{"id":"b749e248-a8ca-4ba0-a3f8-072186a59593","arxiv_id":"1909.00036","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The equivalence groupoid of reduced general Burgers-Korteweg-de Vries equations with space-dependent coefficients is generated by a four-dimensional usual equivalence group together with the equivalence groups of explicitly listed normalized subclasses.","lead":"Equivalence transformations of a wide family of Burgers-Korteweg-de Vries equations with space-dependent coefficients are classified, and the family is decomposed into normalized subclasses. The paper gives explicit formulas for the transformations and new examples of generalized normalization for differential equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's exhaustive list rests entirely on the furcate-splitting solution of (7)–(9) taken as a black box from [6]; an internal sign mismatch in the new parameterizations makes it important to re-verify that solution before accepting completeness.","rationale":"The paper is a continuation of [6] and adds explicit group parameterizations, which is a real contribution. However, the central claim is an exhaustive classification, and exhaustiveness is delegated to [6] rather than proved here. The reader's conditional verdict is therefore appropriate. My reading found no independent verification of the furcate-splitting solution and found a concrete internal reason to doubt the transcription: the sign in the \\tilde u formulas of Propositions 10, 12, and 13 is opposite to Eq. (4), and a direct prolongation shows the minus sign would produce a nonzero u_x coefficient in the target equation, impossible for the class F. This makes the black-box dependency more than an exposition issue. I am not claiming [6] is wrong or that the authors are careless; the proposed checks would settle the question. If an independent re-derivation reproduces Theorem 1's list and the sign is corrected, the paper should be accepted; until then, the conditional verdict stands.","tokens_in":16730,"tokens_out":24141,"duration_ms":207782,"concrete_test":"First, run the cheap sign test: substitute the transformation of Proposition 10 into the prolongation formula used to derive Eq. (4) and compute the coefficient of u_x in the target equation; if it is nonzero, the formula is mis-stated. The decisive test for the central claim is then to re-derive the complete solution of the classifying conditions (7)–(9) independently, using furcate splitting with r as a symbolic parameter and explicitly treating singular cases (α=-2, r=2, a_01=0, a_0=0, degenerate coefficient tuples), and compare the resulting list of maximal subclasses with the nine classes in Theorem 1. If an additional maximal subclass appears, completeness fails; if the lists coincide, the conditional verdict can be upgraded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Theorem 1's completeness: the nine listed subclasses are claimed to exhaust the maximal nontrivial conditional equivalence subgroups, and the equivalence groupoid of F is claimed to be generated by their equivalence groups together with G∼_F. The proof of completeness is not in this paper. Section 4 states only that in [6] the classifying conditions (7)–(9) were solved by the method of furcate splitting, and the present paper supplies explicit parameterizations on top of that solution. If [6] missed any exceptional parameter regime, the theorem overstates the classification. This is not a purely formal worry: the transcription from [6] already shows a sign inconsistency. Propositions 10, 12, and 13 give \\tilde u = X^1/T_t u - X^1_t/T_t x, whereas Eq. (4) of the same paper requires + X^1_t/T_t x for admissible transformations of the class (3) containing F; with the minus sign a standard prolongation gives a nonzero target coefficient of u_x, contradicting the definition of F (A_1=0). If this is a typo, it is still a symptom that the black-box material was not re-derived in this paper; if it is not, the explicit generalized equivalence groups in those propositions are not admissible transformations. Either way, the completeness of the list and the groupoid-generation claim are not independently established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the class F of reduced general Burgers–Korteweg–de Vries equations with space-dependent coefficients, namely ut + uux = Σ_{j=2}^r A_j(x)u_j + A_0(x)u + B(x), for fixed r ≥ 2. The main claim, Theorem 1, is that the usual equivalence group of F is four-dimensional; that the maximal nontrivial conditional equivalence subgroups are exhausted by the generalized equivalence groups of the subclasses FI,1, FI,01, FI,00, FII,1, FIII, FIV,1, FIV,0 (r>2), and FIV,0 (r=2), together with the usual equivalence group of the subclass FII,0; and that the equivalence groupoid of F is generated by the usual equivalence group of F and the equivalence groups of these subclasses. The proof derives the classifying conditions (7)–(9) from the admissible-transformation form of the larger reduced class (3), but then takes the furcate-splitting solution of those conditions from the earlier paper [6], adding explicit parametrizations of the resulting equivalence groups.","tokens_in":16987,"tokens_out":6467,"duration_ms":58647,"significance":"If Theorem 1 is correct, the paper gives a complete structural description of all point equivalences inside a physically relevant class of Burgers–Korteweg–de Vries equations and provides new examples of generalized normalization, including classes whose effective generalized equivalence groups are non-unique. The paper has definite strengths: the explicit transformation formulas for all listed equivalence groups are given in detail; the distinction among usual, generalized, and effective generalized equivalence groups is handled with care; and the non-uniqueness of effective generalized equivalence groups for FII,1 and FIII is argued by conjugation. The central limitation is that the exhaustiveness of the classification is not independently established in this manuscript, and one part of the black-box material transcribed from [6] contains a sign error that affects the admissibility of the listed transformations.","major_comments":[{"comment":"The transformation formulas in these propositions give \\tilde u = \\bar X^1/\\bar T_t u - \\bar X^1_t/\\bar T_t x, but Theorem 7, Eq. (4), requires \\tilde u = X^1/T_t u + X^1_t/T_t x + X^0_t/T_t for every admissible transformation of the superclass (3). Since F is a subclass of (3), every admissible transformation of F must be of this form. With the minus sign, prolongation produces nonzero terms of the form x u_x, which are not present in equations of class (3) and hence not in F. If this is a typo, it must be corrected throughout the affected propositions; if it is not, the listed generalized equivalence groups are not admissible transformations and Theorem 1's list is invalid. The appearance of this error in the material taken from [6] strengthens the need to re-derive rather than quote that portion.","section":"Section 4, Propositions 10, 12, and 13; Eq. (4)"},{"comment":"The exhaustiveness of the list in Theorem 1 and the groupoid-generation claim are load-bearing but are not proved in this manuscript. The text states that the classifying conditions (7)–(9) were solved in [6] by the method of furcate splitting, and the present paper supplies only explicit parametrizations on top of that solution. Because the transcription already shows the sign inconsistency noted above, the completeness claim should not be accepted on the authority of [6] alone. Please include the furcate-splitting solution or a complete, self-contained derivation of the case distinction that leads to the subclasses listed in Theorem 1.","section":"Section 4, Theorem 1"},{"comment":"For the class FI,00, the paper describes the full equivalence groupoid via a third-order ODE and then bypasses quadrature by gauging to the subclass with b0 = 0. The argument that this composition construction indeed exhausts all admissible transformations is only sketched. A precise statement of the domain of definition of the generalized equivalence group and a verification that the gauging-and-composing procedure covers every admissible transformation of FI,00, not merely those with target in the gauged subclass, should be supplied.","section":"Section 4, Propositions 13–15"}],"minor_comments":[{"comment":"The sentence 'Neither the class ¯F nor its superclass ¯F are normalized in any sense' should read 'nor its superclass (1)'.","section":"Section 3"},{"comment":"The phrase 'Classified are admissible transformations' is awkward; suggest 'We classify admissible transformations'.","section":"Abstract and Section 1"},{"comment":"The notational shift between the subclasses with conditions imposed (e.g., 'denote the subclass obtained again by FI,1') is sometimes implicit; please state explicitly which arbitrary elements have been eliminated whenever a subclass name is reused.","section":"Section 4"},{"comment":"In the long formula for \\tilde u, the variables \\hat t and \\bar t are used before their definitions are given in the text; please reorder for readability.","section":"Proposition 15"},{"comment":"The appeal to [1, Corollary 6, p. 97] for smooth dependence of T on parameters should be checked against the specific edition listed in the references.","section":"Remark 11"}],"recommendation":"major_revision","confidential_remarks":"The paper continues the program of [6], which shares an author. This is not circular, because [6] derives the classifying conditions independently, but the current manuscript does not contain enough of that derivation for the exhaustiveness claim to be verified by a reader. The sign inconsistency in the transcription of [6]'s material is a concrete warning that the black-box portion should be re-checked. I would ask the author to provide a self-contained verification of the classification of solutions to (7)–(9) before publication. The paper is within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid continuation of Opanasenko–Bihlo–Popovych (2017). It supplies explicit equivalence groups for the normalized subclasses of the reduced Burgers–KdV class and shows the class F is a union of normalized subclasses, some in the generalized sense. The gauging-to-a-nice-subclass-and-composing-back strategy is sound, and the new examples of generalized normalization are a genuine extension of the earlier paper. If the computations are right, Theorem 1 gives a complete structural description of the equivalence groupoid. That is a useful result for people working on group classification of PDEs.\n\nBut there is a real problem in Section 4.I. Propositions 10, 12, 13, and 14 all write the u-transformation as u_tilde = X^1/T_t u - X^1_t/T_t x. Theorem 7 requires the plus sign for admissible transformations of the class (3), which contains F. With the minus sign, the transformed equation acquires a nonzero coefficient of u_x, so the displayed transformations are not admissible transformations of F. This is either a typo or a substantive error; either way the explicit generalized equivalence groups for FI,1, FI,01, and FI,00 are not correct as printed. I checked the prolongation by hand, and the plus sign is definitely the one that cancels the u_x term. So the author needs to correct the sign and re-verify that the arbitrary-element transformation rules and group compositions still work.\n\nThe other soft spot is completeness. Theorem 1's exhaustive list and the groupoid-generation claim rest entirely on the furcate-splitting solution of (7)–(9) taken from [6]. That is a legitimate citation, and [6] is a published paper, not a rumor. But because this paper does not reproduce that solution or even state the resulting classification in enough detail, a referee cannot independently certify the \"exhausted by\" part without going back to [6] and redoing a lot of work. The sign slip makes me less willing to take that black box on faith.\n\nMinor point: several propositions are asserted without derivations. For a classification paper that is common, but it does slow down verification.\n\nThe paper is aimed at specialists in symmetry analysis, specifically equivalence groupoids and normalization. It is a niche contribution, but a real one. I would send it to peer review, with a major-revision recommendation: fix the sign, re-verify the affected propositions, and ideally state the [6] solution of the classifying conditions more explicitly so the completeness claim can be checked. If the sign correction is clean, the paper is publishable; if it is not clean, the FI part of Theorem 1 collapses.","headline":"A useful continuation of the Opanasenko–Bihlo–Popovych classification, with explicit equivalence groups and new normalization examples, but the FI-family sign error and the black-box completeness from [6] need fixing before I would trust the details.","tokens_in":17482,"tokens_out":4591,"would_cite":false,"duration_ms":42381,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A30","35Q53"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper gives a complete description of all point equivalences inside the class of reduced general Burgers–KdV equations with space-dependent coefficients, identifying exactly nine exceptional subclasses with larger equivalence groups.","keywords":["equivalence groupoid","admissible transformations","generalized equivalence groups","conditional equivalence groups","normalized classes","Burgers–Korteweg–de Vries equations","space-dependent coefficients","furcate splitting"],"falsifier":"Find an equation in $F$ whose coefficient tuple lies outside all listed subclasses and $F_0$ and yet admits a point transformation with non-affine $T(t)$, $X^1(t)$, or $X^0(t)$, which would be a transformation not generated by the four-dimensional usual group. Concretely, for $r=3$, solve (7)–(9) with generic coefficients $A_2,A_3,A_0,B$ allowing $T_{tt}\\neq 0$ or $X^0_t\\neq 0$; any solution outside the theorem's list refutes the claimed exhaustiveness.","tokens_in":16518,"feed_emoji":"","tokens_out":6643,"duration_ms":58866,"temperature":0.7,"pith_summary":"The paper sets out to describe every point transformation that maps one equation of the form $u_t+u u_x=\\sum_{j=2}^{r}A_j(x)u_j+A_0(x)u+B(x)$ to another equation of the same form. It shows that the wider time-dependent Burgers–KdV family can be gauged to this reduced space-dependent class, whose usual equivalence group is only four-dimensional, and then asks whether subclasses admit larger equivalence groups. The main theorem answers this exhaustively: apart from a generic remainder, the class is a union of nine normalized subclasses, each carrying a larger conditional equivalence group (generalized except for the usual case $\\hat F_{II,0}$), and every admissible transformation is generated by these groups together with the four-dimensional group. This closes the structural classification of admissible transformations for the whole family and supplies new examples of classes normalized in the generalized sense.","feed_headline":"All point equivalences of space-dependent Burgers–KdV are mapped","feed_subtitle":"The full equivalence groupoid: a four-dimensional generic group plus nine larger exceptional subclasses.","key_machinery":"The load-bearing object is the equivalence groupoid of $F$: the collection of all triples $(\\theta,\\tilde\\theta,\\phi)$ where $\\phi$ is an invertible point transformation in $(t,x,u)$ sending the equation with coefficient tuple $\\theta$ to the equation with tuple $\\tilde\\theta$. The computations run through the classifying conditions (7)–(9), first-order linear differential equations obtained by differentiating the known equivalence transformations of the reduced class; solving these by furcate splitting, that is, treating monomials in $x$ and derivatives of the coefficients as independent, yields the exceptional subclasses. For subclasses where the resulting transformations depend on arbitrary elements of the equation, the paper uses generalized equivalence groups rather than usual ones, and for two hard cases it gauges a subclass by a family of equivalence transformations to a nicer normalized subclass, describes equivalences there, and composes with the gauging maps to obtain explicit group parameterizations.","core_discovery":"On its own terms, the paper's central claim is Theorem 1: the equivalence groupoid of the class $F$ of reduced general Burgers–KdV equations with space-dependent coefficients is completely described. The usual equivalence group $G^\\sim_F$ is the four-parameter family $\\tilde t=c_1t+c_2$, $\\tilde x=c_3x+c_4$, $\\tilde u=c_3u/c_1$, with corresponding transformations of the coefficients. The maximal nontrivial conditional equivalence subgroups are exactly the generalized equivalence groups of the eight normalized subclasses $\\hat F_{I,1}$, $\\hat F_{I,01}$, $\\hat F_{I,00}$, $\\hat F_{II,1}$, $F_{III}$, $F_{IV,1}$, $F^{r>2}_{IV,0}$, $F^{r=2}_{IV,0}$, together with the usual equivalence group of the normalized subclass $\\hat F_{II,0}$; every admissible transformation of $F$ is generated by these groups and the four-dimensional usual group. The complement $F_0$ is normalized in the usual sense and has the same equivalence group as $F$. This yields a decomposition of the whole class into a union of normalized subclasses and produces new examples of nontrivial normalization in the generalized sense.","pith_inferences":["The gauging-plus-normalization template used for $F_{I,00}$ and $F^{r=2}_{IV,0}$ could be applied to other evolution equations with space-dependent coefficients, such as variable-coefficient KdV or Kuramoto–Sivashinsky equations; the paper does not carry that out.","Because the exhaustiveness of Theorem 1 is imported from the furcate-splitting solution in [6], a computer-algebra re-derivation of the classifying conditions (7)–(9) for a small order such as $r=3$ would be a cheap independent check of the claimed completeness.","The non-uniqueness of effective generalized equivalence groups for $F_{II,1}$ and $F_{III}$ suggests that other classes may admit several different but equivalent parameterizations of the same groupoid; the choice of parameterization is then a modeling convenience rather than an invariant of the class."],"forward_implications":["Every admissible transformation of $F$ is now known: to test whether two equations in $F$ are related by a point transformation, it suffices to check the four-dimensional usual group and the listed exceptional subclasses.","The nine listed subclasses are the only places where the equivalence group is larger than the generic four-dimensional one; any subclass not captured by the list inherits the generic group.","The complement $F_0$ is a normalized class in the usual sense with the same four-dimensional equivalence group, so the generic part of the groupoid is completely understood.","The subclasses $F_{I,00}$ and $F^{r=2}_{IV,0}$ show how effective generalized equivalence groups can be built by gauging to a nicer normalized subclass, composing equivalence transformations there, and then returning through the gauging maps."],"supporting_citations":[{"why":"supplies the initial solution of the classifying conditions (7)–(9) by furcate splitting, whose completeness Theorem 1 inherits","marker":"[6]"},{"why":"establishes the framework of admissible transformations, normalized classes, and generalized equivalence groups used throughout","marker":"[9]"},{"why":"defines the classification problem for admissible transformations and the notion of equivalence groupoids","marker":"[8]"},{"why":"introduces generalized equivalence transformations, the notion whose nontrivial cases the paper studies","marker":"[4]"},{"why":"justifies smooth dependence of the time transformation $T$ on equation parameters and initial conditions, used to parameterize group elements","marker":"[1]"}],"fun_headline_variants":["Equivalence groupoid of space-dependent Burgers–KdV fully mapped","Four-dimensional usual group plus nine exceptions for Burgers–KdV","Every equivalence of space-dependent Burgers–KdV captured","Burgers–KdV equivalence groupoid: complete classification","Nine exceptional subclasses in Burgers–KdV equivalence mapped"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole classification rests on the completeness of an earlier solution of the classifying differential equations, which is quoted as a black box; if that solution overlooked any exceptional case, the list in Theorem 1 would be incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Equivalence groupoid of space-dependent Burgers–KdV fully mapped","Four-dimensional usual group plus nine exceptions for Burgers–KdV","Every equivalence of space-dependent Burgers–KdV captured","Burgers–KdV equivalence groupoid: complete classification","Nine exceptional subclasses in Burgers–KdV equivalence mapped"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000584,"raw_usage":{"total_tokens":2726,"prompt_tokens":907,"completion_tokens":1819,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":1731}},"tokens_in":523,"tokens_out":1819,"duration_ms":13207,"temperature":1.0,"reasoning_tokens":1731,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:03:57.279744+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find an equation in $F$ whose coefficient tuple lies outside all listed subclasses and $F_0$ and yet admits a point transformation with non-affine $T(t)$, $X^1(t)$, or $X^0(t)$, which would be a transformation not generated by the four-dimensional usual group. Concretely, for $r=3$, solve (7)–(9) with generic coefficients $A_2,A_3,A_0,B$ allowing $T_{tt}\\neq 0$ or $X^0_t\\neq 0$; any solution outside the theorem's list refutes the claimed exhaustiveness.","supporting_citations":[{"cited_title":"3, Institute of Mathematics, Kyiv, 2006, 239–254","cited_arxiv_id":null,"evidence_quote":"defines the classification problem for admissible transformations and the notion of equivalence groupoids"},{"cited_title":"Nonlinear Math","cited_arxiv_id":null,"evidence_quote":"introduces generalized equivalence transformations, the notion whose nontrivial cases the paper studies"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"justifies smooth dependence of the time transformation $T$ on equation parameters and initial conditions, used to parameterize group elements"}],"review_version":1}