{"id":"72de1978-74af-4eb7-bfa4-30a884b35ea5","arxiv_id":"2411.18173","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the Klein-Gordon-Boussinesq system, the paper proves local well-posedness and blow-up/global existence conditions, derives solitary wave existence, and numerically confirms the KdV approximation.","lead":"This paper studies a coupled system of equations used to test whether simpler wave equations like KdV give accurate long-time approximations, and proves local existence, uniqueness, global existence or blow-up, plus existence of solitary wave solutions. It also runs numerical simulations that match an earlier KdV approximation theorem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.5's global-existence proof uses ||u||_{L^4} ≤ C||u||_{H^1}, but the energy y(t) does not control ||u_x||_{L^2}; the y^{3/2} estimate is therefore unproved.","rationale":"The paper's central assertion is the local well-posedness theory in Theorem 2.1 and the conditional global-existence and blow-up results in Theorems 2.5 and 2.7. I examined the most load-bearing step in this chain: the global-existence proof. The reader's weakest-assumption analysis correctly identifies that Theorem 2.5's energy estimate assumes H^1 control on u, which the conserved energy (2.11) and the quantity y(t) do not supply. My independent check of the displayed inequality confirms the gap: the terms ∫ u^2 v and ∫ u v^2 require ||u||_{L^4} or ||v||_{L^4} bounds, and for u only ||u||_{L^2} is available. The concrete test with u_ε = ε sin(x/ε) shows that no universal bound of the form |∫ u^2 v| ≤ C y^{3/2} can hold under the stated norms, so the proof as written is internally incomplete rather than merely nonstandard. This does not necessarily falsify the theorem, because a more subtle argument might control ||u||_{L^4} via the equation (2.13), but no such argument appears. The local well-posedness proof in Theorem 2.1 uses the multiplication estimates (2.9) consistently with the hypotheses on r and s, and I found no analogous flaw there. The blow-up proof in Theorem 2.7 algebraically matches the conservation laws under the coefficient condition (2.10), assuming the additional mean-zero condition on ∂_x^{-1}u_0; the notation in its statement is slightly loose but not substantively wrong. The numerical and travelling-wave sections do not bear on the main well-posedness claim, and the positive-operator existence theorem is conditional as stated, which is a weakness but not the decisive one. Because the identified concern is exactly the reader's weakest assumption and does not change my assessment of the overall paper, I recommend keeping the CONDITIONAL verdict without modification.","tokens_in":20138,"tokens_out":13725,"duration_ms":106414,"concrete_test":"Check whether the claimed y^{3/2} bound can be derived from the stated embeddings by testing functions with small L^2 norm and large H^1 norm. Take u_ε(x)=ε sin(x/ε) on [0,1] and v_ε(x)=1; then y_ε ≈ ε^2 (if other terms are zero) while |∫ u_ε^2 v_ε| ≈ ε^2/2, so |∫ u_ε^2 v_ε| / y_ε^{3/2} ≈ (2ε^2)/(ε^3) → ∞ as ε→0, disproving any universal bound of the form |∫ u^2 v| ≤ C y^{3/2}. Alternatively, compute the time derivative of ||u||_{H^1}^2 along the flow of (1.1) using (2.13); if the right-hand side contains terms not bounded by y(t), the missing ||u_x|| control cannot be recovered from the conserved quantities. A positive result would require a new argument (e.g., bounding ||u||_{L^4} via ∂_x^{-1}u_t or the equation) that is absent from the manuscript.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 2.5 (Section 2), the proof defines y(t)=||∂_x^{-1}u_t||_{L^2}^2+α^2||u||_{L^2}^2+||v||_{L^2}^2+||v_t||_{L^2}^2+||v_x||_{L^2}^2 and asserts y(t) ≤ 2E(0) + (2/3)K0 y(t)^{3/2}. The constants C3, C4, C* entering K0 come from the Sobolev embeddings ||f||_{L^p} ≤ C_p||f||_{H^1} (p=3,4,∞). With a_uu=0, the potential F in (2.11) contains the terms B u^2 v and C u v^2. The term ∫ u^2 v is bounded in the proof by ||u||_{L^4}^2||v||_{L^2} ≤ C_4^2 ||u||_{H^1}^2 ||v||_{L^2}. But neither y(t) nor the conserved energy E(t) in (2.11) contains ||u_x||_{L^2}: E(t) includes u_t^2, α^2u^2, (∂_x^{-1}u_t)^2, v^2, v_x^2, v_t^2, and no u_x^2. Hence ||u||_{H^1} is not controlled by y(t), and the displayed estimate does not follow. This is not a minor technicality: the same gap appears whenever u appears with L^p (p>2) exponents in the nonlinear terms, and no additional conservation law or equation (2.13) is used in the proof to supply ||u_x||_{L^2}. The global-existence claim (2.14) is therefore unsupported as written. The local well-posedness in Theorem 2.1 and the blow-up argument in Theorem 2.7 appear consistent with the stated spaces, though they too rely on the same energy framework.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the Klein-Gordon-Boussinesq system (1.1)-(1.2). Its main mathematical claims are: local well-posedness in Sobolev spaces (Theorem 2.1); conservation laws (Theorem 2.2) under the coefficient condition (2.10); a global-existence criterion and finite-time blow-up criteria for the local solution (Theorems 2.5 and 2.7); existence of classical and generalized solitary waves via normal-form and positive-operator methods (Section 3); and a numerical check of the KdV approximation (Section 4). The central analytic result is Theorem 2.5, which asserts that small energy data satisfying (2.14) yield a global solution when a_uu=0.","tokens_in":20627,"tokens_out":22285,"duration_ms":190492,"significance":"The paper addresses a natural and largely open set of questions: a rigorous Cauchy theory and a global/blow-up classification for a model system that has been used in approximation-validity studies. The local well-posedness argument is standard and, if the details are completed, should hold in the stated spaces; the conserved quantities and the Levine-type blow-up mechanism are useful; and the solitary-wave sections give a concrete application of two classical theories together with careful numerics. The numerical test of the KdV approximation is also a valuable illustration. However, Theorem 2.5 currently rests on an unproved H^1 control of u, and the energy space is not covered by the stated local theorem; these gaps affect the principal global-existence claim of the paper. The contribution is therefore potentially significant, but not yet established as written.","major_comments":[{"comment":"The displayed estimate y(t) <= 2E(0) + (2/3)K0 y(t)^{3/2} is not established. The nonlinear terms B u^2 v and C u v^2 appearing in F under the assumption a_uu=0 are controlled in the text by embeddings such as ||u||_{L^4} <= C_4 ||u||_{H^1}; replacing ||u||_{H^1} by y(t)^{1/2} is unjustified because y(t) and the conserved energy E(t) in (2.11) contain only ||u||_{L^2}, ||u_t||_{L^2}, ||partial_x^{-1}u_t||_{L^2}, ||v||_{L^2}, ||v_t||_{L^2} and ||v_x||_{L^2}, and no ||u_x||_{L^2} term. No other conservation law or equation (2.13) is used in the proof to supply control of ||u_x||_{L^2}, so the global existence criterion (2.14) is unsupported as written.","section":"Section 2, proof of Theorem 2.5"},{"comment":"The assumptions of Theorem 2.5 are inconsistent with the energy class used in its proof. Theorem 2.1 requires 1/2 < r <= s <= r+1, so it does not cover the natural energy space X_0 x X_1 needed for E(t) and y(t). Moreover, for r>1/2 the condition u_1 in H^r cap dot{H}^{r-1} does not imply partial_x^{-1}u_1 in L^2, so E(0) need not even be finite. The proof of Theorem 2.5 neither adds an explicit assumption such as partial_x^{-1}u_1 in L^2 nor establishes the required low-regularity local well-posedness; Remark 2.3 only mentions an alternative proof via (2.13) without details. The continuation argument therefore lacks a well-defined starting point.","section":"Section 2, Theorems 2.1 and 2.5"},{"comment":"Even after the missing H^1 control of u is supplied, the application of Lemma 2.4 needs separate verification. With C_1=2E(0) and C_2=(2/3)K_0, the lemma's hypothesis is 2E(0) < (1/3)K_0^{-2}, whereas (2.14) gives only E(0) < (1/6)K_0^{-6}; for 0<K_0<1 the latter is weaker and does not imply the former. The proof also does not explicitly show that y(0) lies in the basin of the smaller root a_1 rather than merely below the intermediate value A, which is what the lemma requires for the conclusion y(t) <= a_1.","section":"Section 2, Lemma 2.4 and condition (2.14)"}],"minor_comments":[{"comment":"Several typographical slips should be corrected, including 'conditions of of Theorem 2.1' in Theorem 2.5 and '(v.vt)' in Theorem 2.1 and its proof.","section":"General"},{"comment":"The numerical claim of O(epsilon^{7/2}) would be easier to verify if the maximum errors at a fixed time were tabulated for successive values of epsilon, since the semilog plot alone does not display the convergence rate explicitly.","section":"Section 4, Figure 5"},{"comment":"The normal-form analysis near C_0 assumes a_uu>0, but the coefficient hypotheses of the section do not state this restriction; the cases a_uu<=0 are not discussed and would require a separate sign or higher-order normal form argument.","section":"Section 3.1"},{"comment":"Theorem 3.3 states that the hypotheses are 'stated above' without collecting them in one place; the coefficient assumptions, in particular the nonnegativity conditions on a_gamma_beta and b_gamma_beta and the convexity conditions in (S1), should be listed explicitly in the theorem statement.","section":"Section 3.2, Theorem 3.3"}],"recommendation":"major_revision","confidential_remarks":"I recommend major revision. The local well-posedness argument and the numerical parts are solid, and the blow-up argument is plausible, but Theorem 2.5 has genuine gaps in the control of ||u_x|| and in matching the energy space with the hypotheses of Theorem 2.1. These issues must be fixed before the main global-existence claim can be accepted. The use of the external theorems from [2, 5, 12, 24, 28, 36] is appropriate and does not raise a circularity concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the paper has real content. The local well-posedness result (Theorem 2.1) is standard and looks correct, and the solitary-wave analysis is careful: the normal-form bifurcation study, the parameter regions, and the explicit wave branches are genuinely useful. The numerical checks of the KdV approximation are honest, and the citation pattern is appropriate. The main soft spot is Theorem 2.5. The proof defines y(t) without any ||u_x||^2 term, then bounds terms like ∫u^2v by ||u||_{L^4}^2||v||_{L^2} and uses the H^1 embedding for u. But neither y(t) nor the conserved energy E(t) contains ||u_x||^2. The equivalent first-order formulation (2.13) confirms this: the Hamiltonian contains u, w, w_x, v, v_x, z, but not u_x. So the claimed estimate y(t) ≤ 2E(0) + (2/3)K0 y(t)^{3/2} does not follow. This is not a minor technicality; it undermines the global-extendability claim (2.14). The blow-up theorem may be salvageable, but the proof is compressed and the same energy framework is used. Give credit where due: the local existence theorem is new for the general coefficient system, the normal-form bifurcation analysis in Section 3 is careful, and the explicit solitary wave formulas in Remark 3.5 are useful. The numerical generation via Petviashvili iteration shows residuals, which is reproducible evidence. Verdict: deserves a serious referee, but not acceptance as is. The authors should find an independent H^1 bound for u or revise Theorem 2.5 under stronger assumptions that include u_x control. As written, I would not rely on the global result, though the local theory and solitary-wave parts are worth keeping.","headline":"Local well-posedness and solitary-wave analysis are solid, but the global-existence theorem has a gap: the energy functional does not control the H^1 norm of u needed for the key estimate.","tokens_in":740,"tokens_out":1695,"would_cite":true,"duration_ms":41971,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76B15","35B35","35C08","65M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Klein-Gordon-Boussinesq system has unique local solutions for Sobolev data with 1/2 < r ≤ s ≤ r+1; under a coefficient condition, small data give global solutions and negative energy forces finite-time blow-up.","keywords":["Klein-Gordon-Boussinesq system","well-posedness","blow-up","solitary waves","KdV approximation","Sobolev spaces","traveling waves","normal form theory"],"falsifier":"Take negative-energy data satisfying the hypotheses of Theorem 2.7 and integrate (1.1) numerically: if $\\|\\partial_x^{-1}u\\|^2 + \\|u\\|^2 + \\|v\\|^2$ does not diverge in finite time, the blow-up claim is false. Separately, in the small-data regime of Theorem 2.5, track $\\|u_x\\|_{L^2}$: if it grows while $y(t)$ stays bounded, the missing $H^1$ control in the proof is confirmed.","tokens_in":19991,"feed_emoji":"🌊","tokens_out":10571,"duration_ms":84920,"temperature":0.7,"pith_summary":"The paper studies the Klein-Gordon-Boussinesq (KGB) system, a pair of one-dimensional wave equations used in the literature as a testbed for whether simpler long-wave models such as KdV, NLS, or Whitham equations faithfully describe the original dynamics. Its central claim is a well-posedness theorem: for regularity exponents with $1/2 < r \\leq s \\leq r+1$, data $(u_0,u_1) \\in H^r \\times (H^r \\cap \\dot H^{r-1})$ and $(v_0,v_1) \\in H^s \\times H^{s-1}$ generate a unique local solution, with time of existence depending only on the norms. Under a coefficient condition relating the quadratic nonlinearities, the paper identifies two alternative fates of this local solution: small data with $a_{uu}=0$ yield global solutions, while negative initial energy, or a threshold inequality, forces finite-time blow-up. The paper also derives the existence of classical and generalized solitary waves by normal-form and positive-operator methods, and it numerically verifies the predicted $O(\\varepsilon^{7/2})$ error of the KdV approximation for soliton initial data. If correct, the KGB system has a rigorous Cauchy theory and an explicit small-data global versus negative-energy blow-up dichotomy, which are prerequisites for making long-wave approximation statements quantitative.","feed_headline":"KGB system: local well-posedness, global data, finite-time blow-up","feed_subtitle":"Under a coefficient condition, small data give global solutions and negative energy forces finite-time collapse.","key_machinery":"The argument runs on three mechanisms. For well-posedness, the linear propagators $U(t)$ and $V(t)$, with Fourier symbols $\\sin(t|\\alpha\\xi|/\\sqrt{1+\\xi^2})\\,\\sqrt{1+\\xi^2}/|\\alpha\\xi|$ and $\\sin(t\\sqrt{1+\\xi^2})/\\sqrt{1+\\xi^2}$, convert the system into a Duhamel integral equation; the smoothing factors bring the nonlinear terms into the algebras $H^r$ and $H^s$ so that contraction mapping applies in $C_T(X_r) \\times C_T(X_s)$. For global existence and blow-up, the central object is the functional $y(t) = \\|\\partial_x^{-1}u_t\\|^2_{L^2} + \\alpha^2\\|u\\|^2_{L^2} + \\|v\\|^2 + \\|v_t\\|^2 + \\|v_x\\|^2$ together with the conserved energy $E$ and the second invariant $F$; the proof attempts to bound $y(t)$ by $2E(0) + (2/3)K_0 y(t)^{3/2}$. For solitary waves, the load-bearing identity is the characteristic equation $\\lambda^4 - B\\lambda^2 + A = 0$ with $A=(c_s^2-\\alpha^2)/(c_s^2(1-c_s^2))$ and $B=(1-\\alpha^2/c_s^2)+1/(1-c_s^2)$; the eigenvalue regions, together with the reversible structure with symmetry $S=\\mathrm{diag}(1,-1,1,-1)$, determine where normal form theory produces homoclinic (classical) or homoclinic-to-periodic (generalized) solitary waves, while the positive-operator route uses the convolution system (3.18) with the kernels (3.19) in a cone.","core_discovery":"On its own terms, the paper's main result is Theorem 2.1: given $1/2 < r \\leq s \\leq r+1$, there is a $T>0$ and a unique solution $(u,v)$ of (1.1)-(1.2) with $(u,u_t) \\in C^1([0,T); X_r)$, $X_r = H^r \\times (H^r \\cap \\dot H^{r-1})$, and $(v,v_t) \\in C^1([0,T); H^s \\times H^{s-1})$. The proof rewrites the system via Duhamel's formula with the linear groups $U$ and $V$, uses the fact that $H^r$ and $H^s$ are Banach algebras in this range, and closes a contraction mapping. With the coefficient condition (2.10) ($b_{uu}=-a_{uv}$, $b_{uv}=-a_{vv}$), the paper proves conservation of the energy $E$ and a second functional $F$; Theorem 2.5 then states that when $a_{uu}=0$ and the data satisfy the smallness conditions (2.14), the local solution extends to all times, while Theorem 2.7 states that under (2.10) the solution blows up in finite time whenever $E(0)<0$ or the ratio condition (2.15) holds, in the sense that $\\|\\partial_x^{-1}u\\|^2 + \\|u\\|^2 + \\|v\\|^2$ diverges. The remainder of the paper concerns traveling waves: the linearized traveling-wave problem has characteristic equation (3.5), and normal form theory near the bifurcation curves $C_0$ through $C_3$ yields classical and generalized solitary waves, while positive operator theory in a cone of even, nonnegative, decreasing functions yields classical solitary waves under nonnegativity and convexity assumptions on the kernels. Finally, the formal KdV approximation (4.3) is validated numerically in the sense that the distance between the numerical solution and the KdV soliton ansatz stays $O(\\varepsilon^{7/2})$ up to time 1000 for the tested values of $\\varepsilon$.","pith_inferences":["If the missing $H^1$ control in the proof of Theorem 2.5 can be supplied, for instance by showing that $E$ and $F$ together bound $\\|u_x\\|_{L^2}$, then the small-data global-existence result becomes unconditional; until such a bound appears, a cautious reading treats that theorem as conditional on that a priori estimate.","The blow-up criterion (2.15) resembles the concavity-method thresholds familiar from semilinear wave equations, and the proof yields an explicit upper bound on the blow-up time; a natural numerical test is whether the actual blow-up time approaches $4I(0)/I'(0)$ in the near-threshold regime.","The well-posedness range $r>1/2$ is natural for one-dimensional Sobolev algebras, but the endpoints $r=1/2$ and $s=r+1$ are left open; determining whether local well-posedness holds or fails at these endpoints would complete the Cauchy theory.","The KdV numerics cover the case $a_{uu}, b_{uu}>0$; testing the same $O(\\varepsilon^{7/2})$ bound for defocusing signs or for the $\\alpha>2$ unstable-resonance case treated in the cited literature would show whether the error constant remains tame outside the favourable coefficient regime."],"forward_implications":["Under the coefficient condition (2.10), both $E(t)$ and $F(t)$ are conserved along energy-space solutions, so the invariant structure of the linearized system survives the quadratic coupling.","If $a_{uu}=0$ and the initial data satisfy the smallness conditions (2.14), the local solution of Theorem 2.1 extends to all times $t \\geq 0$.","If $E(0)<0$, or if $E(0)\\geq 0$ and the threshold inequality (2.15) holds, the solution cannot exist globally: $\\|\\partial_x^{-1}u\\|^2_{L^2} + \\|u\\|^2_{L^2} + \\|v\\|^2_{L^2}$ diverges at some finite time.","For speeds with $\\alpha^2 < c_s^2 < 1$, classical solitary waves exist; near the curve $C_1$ with $|c_s|>1$, generalized solitary waves homoclinic to small periodic orbits exist, and the positive-operator theory gives classical solitary waves under additional nonnegativity and convexity assumptions.","The numerical experiments match the KdV approximation bound (4.6): the maximum error between the numerical solution and the KdV soliton ansatz is $O(\\varepsilon^{7/2})$ and remains bounded in time up to $t=1000$."],"supporting_citations":[{"why":"Supplies the improved Boussinesq/Pochhammer-Chree well-posedness and blow-up framework adapted by the authors.","marker":"[27]"},{"why":"Provides the contraction-mapping fixed-point theorem used to prove local existence in product spaces.","marker":"[21]"},{"why":"Gives the linear Klein-Gordon estimates (2.4)-(2.5) that control the Duhamel terms.","marker":"[20]"},{"why":"Normal-form theory used to derive classical and generalized solitary waves near bifurcation curves.","marker":"[24]"},{"why":"Reversible-systems bifurcation analysis and eigenvalue region classification that locate the solitary-wave regimes.","marker":"[8]"},{"why":"Positive-operator theory for dispersive systems, used to prove solitary wave existence in a cone.","marker":"[5]"},{"why":"States the KdV approximation theorem for the KGB model that the paper validates numerically.","marker":"[2]"},{"why":"Extends the KdV approximation to the general coefficient case, including unstable resonances, and supplies the error framework.","marker":"[36]"},{"why":"Proves the validity of the KdV approximation for particular coefficient values, giving the error bound the numerics test.","marker":"[12]"}],"fun_headline_variants":["KGB system: local well-posedness, blow-up, and solitary waves","Klein-Gordon-Boussinesq: from Duhamel to blow-up","KGB: global data under smallness, collapse for negative energy","KGB: existence, uniqueness, and traveling waves","KGB system: numerical validation of KdV approximation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The global-existence theorem rests on the estimate $y(t) \\leq 2E(0) + (2/3)K_0 y(t)^{3/2}$, which is obtained using $\\|u\\|_{L^4} \\leq C_4\\|u\\|_{H^1}$; the quantity $y(t)$ and the energy space do not control $\\|u_x\\|_{L^2}$, so if no separate $H^1$ bound for $u$ follows from the conserved quantities, the small-data global-existence claim is not established as written.","fun_headline_variants_meta":{"raw":{"variants":["KGB system: local well-posedness, blow-up, and solitary waves","Klein-Gordon-Boussinesq: from Duhamel to blow-up","KGB: global data under smallness, collapse for negative energy","KGB: existence, uniqueness, and traveling waves","KGB system: numerical validation of KdV approximation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000996,"raw_usage":{"total_tokens":4329,"prompt_tokens":1169,"completion_tokens":3160,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":785,"completion_tokens_details":{"reasoning_tokens":3069}},"tokens_in":785,"tokens_out":3160,"duration_ms":20563,"temperature":1.0,"reasoning_tokens":3069,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:26:33.065670+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take negative-energy data satisfying the hypotheses of Theorem 2.7 and integrate (1.1) numerically: if $\\|\\partial_x^{-1}u\\|^2 + \\|u\\|^2 + \\|v\\|^2$ does not diverge in finite time, the blow-up claim is false. Separately, in the small-data regime of Theorem 2.5, track $\\|u_x\\|_{L^2}$: if it grows while $y(t)$ stays bounded, the missing $H^1$ control in the proof is confirmed.","supporting_citations":[{"cited_title":"Liu, Existence and blow up of solutions of a nonlinear Pochhammer-Chree equation, Indiana U","cited_arxiv_id":null,"evidence_quote":"Supplies the improved Boussinesq/Pochhammer-Chree well-posedness and blow-up framework adapted by the authors."},{"cited_title":"Hakkaev, M","cited_arxiv_id":null,"evidence_quote":"Provides the contraction-mapping fixed-point theorem used to prove local existence in product spaces."},{"cited_title":"Ginibre, G","cited_arxiv_id":null,"evidence_quote":"Gives the linear Klein-Gordon estimates (2.4)-(2.5) that control the Duhamel terms."},{"cited_title":"Iooss, K","cited_arxiv_id":null,"evidence_quote":"Normal-form theory used to derive classical and generalized solitary waves near bifurcation curves."},{"cited_title":"Champneys, Homoclinic orbits in reversible system s and their applications in mechanics, fluids and optics, Physica D 112 (1998) 158-186","cited_arxiv_id":null,"evidence_quote":"Reversible-systems bifurcation analysis and eigenvalue region classification that locate the solitary-wave regimes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Positive-operator theory for dispersive systems, used to prove solitary wave existence in a cone."},{"cited_title":"Bauer, P","cited_arxiv_id":null,"evidence_quote":"States the KdV approximation theorem for the KGB model that the paper validates numerically."},{"cited_title":"Schneider, The KdV approximation for a system with un stable resonances, Math","cited_arxiv_id":null,"evidence_quote":"Extends the KdV approximation to the general coefficient case, including unstable resonances, and supplies the error framework."},{"cited_title":"Chong, G","cited_arxiv_id":null,"evidence_quote":"Proves the validity of the KdV approximation for particular coefficient values, giving the error bound the numerics test."}],"review_version":1}