{"id":"1d544992-113f-48dc-bed3-e27120beaa78","arxiv_id":"2608.09553","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For NLS on a strip with an attractive delta interaction, line solitons are orbitally stable below a critical width and unstable above, with a pitchfork branch of new positive solutions at the threshold.","lead":"This paper proves that a soliton in a two-dimensional waveguide with a crack line changes from stable to unstable at a critical width, where new stationary wave shapes appear. It gives rigorous criteria for when the transition happens and when the new wave shapes are stable.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The small-γ stability claim of Theorem 1.5 is not established: the γ=0 limit is singular (translation zero mode), the claimed operator-norm convergence of A_0^{-1} is false, and the final sign d²L/da²<0 contradicts the γ=0 computation and the theorem's own stability criterion.","rationale":"The central claim I stress-tested is not the pitchfork bifurcation itself (which appears carefully executed for fixed γ<0), but the final assertion in Theorem 1.5 that the bifurcating solitons are orbitally stable for sufficiently small |γ| and |a|. That assertion is the advertised payoff of the paper, and it rests entirely on the sign of d²L/da²(0) near γ=0. The proof gives two contradictory signals: the γ=0 calculation via (152), with λ'_*<0, indicates d²L/da²(0)>0, while the concluding line of the γ<0 case states the limit is <0. Since the stability criterion in Theorem 1.5 and Proposition 5.7 make stability equivalent to d²L/da²(0)>0, the text as written is internally inconsistent. The inconsistency is compounded by a genuine spectral degeneracy at γ=0: translation invariance puts φ'_{ω,0} in the kernel of L_{+,0}(0), and because this vector is orthogonal to ξ it is not removed by the projection onto H^2_ort. Hence the operator whose inverse appears in (89) is not invertible at γ=0, and the claimed Kato-type operator-norm convergence of A_0^{-1}(γ) is false. A rigorous repair would need to restrict to the even-in-x subspace and prove a uniform bound on the quadratic form ⟨f_γ, A_0(γ)^{-1}f_γ⟩; this is absent. The proposed concrete check settles the sign question directly and exposes the missing invertibility. I therefore agree with the reader's CONDITIONAL verdict: the bifurcation and threshold results appear structurally sound, but the small-γ stability conclusion is not yet reliably established.","tokens_in":44672,"tokens_out":17561,"duration_ms":157944,"concrete_test":"Recompute the γ=0 case of Eq. (89) using the explicit profiles φ_{ω,0} and χ0 = φ_{ω,0}^{(p+1)/2}: evaluate the two inner products in (89) by direct quadrature for p=3 (and also p=2, p=4) and determine the sign of d²L/da²(0). If the sign is positive, the final '<0' limit in the γ<0 case is a sign error and the stability conclusion could survive once an even-subspace generalized inverse is justified; if the sign is negative, the final 'moreover stable' claim of Theorem 1.5 is false. In either case, exhibit φ'_{ω,0} in the kernel of A_0(0) and in H^2_ort to confirm that the operator-norm convergence argument in the text as written is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The stability of the bifurcating branch for small |γ| is decided by the sign of d²L/da²(0), so the last part of the proof of Theorem 1.5 is load-bearing. Two problems occur in that step. First, at γ=0, L_{+,0}(0) = -∂xx + ω - pφ_{ω,0}^{p-1} has an additional kernel vector φ'_{ω,0} generated by translation invariance. This vector is orthogonal to ξ = √2 cos(πy)χ0 (odd in x, with ∫ cos(πy)dy = 0), so it lies in H^2_ort. Consequently, the operator (P⊥(L_{+,0}-L_*^{-2}∂yy)|_{H^2_ort})^{-1} in Eq. (89) is undefined at γ=0, and the assertion that A_0^{-1}(γ) converges in operator norm to A_0^{-1}(0) is false; the modal formula uses A_0^{-1}(0) on the even source f0, which requires a generalized inverse and a uniform bound that are not supplied. Second, the text concludes lim_{γ→0-} d²L/da²(0) < 0, but the preceding bound (152), together with λ'_* < 0, gives d²L/da²(0) > 0 at γ=0, and the stability statement in Theorem 1.5 requires d²L/da²(0) > 0. Thus the small-γ stability conclusion is neither correctly derived nor consistently stated, and the advertised 'moreover stable' claim is not justified as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the nonlinear Schrödinger equation on a strip R×[0,L] with Neumann boundary conditions and an attractive delta interaction on the x-axis. It establishes local well-posedness in H^1, then identifies a critical width L_* = π/√λ_1 at which the line soliton becomes transversely unstable: orbital stability for 0<L<L_*, orbital instability for L>L_*. At L=L_*, a simple eigenvalue of the linearized operator crosses zero and a C^2 branch of positive stationary solutions bifurcates in a pitchfork. The paper derives an explicit formula for the second derivative d²L/da²(0), which determines the direction of the branch and, through the Grillakis–Shatah–Strauss criterion, the stability of the bifurcating solutions. The final part claims that the bifurcating branch is orbitally stable for sufficiently small |γ| and |a|.","tokens_in":1508,"tokens_out":3140,"duration_ms":86903,"significance":"The main bifurcation and stability results are significant and of current interest: they provide a complete spectral mechanism for transverse instability in a geometrically nontrivial waveguide with a singular potential, and the Lyapunov–Schmidt reduction is carried out in detail. The identification of the critical length L_* and the Morse-index-based stability/instability criterion for the line soliton are clean and, conditional on the cited spectral facts, convincing. The paper also demonstrates sound technique in the well-posedness and instability arguments. However, the advertised small-γ stability of the bifurcating branch rests on a sign-contradictory and analytically singular limiting argument, so that the last theorem's 'moreover' claim is not justified as written.","major_comments":[{"comment":"There is a sign contradiction in the limit passage. Eq. (152) states λ'_* d²L/da²(0) ≤ (negative expression) for γ=0. Since λ'_* = -2L_*^{-3}π² < 0 by Eq. (65), this inequality implies d²L/da²(0) > 0 at γ=0. Yet the text immediately concludes 'lim_{γ→0-} d²L/da²(0) = ... < 0' and then 'd²L/da²(0)<0 for all γ∈(-γ_0,0]'. This contradicts the preceding bound. Moreover, Theorem 1.5 asserts stability for small |γ|, which requires d²L/da²(0)>0; the final '<0' would instead give instability. The sign in the γ=0 bound or in the continuity conclusion must be corrected, and the two parts of the proof currently disagree.","section":"§5.3, proof of Theorem 1.5, Case γ=0 and Case γ<0 (pp. 40–43)"},{"comment":"The perturbation argument from γ=0 is not justified because A_0(γ) is singular at γ=0. For γ=0, L_{+,0}(0) = -∂_xx + ω - pφ_{ω,0}^{p-1} has the translation zero mode φ'_{ω,0}, which is odd in x and therefore orthogonal to ξ = √2 cos(πy)χ_0, so φ'_{ω,0} lies in H^2_ort. Hence the operator P⊥(L_{+,0}(0)-L_*^{-2}∂_yy) restricted to H^2_ort is not invertible, and the claimed operator-norm convergence of A_n^{-1}(γ) to A_n^{-1}(0) fails for n=0. The modal formula uses A_0^{-1}(0) on the even source f0 = φ_{ω,0}^{2p-1}; this requires a generalized inverse on the even subspace and a uniform bound on A_0^{-1}(γ) as γ→0, neither of which is supplied. The assertion that the operators A_n(γ) with A_0 restricted to H^2_ort are uniformly coercive for small |γ| is false because the small eigenvalue of L_{+,0}(γ) tends to 0 as γ→0. Thus the continuity of d²L/da²(0) at γ=0 is not established.","section":"§5.3, Case γ<0 (pp. 42–43) and Eq. (89)"}],"minor_comments":[{"comment":"In the 'Moreover' sentence the family is denoted ϕ(a), while everywhere else the bifurcating family is called φ(a); please unify the notation.","section":"Theorem 1.5 statement"},{"comment":"The sentence 'where ϕ(a) is defined in' is incomplete; it should refer to the definition of φ(a) in Theorem 1.2 or Proposition 4.1.","section":"§5.3, Proposition 5.4"},{"comment":"The notation 'L−2∗∂yy' is a typographical artifact; it should read 'L_*^{-2}∂_yy' consistently with Eq. (42) and the rest of the paper.","section":"Eq. (89) and surrounding text"},{"comment":"References [24] and [26] are duplicated (the same Grillakis–Shatah–Strauss paper); please collapse them.","section":"References"},{"comment":"The statement 'For any n∈R, n≥0' should specify n integer; the proof treats Fourier modes n∈N∪{0}.","section":"Appendix B, Lemma B.2"}],"recommendation":"major_revision","confidential_remarks":"The reader's stress test correctly identifies a genuine flaw in the proof of Theorem 1.5: the small-γ stability claim is undermined by both a sign contradiction and the singular behavior of A_0^{-1} at γ=0. The main bifurcation theorem (Theorem 1.2) and the line-soliton stability theorem (Theorem 1.4) appear sound and are well argued, but the final 'moreover' statement of Theorem 1.5 needs either a corrected asymptotic argument avoiding the γ=0 singular limit or a qualification that the sign of d²L/da²(0) can only be determined for fixed γ<0 by direct computation. The paper is otherwise carefully written and the results are of sufficient interest for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you work on NLS in waveguides with point defects. The genuinely new material is the attractive-delta case gamma<0: the delta potential breaks the full scaling, so the authors use a partial transverse rescaling and a modified Lyapunov-Schmidt reduction to find the critical width L*=pi/sqrt(lambda1) at which a transverse eigenvalue crosses zero. The bifurcation construction, the well-posedness proof, and the line-soliton stability threshold for L<L* vs L>L* are detailed and largely follow standard Crandall-Rabinowitz and Grillakis-Shatah-Strauss machinery; the Strichartz estimates for the delta-perturbed operator are handled carefully.\n\nThe soft spot is the advertised stability of the bifurcating branch for small |gamma|. The last part of Theorem 1.5 is not established. There are three concrete problems. First, sign: at gamma=0 the paper derives a negative upper bound (152), and since lambda'_*<0 that gives d^2L/da^2(0)>0, yet the next displayed line states lim_{gamma->0-} d^2L/da^2(0)<0. That contradicts the theorem's own stability criterion, which requires d^2L/da^2(0)>0 for stability. Second, the perturbation argument for the inverse operator is not justified: at gamma=0 the linearized operator has the translation kernel phi'_omega,0, which lies in H^2_ort, so A_0(0) is not invertible on that space; the modal formula with <f_gamma, A_0^{-1}(gamma)f_gamma> needs a generalized inverse and a uniform bound that are not supplied. The claimed operator-norm convergence of A_0^{-1}(gamma) to A_0^{-1}(0) is false. Third, Proposition 5.4 assumes omega>gamma^2/2 while Theorems 1.4 and 1.5 state omega>gamma^2/4; this mismatch is unexplained. A smaller consistency issue: Appendix C normalizes chi with L^2-norm sqrt(2), contradicting ||chi||=1 used earlier.\n\nNone of this undermines the pitchfork construction or the L<L*/L>L* dichotomy; those parts look sound. But Theorem 1.5's 'moreover stable' claim should be either proven with a correct sign and a proper generalized inverse, or withdrawn. I would send this to a serious referee: the core analysis deserves scrutiny. I would not rely on the small-gamma stability conclusion as written.","headline":"Solid bifurcation analysis for the attractive delta strip, but the small-gamma stability conclusion in Theorem 1.5 has a sign contradiction and an unjustified inverse limit; the core pitchfork and line-soliton threshold still deserve review.","tokens_in":45577,"tokens_out":4856,"would_cite":true,"duration_ms":42875,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","35A15","35B38"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the nonlinear Schrödinger equation on a strip with an attractive line defect, the paper proves that the one-dimensional line soliton is orbitally stable below the critical width $L_*=\\pi/\\sqrt{\\lambda_1}$, unstable above it, and that…","keywords":["nonlinear Schrödinger equation","standing waves","action ground state","energy ground state","nonlinear quantum graphs","dimensional reduction","transverse instability","pitchfork bifurcation"],"falsifier":"A direct numerical test is to compute the lowest eigenvalues of $L_{+,0}$ for chosen $\\gamma<0$, $\\omega>\\gamma^2/4$, and $2\\le p<5$, then simulate the stationary strip equation (or the full NLS flow) for widths around $L_*=\\pi/\\sqrt{\\lambda_1}$: if the line soliton remains orbitally stable for some $L>L_*$, or if the bifurcating branch appears on the opposite side from the sign of $d^2L/da^2(0)$ — equivalently, if the second eigenvalue $\\lambda_2(a)$ of the linearized operator does not behave like $-d^2L/da^2(0)\\,\\lambda'_* a^2$ with $\\lambda'_*<0$ — then the threshold or the stability criterion fails. A cheaper spectral check consists in verifying numerically that $-\\lambda_1$ is simple and that the first transverse mode crosses zero exactly at $L_*$.","tokens_in":44379,"feed_emoji":"🌊","tokens_out":13824,"duration_ms":108281,"temperature":0.7,"pith_summary":"The paper studies the nonlinear Schrödinger equation on a two-dimensional strip $\\mathbb{R}\\times[0,L]$ with an attractive delta interaction along the $x$-axis, modelling a waveguide with a fracture, and asks how the strip width $L$ affects the standing waves inherited from the one-dimensional problem. Its central claim is that the line soliton — the one-dimensional ground state of the delta-potential NLS, extended constantly in the transverse direction — is orbitally stable for $0<L<L_*=\\pi/\\sqrt{\\lambda_1}$ and is not orbitally stable for $L>L_*$, where $-\\lambda_1$ is the unique negative eigenvalue of the one-dimensional linearized operator at the soliton. At the critical width this eigenvalue crosses zero, and the paper constructs, by the standard reduction to a scalar bifurcation equation, a $C^2$ branch of positive two-dimensional stationary solutions that emerges from the line soliton in a pitchfork. The direction of the pitchfork and the orbital stability of the emerging states are both controlled by one explicitly computed coefficient, $d^2L/da^2(0)$, whose sign the paper evaluates for small attractive coupling and finds to be positive. A sympathetic reader should care because the paper turns a geometric parameter (strip width) into a sharp, computable stability threshold and predicts that the unstable line soliton is replaced by a genuinely two-dimensional stable profile.","feed_headline":"One critical width decides soliton stability on a strip","feed_subtitle":"Below the width the line soliton holds; above it a stable 2D branch bifurcates.","key_machinery":"The load-bearing object is the one-dimensional linearized operator $L_{+,0}=-\\partial_{xx}+\\omega+\\gamma\\delta_0-p\\phi_{\\omega,\\gamma}^{p-1}$ around the ground state $\\phi_{\\omega,\\gamma}$; its lowest eigenvalue $-\\lambda_1$ is assumed simple and spectrally isolated, and the whole argument is organized around that single number. After separating variables in the transverse direction, the $n$-th Fourier mode of the linearized operator on the strip is $L_{+,0}+(n\\pi/L)^2$, so the first mode crosses zero exactly when $L$ equals $L_*=\\pi/\\sqrt{\\lambda_1}$. The bifurcation analysis reduces the stationary problem $H_L u+\\omega u-|u|^{p-1}u=0$ to a scalar equation on the kernel span of $\\xi=\\sqrt{2}\\cos(\\pi y)\\chi_\\gamma$; the auxiliary (infinite-dimensional) part is solved by the implicit function theorem, and the remaining scalar bifurcation function $g(a,L)$ satisfies $g(0,L_*)=0$, $\\partial_a g(0,L_*)=0$, and $\\partial_L g(0,L_*) = \\lambda'_* = -2\\pi^2 L_*^{-3}<0$, which is exactly the transversality condition for a pitchfork. The second derivative of the width, $d^2L/da^2(0)$, is computed by differentiating the reduced equation twice; its explicit formula contains an integral term and an inner product with the inverse of $L_{+,0}+(2\\pi/L_*)^2$ on the orthogonal complement of the kernel, and its sign is fixed by continuity from the exactly solvable case $\\gamma=0$, where the sign is positive.","core_discovery":"The discovery is a transverse pitchfork bifurcation with a sharp stability threshold. For any attractive coupling $\\gamma<0$, frequency $\\omega>\\gamma^2/4$, and power nonlinearity with $1<p<5$, the line soliton $\\phi_{\\omega,\\gamma}(x)$ solves the strip equation because it is independent of $y$. The paper proves that this solution is orbitally stable exactly for widths $0<L<L_*=\\pi/\\sqrt{\\lambda_1}$ and is orbitally unstable for $L>L_*$; here $-\\lambda_1$ is the smallest, simple, isolated eigenvalue of the one-dimensional linearized operator $L_{+,0}=-\\partial_{xx}+\\omega+\\gamma\\delta_0-p\\phi_{\\omega,\\gamma}^{p-1}$. At $L=L_*$ the first transverse mode $\\xi=\\sqrt{2}\\cos(\\pi y)\\chi_\\gamma$ (with $\\chi_\\gamma$ the normalized eigenfunction of $L_{+,0}$) enters the kernel of the linearized operator on the strip, and a standard reduction to the one-dimensional kernel yields a $C^2$ family of positive stationary solutions $\\varphi(a)$ with width $L(a)=L_*+\\frac12\\frac{d^2L}{da^2}(0)a^2+o(a^2)$ and profile $\\varphi(a)=\\phi_{\\omega,\\gamma}+a\\xi+O(a^2)$. The sign of the coefficient $d^2L/da^2(0)$, which the paper computes in closed form and shows to be positive for all sufficiently small attractive couplings, determines both the side on which the branch opens and, through the Morse index of the linearized action, whether the bifurcating solitons are orbitally stable or unstable.","pith_inferences":["The threshold mechanism is purely geometric — only the spacing of Neumann transverse modes and the position of the single negative eigenvalue of $L_{+,0}$ enter — so an analogous stability threshold should appear for any confinement with a discrete transverse spectrum, not just the flat strip, whenever the longitudinal linearized operator has one simple negative eigenvalue.","Because the stability conclusion for small $|\\gamma|$ is obtained by perturbation from $\\gamma=0$, a natural numerical test is to evaluate the closed-form formula for $d^2L/da^2(0)$ at moderate couplings; a sign change away from $\\gamma=0$ would mark a regime where the bifurcating branch becomes unstable, which is not covered by the paper's small-$\\gamma$ statement.","The result suggests a concrete observable in waveguide experiments: as the transverse width is tuned past $L_*$, the otherwise uniform (in $y$) soliton should spontaneously develop a $\\cos(\\pi y/L)$ modulation, since that is the kernel mode that drives the pitchfork.","The same reduction could be applied to the repulsive case $\\gamma>0$, although the line soliton is already unstable on the line; the bifurcation analysis would then describe additional branches rather than a stability threshold."],"forward_implications":["For $0<L<L_*$ the line soliton is orbitally stable; for $L>L_*$ it is linearly unstable and the instability persists nonlinearly, so the threshold is a true dynamical transition, not merely a spectral event.","At $L=L_*$ the linearized operator has a one-dimensional kernel spanned by the first transverse mode, so the bifurcation is necessarily a steady pitchfork with width changing quadratically in the amplitude.","The sign of $d^2L/da^2(0)$ decides everything: it selects the side of $L_*$ on which the new branch exists, fixes the Morse index of the linearized action, and therefore determines whether the bifurcating two-dimensional solitons are orbitally stable.","For sufficiently small attractive coupling and small amplitude, the coefficient is positive, so the bifurcating branch is made of orbitally stable two-dimensional solitons that replace the unstable line soliton.","The paper also closes the local well-posedness loop for the model: well-posedness in $H^1$, mass/energy conservation, and global existence for $p<3$ follow from Strichartz estimates adapted to the strip."],"supporting_citations":[{"why":"Constructs ground states on the fractured strip and establishes the length-dependent dimensional reduction that the line-soliton analysis starts from.","marker":"[13]"},{"why":"Supplies the spectral properties of the one-dimensional linearized operator $L_{+,0}$: the simple isolated negative eigenvalue $-\\lambda_1$ and the spectral gap used to locate the critical width.","marker":"[33]"},{"why":"Provides the symmetry-breaking bifurcation method that the paper adapts to the strip geometry.","marker":"[32]"},{"why":"Provides the normal-form computation for the symmetry-breaking bifurcation, used to derive the second derivative of the width.","marker":"[34]"},{"why":"Gives the variational orbital-stability criterion (Morse index plus slope condition) used to prove stability and instability of the line soliton.","marker":"[24]"},{"why":"Supplies the framework to upgrade linear instability to nonlinear orbital instability for the supercritical-width regime.","marker":"[19]"},{"why":"Provides the exponential decay estimates for positive solutions in cylindrical domains used to justify the dominated-convergence steps in the second-derivative computation.","marker":"[6]"},{"why":"The transversality theorem for bifurcation from a simple eigenvalue that justifies the pitchfork branch.","marker":"[14]"},{"why":"Treats the unperturbed $\\gamma=0$ case and supplies the anchor computation from which the sign of $d^2L/da^2(0)$ is obtained by continuity.","marker":"[40]"}],"fun_headline_variants":["Critical width flips soliton stability on strip","Strip width controls line soliton stability","Bifurcation at critical width for NLS solitons","Line soliton destabilizes past critical width","Pitchfork bifurcation sets soliton stability limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the one-dimensional stability problem has exactly one negative direction — the eigenvalue $-\\lambda_1$ — and that this direction is isolated from the rest of the spectrum; if a second nearly-crossing direction existed near the critical width, the reduction to a single bifurcation equation and the predicted stability threshold would both fail.","fun_headline_variants_meta":{"raw":{"variants":["Critical width flips soliton stability on strip","Strip width controls line soliton stability","Bifurcation at critical width for NLS solitons","Line soliton destabilizes past critical width","Pitchfork bifurcation sets soliton stability limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000518,"raw_usage":{"total_tokens":2598,"prompt_tokens":1121,"completion_tokens":1477,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":737,"completion_tokens_details":{"reasoning_tokens":1404}},"tokens_in":737,"tokens_out":1477,"duration_ms":9526,"temperature":1.0,"reasoning_tokens":1404,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:14:39.419114+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical test is to compute the lowest eigenvalues of $L_{+,0}$ for chosen $\\gamma<0$, $\\omega>\\gamma^2/4$, and $2\\le p<5$, then simulate the stationary strip equation (or the full NLS flow) for widths around $L_*=\\pi/\\sqrt{\\lambda_1}$: if the line soliton remains orbitally stable for some $L>L_*$, or if the bifurcating branch appears on the opposite side from the sign of $d^2L/da^2(0)$ — equivalently, if the second eigenvalue $\\lambda_2(a)$ of the linearized operator does not behave like $-d^2L/da^2(0)\\,\\lambda'_* a^2$ with $\\lambda'_*<0$ — then the threshold or the stability criterion fails. A cheaper spectral check consists in verifying numerically that $-\\lambda_1$ is simple and that the first transverse mode crosses zero exactly at $L_*$.","supporting_citations":[{"cited_title":"Ground states on a fractured strip and one dimensional reduction","cited_arxiv_id":"2411.18187","evidence_quote":"Constructs ground states on the fractured strip and establishes the length-dependent dimensional reduction that the line-soliton analysis starts from."},{"cited_title":"Le Coz, R","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral properties of the one-dimensional linearized operator $L_{+,0}$: the simple isolated negative eigenvalue $-\\lambda_1$ and the spectral gap used to locate the critical width."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the symmetry-breaking bifurcation method that the paper adapts to the strip geometry."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the normal-form computation for the symmetry-breaking bifurcation, used to derive the second derivative of the width."},{"cited_title":"Grillakis, J","cited_arxiv_id":null,"evidence_quote":"Gives the variational orbital-stability criterion (Morse index plus slope condition) used to prove stability and instability of the line soliton."},{"cited_title":"Georgiev and M","cited_arxiv_id":null,"evidence_quote":"Supplies the framework to upgrade linear instability to nonlinear orbital instability for the supercritical-width regime."},{"cited_title":"Berestycki and L","cited_arxiv_id":null,"evidence_quote":"Provides the exponential decay estimates for positive solutions in cylindrical domains used to justify the dominated-convergence steps in the second-derivative computation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The transversality theorem for bifurcation from a simple eigenvalue that justifies the pitchfork branch."},{"cited_title":"Yamazaki","cited_arxiv_id":null,"evidence_quote":"Treats the unperturbed $\\gamma=0$ case and supplies the anchor computation from which the sign of $d^2L/da^2(0)$ is obtained by continuity."}],"review_version":1}