Papers

We revisit the pathwise global well-posedness issue of the modulated Korteweg-de Vries equation (KdV) on the circle. In the previous work (2024), by combining the $I$-method and the sewing lemma, the second and fourth authors with C. Chouk, G. Li, and J. Li proved its global well-posedness in negative Sobolev spaces. This result was, however, restricted to the scaling subcritical regime $s > - \frac 32$ due to the use of the classical KdV scaling. In this paper, by noting that the modulated KdV enjoys additional one degree of freedom in its scaling symmetry thanks to the modulation term, we apply a non-KdV scaling to the unknown and prove that, given any $s \in \mathbb R$, the modulated KdV on the circle with a sufficiently irregular modulation is globally well-posed in $H^s(\mathbb T)$, thus going beyond the barrier of the scaling critical regularity $s = - \frac 32$.

We formalize a research result in the Lean 4 proof assistant by having a mathematician direct an AI system, and frame the activity as a formalization game. The objective is to turn a LaTeX document into Lean. The game is won when the development compiles, contains no sorry, and a machine check shows the target theorems rest on Lean's foundational axioms alone. Reuse is a second check, by a definition we introduce: whether the development yields a self-contained layer of general mathematics the wider library could absorb. The case study is a complete, axiom-clean formalization of well-posedness for the nonlinear Vlasov equation via Dobrushin's mean-field route – existence, uniqueness, the stability estimate and mean-field limit, and a short-window superposition principle (weak solutions are Lagrangian). The human's role was to direct, not to write proofs: to scope the definitions, steer the decompositions, and triage the library's gaps; the AI agent executed. The formalization certifies the proof of each statement as written; whether the written statement is the intended theorem stays the mathematician's judgment. The optimal-transport machinery that fell out of the build (in particular, properties of the Wasserstein-1 metric and the Kantorovich-Rubinstein duality theorem) separates into a self-contained layer that compiles against Mathlib alone: about a sixth of the development (49 of 299 declarations), behind a 22-declaration interface with no reverse dependency. The headline theorems ran in about a week, the full development in about a month. We report the quantitative claims as observations of one game, not as general laws. The game's rules name no particular system, so the methodological framing is meant to outlast the tools of any one run.

We determine the asymptotics of a game inspired by classic vector balancing problems in combinatorial discrepancy theory. In this game, which we call the online Komlós game, two players, Paul and Carol, update the state vector $y$ in $\mathbb{R}^m$, initially placed at $0$. At each round, Paul chooses freely a set of $n$ vectors in the Euclidean unit ball, and Carol chooses, for each such vector, whether to leave it unchanged or reverse its sign. The resulting vectors are all added to $y$, and the game proceeds to a new round. After $T$ rounds, the game ends, and the $\ell_\infty$ norm of the state vector $y$ is determined. Paul's objective throughout the game is to maximize this norm, and Carol's objective is to minimize it. As $T$ gets large, we establish that the leading order term of the value of this game is $\sqrt{T/2\tau}$, where $\tau$ is the extinction time of the unit cube in $\mathbb{R}^m$ under a curvature-based flow characterized by the values of $m$ and $n$. When $n\geq m-1$, this flow is the mean curvature flow, and we show that $1/\sqrt{2\tau} =\Theta(\sqrt{\log m})$. Our results build upon the work of Kohn and Serfaty on deterministic games and mean curvature flow, combined with Banaszczyk's $\ell^2$ analogue of the Beck-Fiala theorem. As the large $T$ limit of the online Komlós game amounts to a localization of the classic Komlós problem, we hope this work can shed light on this and other vector balancing problems. Our results generalize to the version of the online Komlós game with the final value given by an arbitrary norm in $\mathbb{R}^m$.

We develop the regularity theory of viscosity solutions to degenerate fully nonlinear free transmission problems with Hamiltonian terms. By framing the equation in the context of viscosity inequalities, we establish local Hölder regularity of the gradient. In addition, based on a new improved oscillation-type estimate combined with a localized analysis, we obtain sharp pointwise $C^{1,\alpha}$ regularity.

In this manuscript we consider the steady state problem for a chemotaxis-consumption model with positive inflow of a nutrient across the boundary. We show that in any bounded smooth domain there exists a unique nonconstant steady state provided that the mass of bacteria is sufficiently large. We further prove that, if the domain is a ball, this condition can be dropped and solutions exist for any positive bacterial mass.

This paper is devoted to the sharp well-posedness and ill-posedness of the Cauchy problem for the Camassa-Holm (CH) equation in critical Triebel-Lizorkin spaces $F^{1+\frac{1}{p}}_{p,q}(\mathbb{R})$ with $(p,q)\in[1,\infty)\times[1,\infty]$ or $p=q=\infty$. On the one hand, we establish the local well-posedness in the sense of Hadamard in $F^2_{1,q}(\mathbb{R})$ for $1\leq q<\infty$ via Lagrangian coordinate transformation. On the other hand, by means of smooth atomic decomposition, strong ill-posedness is then proved in $F^{1+\frac{1}{p}}_{p,q}(\mathbb{R})$ with $(p,q)\in(1,\infty)\times[1,\infty]$ or $p=q=\infty$ in the sense of norm inflation, which in particular yields the ill-posedness of CH in critical Sobolev spaces $W^{1+\frac{1}{p},p}(\mathbb{R})$ with $1<p<\infty$, and provides a new perspective on the ill-posedness of CH in $H^{\frac{3}{2}}(\mathbb{R})$.

The well-known Shallow Water Equations (SWE) are used for modeling incompressible free-surface flows whenever the shallowness allows for a vertical-averaging; i.e., vertical effects are negligible in comparison to horizontal ones. But vertical averaging comes with the price of losing information along the vertical axis. Moment models for shallow flow contain information on the vertical velocity and pressure profile despite being dimensionally reduced. A class of these models incorporating a non-hydrostatic pressure have been introduced before as Dispersive Shallow Moment Models (DSM). However, no method for solving the non-stationary equations has been presented yet, mainly because it was unclear how to compute the pressure equation in the form of the divergence-free constraint. We rewrite the pressure equations of the DSM models in the form of a Poisson-like problem to enable their solution with a projection-type splitting scheme. For the linear equations, we present the calculations for the generalized model and discuss the non-linear case. We state the first two linear models and the corresponding nonlinear counterparts. Finally, we introduce a hybrid Finite-Volume Finite-Difference method and discuss the non-stationary numerical results for an experiment with periodic boundary and uneven bottom topography.

We present a geometric-analytic mechanism for the suppression of finite-time singularities in the 3D incompressible (unforced) Navier-Stokes equations for critical point singularities exhibiting $L^{3/2, \infty}$ spatial concentration of vorticity. We demonstrate that if the vorticity direction resides locally in a logarithmically weighted space of bounded mean oscillations, $\mathrm{bmo}_{1/|\log r|}$ – a space failing the Dini condition and thus permitting wild oscillatory defects – the nonlinear vortex stretching is fundamentally depleted. By isolating a unidirectional geometric cancellation, we recast the stretching eigenvalue as a singular integral commutator. Utilizing a localized Coifman-Rochberg-Weiss estimate coupled with dyadic BMO tail bounds, we prove the stretching potential vanishes as a logarithmic envelope on shrinking super-level sets. This depletion forces the vorticity magnitude into a sub-critical Lorentz-Zygmund space via interpolated De Giorgi energy method. The logarithmic gain is subsequently transferred to the velocity field, forcing the geometric scale of local 1D sparseness below the uniform radius of spatial analyticity, ultimately averting the finite time blow-up via the harmonic measure maximum principle.

We give a self-contained short proof of optimal regularity for minimizers of the Alt-Phillips Free Boundary Problem for $\gamma \in (0, 1)$. We adopt a dichotomy argument that originates from [DS20].

We prove that $S^3$ endowed with an arbitrary Riemannian metric $g$ admits at least two embedded minimal spheres. The proof is based on an iterative scheme of relative min-max constructions.

This work investigates time-harmonic electromagnetic scattering governed by the Maxwell system, where bounded anisotropic scatterers are embedded in a homogeneous electromagnetic background. We focus on the localized enhancement of the gradients of the total electric and magnetic fields in small boundary-attached neighborhoods of finitely many prescribed points near boundaries of anisotropic electromagnetic scatterers. We show that, through a suitable construction of incident electromagnetic waves, the gradients of both the total electric field and the total magnetic field can be made arbitrarily large in these neighborhoods. The main strategy is based on the introduction of auxiliary boundary-attached electromagnetic neighborhoods and the associated electric and magnetic fields, which exhibit strong gradient variation near the prescribed points. Using the approximation property of Maxwell Herglotz wave functions, these auxiliary fields are then approximated by physically admissible incident waves in the neighborhood of the scatterers. Together with the well-posedness and continuous dependence of the anisotropic scattering problem, this implies that the corresponding scattered field can be controlled to be sufficiently weak in the relevant region. Consequently, the total field is dominated by the incident field near the prescribed points and inherits its large-gradient behavior. The result provides a theoretical mechanism for localized gradient enhancement in anisotropic electromagnetic scattering and may have implications for field concentration, high-resolution probing, and sensitivity analysis of electromagnetic responses in complex media.

This paper investigates the behaviour of a fluid characterized by a viscosity simultaneously depending on pressure and shear rate within a Hele-Shaw cell featuring a sharp corner geometry. The study extends previous analyses conducted on purely pressure-dependent (piezo-viscous) and yield-stress fluids, providing a new perspective on confined complex flows. Motivated by practical applications related to designing biomedical devices and flows of relevance to biomedicine area, thin film technologies, injection molding – to name only a few – the flow configuration considered here can highlight essential features of complex fluid behavior in narrow-gap geometries around a sharp edge. Starting from the governing equations for an incompressible generalized Newtonian fluid and employing an appropriate rheological model, we derive the modified flow equations adapted to the Hele-Shaw flow. A particular solution is obtained near the corner region. Numerical simulations complement the theoretical results, illustrating the influence of the rheological parameters on the flow behavior.

The dynamics of a mathematical model for a chemostat-type reactor is investigated. The model describes the temporal evolution of suspended and wall-attached bacterial populations, with the latter represented as a one-dimensional biofilm, subject to a non-reproducing growth-limiting substrate and a reaction product formed through bacterial utilization of the substrate. In particular, it is shown that, in the regime where the trivial (washout) equilibrium is unstable, there exists a unique nontrivial equilibrium that is locally asymptotically stable. Under slightly stronger assumptions, uniform persistence and global asymptotic stability of the nontrivial equilibrium are established.

The SIGEST paper Nielsen and Strakoš (2024) characterized the spectrum of the preconditioned operator $\Delta^{-1}[\nabla \cdot (K\nabla u)]$ in a bounded open two-dimensional domain $\Omega$, where $\Delta$ denotes the Laplacian and $K(x,y)$ is a continuous symmetric matrix-valued function. An important part of the analysis states that for a diagonal tensor $K$ constant in an open subdomain $S \subset \Omega$, the closed interval defined by its diagonal elements belongs to the spectrum of the preconditioned operator. This result is correct, but the proof in Nielsen and Strakoš (2024) must be refined. This paper presents a refined proof and extends the previous work. As shown in the cited papers, for any point $\lambda$ in the open interval defined by the elements of the diagonal tensor constant in $S$ and any point $(x_0,y_0)\in S$, a rectangular subdomain $\Sigma_l\subset S$ can be constructed such that the generalized eigenvalue problem associated with the preconditioned operator restricted to $\Sigma_l$, of arbitrarily small size, has the eigenvalue $\lambda$ and infinitely many eigenfunctions. These are given by solutions of a locally defined wave equation. However, such solutions of the locally restricted generalized eigenvalue problem cannot be extended to the whole domain $\Omega$. Using instead rectangular subdomains whose size shrinks to zero, the present paper constructs a Weyl singular sequence of approximate eigenfunctions associated with $\lambda$, proving that $\lambda$ belongs to the spectrum of the preconditioned operator. Since self-adjoint operators in a separable Hilbert space can have at most a countable set of eigenvalues, this shows that the eigenvalues of the locally defined operator converge to points of the continuous spectrum of the preconditioned operator on the entire domain.

We initiate the study of Lyapunov-type inequalities for Dirichlet problems driven by the discrete p-Laplacian on weighted graphs. The approach is capacitary and is based on point p-capacities and the associated capacitary radii. First, we prove general Lyapunov-type inequalities on arbitrary connected locally finite weighted graphs. These inequalities provide intrinsic lower bounds, expressed in terms of the capacitary radii, for the positive part of the potential whenever the corresponding Dirichlet problem admits a nontrivial solution. Next, we estimate these capacitary radii in several geometric settings and prove the sharpness of the resulting Lyapunov-type inequalities. As an application, we derive lower bounds for the first weighted Dirichlet eigenvalue of the discrete p-Laplacian.

We investigate the stability and asymptotic behavior of spatially periodic cnoidal waves in the Korteweg-de Vries equation subject to localized perturbations. Standard stability arguments in Hamiltonian systems break down in this setting, since localized perturbations preclude a characterization of stable periodic waves as strict minimizers of a suitable energy functional subject to finitely many constraints. As a result, the nonlinear stability of periodic waves under localized perturbations has remained a long-standing open problem in Hamiltonian systems, with previous results only addressing plane waves that can be reduced to constant states by passing to polar coordinates. In this paper, we develop a novel method that resolves this obstruction by combining variational arguments, Floquet-Bloch theory, and Duhamel-based estimates with spatiotemporal modulation. Our framework applies to general periodic waves in Hamiltonian systems with symmetry and reduces the nonlinear stability problem to verifying diffusive spectral stability conditions for the second variation of a suitable conserved energy. Applying our approach to cnoidal waves in the Korteweg-de Vries equation, we obtain the first nonlinear stability result for periodic waves in Hamiltonian systems under localized perturbations that cannot be reduced to constant states.

We investigate pathwise well-posedness of the stochastic modulated Korteweg-de Vries equation (KdV) on the circle with a multiplicative noise, where a time non-homogeneous modulation acts on the linear dispersion term. (i) In the Young case (= fractional-in-time case with Hurst parameter greater than $\frac 12$), we establish a new regularization-by-noise phenomenon on the stochastic convolution in a pathwise manner, where a gain of spatial regularity becomes (arbitrarily) larger for more irregular modulations. We then prove that, given any $s \in \mathbb R$ and any multiplicative Young noise, however rough it is in space, the stochastic modulated KdV is pathwise locally well-posed in $H^s(\mathbb T)$, provided that the modulation is sufficiently irregular. (ii) In the rough case (= white-in-time case), irregularity of the modulation does not induce any smoothing on the stochastic convolution, and in fact, there is a slight loss in the spatial regularity. In this case, by slightly regularizing the multiplicative noise term, we prove pathwise local well-posedness in $H^s(\mathbb T)$ for any given $s \in \mathbb R$, provided that the noise is sufficiently smooth in space. We achieve these goals by combining (i) the sewing lemma approach to the nonlinear Young integration theory, introduced by Chouk and the second author (2014), and (ii) the pathwise construction of stochastic convolutions as Young or rough integrals via the random tensor estimate and the sewing lemma, introduced by the first, fourth, and fifth authors (2026). In the appendix, we also present an example of regularization by noise for a stochastic modulated Schrödinger equation with a multiplicative Young noise.

We study the existence, multiplicity, and concentration of normalized semiclassical states for a magnetic Schrödinger–Poisson system in $\mathbb{R}^3$ featuring both the Sobolev-critical local nonlinearity $|u|^4u$ and a critical nonlocal Poisson interaction. The problem is considered under the prescribed mass constraint $\int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3,$ where $a>0$ denotes the prescribed mass and $\varepsilon>0$ is the semiclassical parameter. By combining constrained variational methods, a suitable penalization scheme, concentration–compactness arguments, and Ljusternik–Schnirelmann theory, we first prove the existence of a normalized semiclassical solution for sufficiently small $a$ and $\varepsilon$. We then establish a multiplicity result showing that, for every sufficiently small $\varepsilon>0$, the number of distinct normalized solutions is bounded from below by the Ljusternik–Schnirelmann category of the minimum set \[ \mathcal M = \{x\in\mathbb{R}^3:V(x)=\min_{\mathbb{R}^3}V\}. \] Finally, we describe the semiclassical concentration phenomenon by showing that the maximum points of the resulting solutions approach $\mathcal M$ as $\varepsilon\to0$.

We study the geometric regularization of a positive closed current by the (twisted) Kähler-Ricci flow on a compact Kähler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete Kähler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.

Let $n\in\mathbb N\cap[2,\infty)$ and $\Omega\in L^1(\mathbb S^{n-1})$ with $\Omega\not\equiv 0$. In this article, we introduce a new family of lifted rough maximal operators $\{\mathcal{M}_\theta^\Omega\}_{\theta\in(0,\infty)}$ in the upper-half plane and establish their optimal weak-type estimates. Specifically, we prove that, for any $p \in (1, \infty)$, the estimate, with the positive equivalence constants independent of $f$, \[ \sup_{\theta,\lambda\in(0,\infty)}\lambda^p \underset{{\mathcal M}^\Omega_\theta(f)(x,t) > \lambda t^\frac{\gamma}{p}} {\int_{\mathbb R^n}\int_0^\infty} t^{\gamma-1}\,dt\,dx \sim \|f\|_{L^p(\mathbb{R}^n)}^p \] holds for all $f\in L^p(\mathbb R^n)$ if and only if $\gamma\in\mathbb R\setminus\{0\}$. For the endpoint case $p=1$ and $\Omega \in L(\log L)(\mathbb{S}^{n-1})$, we prove that the above estimate holds if and only if $\gamma \in (-\infty, -n) \cup (0, \infty)$. As applications, we obtain weak-type estimates for generalized Poisson integrals without any logarithmic integrability assumptions, which gives an affirmative answer to the question posed by Sjögren and Soria in page 228 of [Israel J. Math. 95 (1996)]. Moreover, although the operator $M^\ast_\Omega$, arising from the method of rotation of Calderón and Zygmund, is not of weak type $(1,1)$, we find that its lifted variant is weak type $(1,1)$. In addition, we establish a new characterization of Hardy spaces in terms of truncated rough singular integrals.

The proximal operator is a fundamental tool in variational analysis and optimization. In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous. In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and by Adve and Mészáros, among others, and are not completely understood. In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.

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.

We consider a nonlinear PDE describing a nonlinear electrostatic medium with nonlocal dielectricity. The existence proof for the corresponding equation is based on Schauder's theorem and a new compactness theorem for moving coefficients (“Helga's Theorem”). This technique uses insights from (operator-theoretic/topological) homogenisation theory. Surprisingly, even though monotonicity assumptions are neither used nor valid, the underlying domain is only required to be weak Lipschitz and no assumption on the derivatives of the nonlinearity is needed.

This work is concerned with an inverse medium problem for elastic waves, in which unknown inhomogeneities are reconstructed from time-resolved boundary measurements. We propose a novel time-domain direct sampling method for locating scatterers from a single incident source, without imposing specific assumptions on the temporal profile of the excitation. In particular, the imaging functional introduces a time-shifted correlation strategy that replaces the traditional $P$-$S$ wave decomposition with a travel-time alignment mechanism, thereby enabling direct imaging from the coupled elastic wave field. To analyze the proposed time-domain imaging functional, we employ Parseval's identity for the Fourier–Laplace transform and reformulate the functional in the frequency domain. By exploiting properties of modified Bessel functions, we characterize the asymptotic behavior of the imaging functional and show that it attains its maximum at the target location, which enables reliable identification of the scatterer. Rigorous theoretical justifications are provided to substantiate the effectiveness of the proposed method. Numerical experiments are also presented to demonstrate its performance and applicability.

We consider a family of semilinear parabolic equations with homogeneous Neumann boundary conditions on a family of varying non-smooth domains $\{\Omega_\mu\}_{\mu \in \Lambda} \subset \mathbb{R}^n $. Assuming only that the domains have uniformly bounded volumes, bounded away from zero, and satisfy a uniform Jones condition, together with the usual structural requirements on the linear and nonlinear terms, we establish the well-posedness of the problem in an appropriate scale of fractional Banach spaces and prove the existence of global attractors. Using a Moser-Alikakos bootstrap iteration in conjunction with the uniform Gronwall lemma and the uniform properties of the Jones extension operator, we show that bounded sets of $L^{q}(\Omega_\mu)$ are absorbed, in finite time and uniformly in $\mu$, into a fixed ball of $L^\infty(\Omega_\mu)$; the iteration uses only first-order Sobolev embeddings, which is what makes it available on domains where the domain of the elliptic operator is not contained in $H^{2}$. Finally, assuming the volume convergence of the domains, $|\Omega_\mu \triangle \Omega_0| \to 0$, we construct a framework of connecting maps and prove that the family of attractors is upper semicontinuous at $\mu = 0$ in $L^2(\Omega_\mu)$ and hence, by interpolation, in $H^s(\Omega_\mu)$ for every $s<1$.

We study finite-time breakdown of classical solutions to the Euler-alignment system through the degeneration of the associated Lagrangian flow. This approach allows us to characterize singularity formation in terms of the loss of local invertibility of the flow and the resulting concentration of density along characteristics. For the case of constant communication kernels, we derive an explicit formula for the flow and obtain an exact pointwise breakdown criterion in arbitrary dimension. In two dimensions, this criterion admits a closed-form reformulation in terms of the symmetric part of the initial velocity gradient and the initial vorticity. For general non-constant kernels, we derive sufficient conditions for finite-time degeneracy by combining a leading compressive mechanism with perturbative control of the nonlocal remainder. These conditions provide quantitative supercritical breakdown criteria in arbitrary dimension, complementing the existing subcritical global-regularity theory for multidimensional Euler-alignment systems.

In this paper, we study the following elliptic system \begin{equation}\label{main_1} \begin{cases} -\Delta u = |v|^{p-1} v + \epsilon (\alpha u + \beta_1 v), & \text{in } \Omega, \\ -\Delta v = |u|^{q-1} u + \epsilon (\beta_2 u + \alpha v), & \text{in } \Omega, \\ u = v = 0, & \text{on } \partial \Omega, \end{cases} \tag{*} \end{equation} where \(\Omega\) is the unit ball in $\mathbb{R}^N$, \(\epsilon\) is a small parameter, \(\alpha\), \(\beta_1\) and \(\beta_2\) are real numbers, \((p, q)\) is a pair of positive numbers lying on the critical hyperbola \begin{equation} \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}.\nonumber \end{equation} Under suitable assumptions and suitable restrictions on $(p,q)$ and $N$, we construct infinitely many sign-changing solutions to \eqref{main_1} which look like a positive radial solution to \eqref{main_1} crowned by $k$ negative bubbles arranged on a regular polygon of a suitable radius, whose energy can be arbitrarily large.

We extend some results of weak KAM theory to Lagrangians that are defined only on the horizontal distribution of a subriemannian manifold and depend on the unknown function.

The framed Beltrami–Vekua equation $\Phi(w_{\bar z} - \mu w_z) + \Psi(\overline{w_z} - \mu\overline{w_{\bar z}}) + \mathfrak{a}w + \mathfrak{b}\bar w = \mathfrak{f}$, with $|\mu|<1$ and $|\Phi|>|\Psi|$, carries a numerator field $N = \Phi\mathfrak{b} - \Psi\mathfrak{a} - W_L(\Phi,\Psi)$ whose weighted modulus integrates to the pseudo-analytic mass. This paper extracts the integer carried by the same field. When the zero set of $N$ is compactly contained in a bounded simply connected domain, the winding number of $N$ along any enclosing curve – the pseudo-analytic charge $n \in \mathbb{Z}$ – is invariant under every recombination $w = \varphi w' + \psi\bar w'$ of the unknown, every scaling of the equation, and every orientation-preserving $C^1$ change of variables: recombinations multiply $N$ by the positive factor $|\varphi|^2 - |\psi|^2$, so their invariance is exact, while on multiply connected domains the other two actions fix the component charges only in $\mathbb{Z}/2\mathbb{Z}$ and the total charge exactly. The charge is a Brouwer degree: it localizes at the zeros of $N$, vortices which no action of the class creates or destroys; an isolated vortex persists under perturbation of the data precisely when its local charge is non-zero. It involves the Beltrami coefficient only through the $L$-Wronskian of the frame, and is $\mu$-independent wherever $W_\partial(\Phi,\Psi) \equiv 0$ – in particular at the trivial frame, where $N = \mathcal{B}$ and the charge is the gauge-invariant winding of the coefficient of the Beltrami–Vekua equation. Mass and charge are independent: every pair in $(0,\infty)\times\mathbb{Z}$ is realized.

We prove rates of decay at infinity for solutions to variable-coefficient Schrödinger equations of the form $-\text{div}(A \nabla u) + W \cdot \nabla u + V u = \lambda u$ in cylinders, $\mathbb{T}^d \times \mathbb{R}^m$. We assume that $W$ and $V$ are bounded and that $\lambda \in \mathbb{C}$. Our rates depend on the decay of $|\nabla A|$ at infinity. In particular, we prove a range of quantitative unique continuation-type results at infinity when $|\nabla A(\theta, x)| \le C (1 + |x|)^{-\tau}$ for $\tau \in [0,1]$. By adapting the methods in [KLP25], we construct explicit solutions to demonstrate the sharpness of our estimates for each such $\tau$.