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Instability of gray solitons in a Gross-Pitaevskii model with a moving impurity

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Pith's one-line read Gray modes that sit at maxima of the impurity's effective potential are spectrally unstable for every subsonic nonzero speed.

desk verdict Solid Evans-function proof that gray modes bifurcating from maxima of the effective potential are spectrally unstable for small positive ε; first rigorous treatment of the non-off-diagonal case. read the letter →

arxiv 2607.08190 v1 pith:7SBASU7Z submitted 2026-07-09 math.AP math-phmath.MP

Paolo Antonelli , Martino Caliaro This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35Q5535B3535B3237K4537K50
keywords Gross-PitaevskiiequationgraysolitonmovingimpurityEvansfunctionspectralinstabilityeffectivepotentialsuperfluidityedgebifurcation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies a one-dimensional Gross-Pitaevskii condensate that contains a localized impurity moving at constant subsonic speed. In the co-moving frame it constructs a family of steady gray modes that bifurcate from a translated gray soliton; the translation is fixed by a critical point of an effective potential built from the impurity profile. Using an Evans-function expansion around the origin, the authors then prove that whenever that critical point is a non-degenerate maximum and the impurity strength is small and positive, the linearized operator about the gray mode possesses a pair of eigenvalues with positive real part. The same formal calculation applied to a repulsive delta impurity yields the identical instability. The result therefore supplies a rigorous spectral obstruction to frictionless motion of the impurity once the gray mode has formed.

What carries the argument

The Evans function E(λ,ε) associated with the first-order spatial system of the linearized operator L_ε; after analytic continuation through a neighborhood of the origin it admits the expansion E(λ,ε) = -2P_r'(v)λ^{3} - 2M''(s0)ελ + higher-order terms whose zeros locate the unstable eigenvalues.

What would settle it

Compute the spectrum of the linearized operator about an explicit gray mode for a smooth, even, repulsive Gaussian potential of small amplitude and check whether a conjugate pair of eigenvalues with positive real part appears precisely when M''(0)<0.

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Extended reading notes

Core claim

For every nonzero subsonic velocity and every simple critical point s0 of the effective potential M at which M''(s0)<0, the family of gray modes that bifurcates from the gray soliton centered at s0 is spectrally unstable for all sufficiently small positive impurity strengths; the unstable eigenvalues satisfy λ² = -(M''(s0)/P_r'(v))ε + O(ε^{3/2}).

Load-bearing premise

The Evans function can be continued analytically through a neighborhood of zero that intersects the essential spectrum, so that its Taylor expansion remains valid for the edge bifurcation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies the one-dimensional Gross-Pitaevskii equation with a smooth, exponentially localized potential moving at constant subsonic speed v eq 0. In the co-moving frame it proves, via Lyapunov-Schmidt reduction on the hydrodynamic formulation, the existence of a family of gray modes ϕ_ε,v(x-s_ε) that bifurcate from a translated gray soliton ϕ_0,v(x-s_0), where s_0 is a simple critical point of the effective potential M(s)=∫V(x)[1-|ϕ_0,v(x-s)|^{2}]dx (Proposition 1.1 / 2.1). Spectral stability is analyzed by linearization about these modes and construction of an Evans function E(λ,ε). After establishing analytic continuation of E through a neighborhood of the origin that intersects the essential spectrum (Lemmas 4.2–4.7 and Appendix A), a third-order Taylor expansion is obtained: E(λ,ε)=-2P_r'(v)λ^{3}-2M''(s_0)ελ+O(λ^{4},λ^{2}ε,λε^{2}). Consequently, when M''(s_0)<0 and ε>0 is small, a pair of eigenvalues satisfies λ^{2}=-(M''(s_0)/P_r'(v))ε+O(ε^{3/2}), proving spectral instability (Corollary 1.3 / 5.4). The same formal expansion is applied to a repulsive delta impurity.

Significance. The work supplies the first rigorous spectral-instability proof for gray modes bifurcating from gray solitons under a moving impurity, extending the black-soliton analysis of Pelinovsky–Kevrekidis to the genuinely complex, non-off-diagonal case v eq0. The effective-potential criterion M''(s_0)<0 is explicit, parameter-free, and immediately applicable to the physically relevant Gaussian laser potential. The Evans-function construction, including the verification that the asymptotic eigenvalues form period-one cycles and the Gap-Lemma continuation, is carried out with complete coefficient computations; these technical ingredients are of independent interest for edge bifurcations in non-self-adjoint Hamiltonian systems. The formal delta-potential application recovers the leading-order formula previously observed numerically, closing a gap between rigorous analysis and the physics literature on superfluid drag.

minor comments (4)
  1. In the expansion of Ep(λ) at v=0 (Remark 4.7) the authors note a factor-of-two discrepancy with Kapitula–Rubin; a short sentence clarifying that the missing contribution is precisely the term (118) would help readers reconciling the two calculations.
  2. Figure 1 shows M''(0) versus σ only for v=0.5; a brief remark that the sign remains negative for all v otin{0} (by the change of variables already mentioned) would make the Gaussian example self-contained.
  3. The formal transfer to the delta potential (Section 6) is clearly labeled non-rigorous; a single sentence indicating that a rigorous justification would require eigenvalue convergence under approximation of δ by smooth potentials would forestall any misreading.
  4. Typographical: several instances of “Frech´et” and “Mari¸s” retain residual encoding artifacts; a global clean-up of accents would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: existence via Lyapunov-Schmidt and instability via Evans-function expansion are self-contained first-principles calculations.

full rationale

The paper derives existence of gray modes by a standard Lyapunov-Schmidt reduction applied to the hydrodynamical form of the stationary GP equation (Proposition 2.1 / 1.1). The only non-degeneracy condition is that s0 is a simple root of the explicitly defined effective potential M'(s)=0; this is a Fredholm alternative, not a fitted parameter. Spectral instability is obtained by constructing the Evans function for the linearized operator L_ε, proving its analytic continuation through a neighborhood of λ=0 (Lemmas 4.2–4.5, 4.7 and Appendix A, using the Gap Lemma of Kapitula–Sandstede), and computing the leading Taylor coefficients explicitly from the unperturbed gray soliton and the given potential V (Propositions 4.10 and 5.3). The resulting eigenvalue asymptotics λ^{2} = −(M''(s0)/P_r'(v))ε + O(ε^{3/2}) (Corollary 5.4) therefore follow by direct expansion; none of the coefficients is fitted to data or assumed a priori. Self-citations ([4], [32]) supply background orbital-stability results or the black-soliton case and are not load-bearing for the new gray-soliton instability. The formal transfer to a delta impurity is clearly labeled non-rigorous and does not enter the main theorems. Consequently the derivation chain contains no self-definitional step, no fitted-input-called-prediction, and no uniqueness imported solely from the authors.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The work rests on standard functional-analytic and ODE tools plus the modeling assumptions of the 1D Gross-Pitaevskii equation with exponentially localized potential. No free parameters are fitted; the only 'invented' objects are the effective potential M and the gray-mode class, both defined explicitly from existing objects.

assumptions (4)
  • domain assumption The external potential V is smooth and decays exponentially at infinity.
    Stated in the introduction and used throughout to guarantee essential spectrum and exponential decay of solutions; without it the asymptotic matrices and Gap-Lemma arguments fail.
  • domain assumption The velocity v lies in the open subsonic interval (-1,1)\{0}.
    Required for the existence of gray solitons and for the spectral gap structure of the unperturbed operator; the paper never claims results outside this regime.
  • standard math Zero is a simple eigenvalue of the linearized hydrodynamic operator L (Kerp L = span{∂x ρ0}).
    Invoked in the Lyapunov-Schmidt reduction (after (23)); justified by reference to standard results on the black/gray soliton linearization.
  • standard math The four eigenvalues of the asymptotic matrices M±(λ) form cycles of period one near λ=0, allowing holomorphic extension of eigenvectors and of the Evans function.
    Proved in Lemmas 4.4-4.5 by contradiction using the characteristic polynomial; this is the key technical hypothesis for analytic continuation across the essential spectrum.
invented entities (2)
  • effective potential M(s) = ∫ V(x)[1-|ϕ0,v(x-s)|²] dx independent evidence
    purpose: Locates the bifurcation points s0 of gray modes and supplies the sign that decides stability.
    Defined explicitly from the unperturbed soliton and the given V; its critical points are the only free geometric data of the problem.
  • gray mode (Definition 1.1) independent evidence
    purpose: Class of nowhere-vanishing steady states with well-defined phase limits at ±∞ that contains the bifurcating family.
    A convenient name for solutions already known to exist formally; no new physical object is postulated.

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Pith. "Pith review of Instability of gray solitons in a Gross-Pitaevskii model with a moving impurity." pith.science (2026). https://pith.science/paper/7SBASU7Z

@misc{pith2026260708190,
  author       = {Pith},
  title        = {Pith review of: Instability of gray solitons in a Gross-Pitaevskii model with a moving impurity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7SBASU7Z}},
  note         = {Machine review of arXiv:2607.08190}
}
abstract

The effect of a moving impurity in a dilute Bose-Einstein condensate is investigated by means of the one-dimensional Gross-Pitaevskii model (GP) with non-zero boundary conditions at infinity. The impurity is modeled as a localized external potential, that travels at constant speed $v \in \mathbf{R}$. In a co-moving reference frame, we study the existence and stability of time-independent solutions. The latter are of physical relevance, being associated with the superfluid behavior of the condensate. For every non-zero velocity $v$ in the subsonic regime, we show the existence of a family of time-independent solutions which bifurcates from a (displaced) gray soliton $\phi_{0,v}(x-s_0)$, with $s_0 \in \mathbf{R}$, of the GP equation. The position $s_0$ is determined as an extremal point of an effective potential explicitly defined. Moreover, we study the spectral stability of these states. For small values of the potential strength, we show that the families originating from the maxima of the effective potential are spectrally unstable. For this last result, we employ an Evans function approach. Finally, we formally apply the instability result to the case of a repulsive delta potential.

Figures

Figures reproduced from arXiv: 2607.08190 by the authors.

Figure 1
Figure 1. Plot of M2 p0q as a function of σ for the case of the gaussian potential V1, and for a velocity v “ 0.5. We conclude this section by considering the case of the Gaussian external potential V1 defined in (13). As we mentioned in the introduction, this kind of potential is of physical relevance, as it models the repulsive effect of a laser beam moving on the BEC. As we will see below, the gray modes which bifurcate fr… view at source ↗

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Works this paper leans on

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