual study linking (34) to (9) is provided. Moreover, the threshold times for max_n |u_n| = 100 increase from 11.8075 (N=32) to 81.5007 (N=64) and 163.4591 (N=128), which is the opposite of the behavior one expects for a converged finite-time singularity as h→0. The statement in §4 that the solutions 'become quasiperiodic before the solution blows up with the mesh refinement increasing' strengthens this concern. Please either add a genuine convergence study with a threshold-independent singularity diagnostic, or explicitly reframe the claim as a property of the discrete nonintegrable model (34).","section":"§4, Eq. (34), Figs. 7–9"},{"comment":"The convergence table reports only successive-iterate differences (e.g., |||F_e,n| − |F_e,s|||_L2) and |α/β|; it does not report the residual of the stationary equation (11) or (23) at the computed fixed point. Because the linear stability analysis in §3 starts from this numerical stationary solution, an unverified fixed point would propagate directly into the eigenvalue conclusion. Please report the maximum and L2 residuals of (23) at the working truncation, and state the lattice cut-off N used for the fixed-point computation.","section":"§2, table on p.7, Eq. (18)"},{"comment":"The stability calculation is not fully reproducible as written: the text does not specify the cut-off N or the number of lattice sites used in the finite truncation of the infinite eigenvalue problem, nor does it report convergence checks in N. The statement that 'this stability problem contains eight eigenvalues with nonzero imaginary part' (p.8) is therefore hard to verify. Please add the truncation details and a brief check that the relevant unstable eigenvalues converge as N increases.","section":"§3, Eq. (28), Figs. 3–4"}],"minor_comments":[{"comment":"The match between the numerical stationary mode and the continuum solution (22) is asserted visually only; please report a quantitative error norm (e.g., L2 or L∞ difference) at the working resolution.","section":"§2, Fig. 2, Eq. (22)"},{"comment":"There are several typographical errors: 'extensely' on p.2, 'wherw' on p.7, 'envolution' on p.10, and 'let us we discuss' on p.8.","section":"General"},{"comment":"The sentence that 'the discrete solution of Cauchy problem (35) converges to the one of Cauchy problem (9)' should be sharpened: the numerical reference u^256_n is itself a discrete solution, so the plotted errors compare two discrete schemes rather than verifying convergence to the PDE (9).","section":"§4, p.10, Figs. 5–6"},{"comment":"The convolution star is used in expressions such as Q1 * F_e(q) without explicitly defining the two-function convolution; the definition of the triple convolution is given, but the binary case should be stated as well.","section":"§2, Eq. (16)"},{"comment":"Captions (d) say '|u0(t)| of the Cauchy problem (36)' while the text describes the maximum module over n; please clarify whether the plotted quantity is |u0(t)| or max_n |u_n(t)|.","section":"§4, Figs. 7–9"},{"comment":"Reference [44] appears in the bibliography but is not cited in the body of the paper.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the framing in Section 4: the paper's headline 'blow-up' statement is attached to the continuum Cauchy problem (9) but all simulations are for discrete schemes, and the threshold times grow strongly with mesh refinement. I think this is fixable by reframing the claim as a property of the discrete nonlocal model or by adding a genuine convergence study. I do not see grounds for rejection if that is done and the requested residuals and numerical details are supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper numerically explores a nonintegrable discrete version of the nonlocal NLS, Eq. (6). That equation is the obvious nonlocal analogue of the standard discrete NLS, and I don't know of a prior numerical study of it. The stationary-soliton construction via discrete Fourier transform and a Petviashvili-type iteration is a sensible adaptation of established methods, and the linear-stability eigenvalue computation follows the standard route. If the stationary waves are genuine, the finding that they are linearly unstable for the nonintegrable model, while the corresponding integrable ones are stable in the symmetric case, is a reasonable and interesting contrast.\n\nThe soft spots are real, though. The iteration convergence table reports only differences between successive iterates, not the residual of the discrete equation (11). So a spurious fixed point is not excluded. More serious is the 'blow-up' claim in Section 4. The authors solve the discrete system (34), not the continuum Cauchy problem (9), and they call reaching |u|=100 'blow-up.' No convergence study of (34) to (9) is shown. The stress-test note is right: for fixed initial data, the time to reach |u|=100 grows from about 11.8 for N=32 to 81.5 for N=64 to 163.5 for N=128. That is the opposite of what one expects from a resolved finite-time singularity. The paper even says the solutions become quasiperiodic before the amplitude grows, which weakens the claim further. The authors' actual observation—phase-dependent large-amplitude growth that is sensitive to the discrete model—may be true and worth reporting, but it is not a validated blow-up of the PDE.\n\nThere are also smaller issues: the comparison to the continuum soliton is only visual, no code or data are provided, and there are typos ('wherw', 'envolution'). None of these are fatal, but they add to the impression of a rushed manuscript.\n\nWho is this for? Someone working on discrete nonlocal NLS or PT-symmetric lattices might find the model and the numerical observations a useful starting point. The paper is not ready as is. It needs a careful revision: re-frame the blow-up language as large-amplitude growth in the discrete model, add residual checks for the stationary solutions, and show at least one mesh-refinement study for the Cauchy problem that distinguishes discrete artifacts from genuine dynamics. A serious referee can help the authors do that, and the underlying idea is worth the effort.","headline":"A plausible but over-claimed numerical study of a new nonintegrable discrete nonlocal NLS; the stability observations are worth checking, but the 'blow-up' claim does not survive contact with the data.","tokens_in":13639,"tokens_out":2072,"would_cite":false,"duration_ms":23073,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","37K60","65M70"],"pacs":[],"model":"deepseek-v4-flash","headline":"A nonintegrable discrete nonlocal NLS has linearly unstable solitary waves and phase-controlled blow-up.","keywords":["nonlocal nonlinear Schrödinger equation","discrete NLS","PT symmetry","stationary solitary waves","linear stability","blow-up","Cauchy problem","modified Neumann iteration"],"falsifier":"Evaluate the residual of the stationary advance-delay system (11) at the converged iterate $\\hat F_{e,16},\\tilde F_{o,16},\\hat G_{e,16},\\tilde G_{o,16}$ on a lattice wider than the soliton; if its $L^2$ norm does not tend to zero as the iteration count and lattice cutoff increase, the claimed stationary wave is a numerical artifact and the instability eigenvalues do not describe the true equation. For the Cauchy problem, check whether the time for $\\max_n |u_n|$ to exceed 1000, then $10^4$, continues to decrease as $N$ grows; if it does not, the apparent blow-up is a discretization effect.","tokens_in":12609,"feed_emoji":"💥","tokens_out":18237,"duration_ms":172911,"temperature":0.7,"pith_summary":"The paper studies the nonintegrable discrete nonlocal nonlinear Schrödinger equation $i u_{n,t}+h^{-2}(u_{n+1}-2u_n+u_{n-1})+u_n^2u_{-n}^*=0$, a parity-time-symmetric lattice discretization of the reverse-space nonlocal NLS. It constructs stationary solitary-wave solutions using the discrete Fourier transform and a modified Neumann iteration, and then analyzes their linear stability; the central finding is that these waves are linearly unstable. For the periodic Cauchy problem with initial data $a[1+\\varepsilon\\cos(\\mu x+\\phi_0)]$, the numerical solutions exhibit what the paper calls blow-up: the maximum modulus reaches 100 at a lattice site and time that depend strongly on the phase $\\phi_0$. The paper contrasts this with the classical NLS, whose nonintegrable discretization stays bounded for the same data, and with the integrable discrete nonlocal NLS, which blows up much later and only at the site $n=0$.","feed_headline":"Nonlocal discrete NLS blows up at a site set by the initial phase","feed_subtitle":"Numerics show the initial phase decides where and when the singularity forms, unlike the classical discrete NLS.","key_machinery":"The argument rests on the stationary-wave ansatz $u_n(t)=[F(nh)+iG(nh)]e^{i\\omega t}$, which converts equation (6) into the advance-delay system (11) with the parity operator $P X(nh)=X(-nh)$. The paper decomposes $F,G$ into even and odd parts, applies the discrete Fourier transform to obtain the nonlinear integral system (16), and solves it by a modified Neumann iteration (18), a fixed-point iteration equipped with the stabilizing factor $|\\alpha/\\beta|^{3/2}$; the converged fixed point is inverse-transformed to give the approximate solitary wave. Linear stability is then decided by the matrix eigenvalue problem (28) built from a tridiagonal matrix $A$ and a diagonal matrix $B$; eigenvalues with nonzero imaginary part are the criterion for linear instability. The same eigenvalue construction is applied to the integrable discrete nonlocal NLS to produce the comparison stable and unstable cases.","core_discovery":"On the paper's own terms, the discovery is that the nonintegrable spatial discretization (6) of the reverse-space nonlocal NLS has two dynamical signatures that the integrable and classical versions do not share. Stationary solitary waves obtained as fixed points of the Fourier-domain iteration (18) are linearly unstable: the eigenvalue problem (28) produces eight eigenvalues with nonzero imaginary part, so small perturbations grow. In the periodic Cauchy problem (9), the numerical solution reaches large amplitude (what the paper calls blow-up, in the operative sense of $\\max_n|u_n|=100$) at a site selected by the phase $\\phi_0$ of the initial modulation: for $\\phi_0\\in(-\\pi/2,\\pi/2)$ the maximum first grows at $n=0$, for $\\phi_0\\in(\\pi/2,3\\pi/2)$ at $n=L/(2h)$, and for $\\phi_0=\\pm\\pi/2$ at both sites. The time required to reach that amplitude increases as $\\phi_0$ runs from $0$ to $\\pi/2$, and with mesh refinement the solution oscillates quasiperiodically before the large-amplitude growth. The integrable discrete nonlocal NLS (5), run with the same initial data, takes far longer to reach the same amplitude and always does so at $n=0$.","pith_inferences":["The paper leaves implicit that the two blow-up sites are exactly the sites fixed by the lattice symmetry $n\\mapsto -n$ under periodic boundary conditions; at $n=0$ and $n=N/2$ the nonlocal term $u_n^2u_{-n}^*$ reduces to the local term $|u_n|^2u_n$, which may explain why the singularity is pinned to those sites.","A testable extension is to raise the amplitude threshold from 100 to 1000 or $10^4$ and track whether the time-to-threshold follows a consistent finite-time blow-up law as $N$ grows; if the delay grows without bound, the reported blow-up is a property of the nonintegrable discretization rather than of the nonlocal NLS itself.","The linear instability of the stationary waves and the phase-dependent growth in the Cauchy problem may share a mechanism: the eigenvalue with nonzero imaginary part from the stability spectrum should set the growth rate seen in the time simulations, and comparing the two rates would test that connection."],"forward_implications":["Stationary solitary waves of the nonintegrable discrete nonlocal NLS (6) are linearly unstable, so small perturbations grow exponentially; stable discrete solitons in this model would require a different nonlinearity or parameter regime.","For the periodic Cauchy problem, the nonintegrable discretization predicts large-amplitude growth whose lattice site is fixed by the initial phase $\\phi_0$: $n=0$ for $\\phi_0\\in(-\\pi/2,\\pi/2)$, $n=L/(2h)$ for $\\phi_0\\in(\\pi/2,3\\pi/2)$, and both sites at $\\phi_0=\\pm\\pi/2$.","Mesh refinement from $N=32$ to $64$ to $128$ delays the moment at which $\\max_n|u_n|$ reaches 100, from about 11.8 to 81.5 to 163.5 time units, and makes the pre-growth dynamics quasiperiodic, whereas the classical nonintegrable discrete NLS gives bounded solutions for the same initial data.","The integrable discrete nonlocal NLS (5) reaches the same large amplitude much later and always at $n=0$, so integrability changes both the location and the time scale of the singularity-like growth."],"supporting_citations":[{"why":"defines the nonintegrable discrete NLS scheme (2) and the numerically induced chaos baseline that the Cauchy-problem comparison extends.","marker":"[24]"},{"why":"introduces the continuous reverse-space nonlocal NLS (4) whose spatial discretization is the subject of the paper.","marker":"[30]"},{"why":"supplies the integrable discrete nonlocal NLS (5) used as the comparison scheme and as the source of the benchmark stationary soliton (31).","marker":"[40]"},{"why":"provides the N-soliton solution (32), from which the second stationary solution (33) used in the stability comparison is derived.","marker":"[41]"},{"why":"supplies the modified Neumann iteration scheme used to converge the stationary solutions from the Fourier integral system.","marker":"[42]"},{"why":"gives the exact stationary solution (22) of the continuous nonlocal NLS against which the numerical solitary wave is checked.","marker":"[43]"},{"why":"supplies the adaptive Runge-Kutta solver used for the time integrations in the Cauchy-problem simulations.","marker":"[45]"}],"fun_headline_variants":["Phase sets blow-up site in nonlocal discrete NLS","Nonlocal discrete NLS blow-up site chosen by initial phase","Stationary waves unstable in nonlocal discrete NLS","Phase controls blow-up location in nonlocal discrete NLS","Discrete nonlocal NLS: blow-up site tied to initial phase"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper relies on the numerical fixed point reached by the modified Neumann iteration at step 16 being a true stationary solution of the infinite-lattice equation; the table reports only how much the solution changed between iterations, not how well it satisfies the equation itself.","fun_headline_variants_meta":{"raw":{"variants":["Phase sets blow-up site in nonlocal discrete NLS","Nonlocal discrete NLS blow-up site chosen by initial phase","Stationary waves unstable in nonlocal discrete NLS","Phase controls blow-up location in nonlocal discrete NLS","Discrete nonlocal NLS: blow-up site tied to initial phase"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000207,"raw_usage":{"total_tokens":1423,"prompt_tokens":994,"completion_tokens":429,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":345}},"tokens_in":610,"tokens_out":429,"duration_ms":4498,"temperature":1.0,"reasoning_tokens":345,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:57:43.198733+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the residual of the stationary advance-delay system (11) at the converged iterate $\\hat F_{e,16},\\tilde F_{o,16},\\hat G_{e,16},\\tilde G_{o,16}$ on a lattice wider than the soliton; if its $L^2$ norm does not tend to zero as the iteration count and lattice cutoff increase, the claimed stationary wave is a numerical artifact and the instability eigenvalues do not describe the true equation. For the Cauchy problem, check whether the time for $\\max_n |u_n|$ to exceed 1000, then $10^4$, continues to decrease as $N$ grows; if it does not, the apparent blow-up is a discretization effect.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the nonintegrable discrete NLS scheme (2) and the numerically induced chaos baseline that the Cauchy-problem comparison extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the continuous reverse-space nonlocal NLS (4) whose spatial discretization is the subject of the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the integrable discrete nonlocal NLS (5) used as the comparison scheme and as the source of the benchmark stationary soliton (31)."},{"cited_title":"Ma and Z","cited_arxiv_id":null,"evidence_quote":"provides the N-soliton solution (32), from which the second stationary solution (33) used in the stability comparison is derived."},{"cited_title":"Petviashvili, Equation of an extraordinary soliton , Sov","cited_arxiv_id":null,"evidence_quote":"supplies the modified Neumann iteration scheme used to converge the stationary solutions from the Fourier integral system."},{"cited_title":"Huang and L","cited_arxiv_id":null,"evidence_quote":"gives the exact stationary solution (22) of the continuous nonlocal NLS against which the numerical solitary wave is checked."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the adaptive Runge-Kutta solver used for the time integrations in the Cauchy-problem simulations."}],"review_version":1}