arXiv subjects

Sohrab Shahshahani

Publications and source records attributed to Sohrab Shahshahani.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Spectral Theory and Numerics

This is the first of three papers proving asymptotic stability of the degree-one vortex under equivariant perturbations in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at self-dual coupling. In the orthogonal gauge, the linearized dynamics are governed by a selfadjoint matrix Schrödinger operator $\mathbf{M}$. The super-symmetric partner operator is a diagonal matrix whose diagonal entries are strongly singular radial Schrödinger operators on $\mathbb{R}^2$. After a conjugation, this reduces the spectral problem to the analysis of two strongly singular scalar half-line operators. Combining analysis with rigorous interval arithmetic, we prove absence of threshold resonances and show that the discrete spectrum of $\mathbf{M}$ consists of exactly one positive gap eigenvalue (internal mode) with a two-dimensional eigenspace. We also certify that the relevant nonlinear Fermi Golden Rule coefficients form a definite quadratic form, yielding effective nonlinear damping of the internal mode. These spectral inputs form the basis of the stability analysis in the subsequent papers.

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arXiv2026

Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Linear Theory

We study the linearized dynamics near the degree-one vortex of the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, restricted to equivariant perturbations in the orthogonal gauge. The linearized operator is a selfadjoint matrix Schrödinger operator $\mathbf{M}$ on radial $L^2_{\mathrm{rad}}(\mathbb{R}^2;\mathbb{R}^4)$ with continuous spectrum $[1,\infty)$ and a two-dimensional internal mode at a unique gap eigenvalue $λ^2 \in (0,1)$, as established in Part I of our three-paper series on asymptotic stability of the ground state vortex. In this second part of the series, we prove linear estimates for $\mathbf{M}$ for applications in Part III. Specifically, we prove dispersive and local-energy decay estimates, as well as a transference relation which allows us to implement the space-time resonance method with respect to the flat Klein-Gordon operator in the nonlinear analysis in Part III. The engine for proving linear estimates for $\mathbf{M}$ in our approach is the distorted Fourier transform associated with $\mathbf{M}$. The construction of the distorted Fourier transform together with a detailed analysis of the underlying generalized eigenfunctions occupy the first half of this paper. For this we exploit the super-symmetric factorization of $\mathbf{M}$, and the diagonal structure of the super-symmetric partner operator, to relate the problem to the Weyl-Titchmarsh theory of two strongly singular scalar half-line Schrödinger operators.

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arXiv2026

Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model

We prove asymptotic stability of the degree-one vortex in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, for small equivariant perturbations in weighted Sobolev spaces. The abelian Yang-Mills-Higgs model is a classical relativistic field theory on $(1+2)$-dimensional Minkowski space, describing a complex-valued field coupled to an electromagnetic potential and admitting topological solitons known as vortices. This paper is the final and main part of a three-paper series. Under the orthogonal gauge condition used here, perturbations of the vortex are governed by a system of nonlinear Klein-Gordon equations for the dynamical variables, coupled to an elliptic equation for the temporal component of the electromagnetic potential. The linearized operator has continuous spectrum $[1,\infty)$ and a single positive gap eigenvalue (internal mode) of multiplicity two, whose spectral properties, associated distorted Fourier theory, and linear decay estimates are developed in the two companion papers. The main difficulty is the long-time analysis of the coupled radiation--internal-mode dynamics. In two space dimensions the dispersive decay of the Klein-Gordon radiation is relatively weak, while the internal mode decays only on the long time scale dictated by nonlinear radiation damping. At the same time, the Klein-Gordon equations for the radiation contain non-spatially localized variable coefficient quadratic interactions, which cannot be treated perturbatively and require a normal form analysis. We prove decay of the radiation by combining a good-bad decomposition, a flat-sharp decomposition, and a space-time resonance analysis carried out relative to the flat Klein-Gordon flow. The passage between the flat analysis and the Klein-Gordon flow with potential is achieved through ILED and transference estimates derived from the distorted Fourier theory.

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arXiv2025

The distorted Fourier transform for the linearized Gross-Pitaevskii equation in the Hyperbolic plane

Motivated by the stability problem for Ginzburg-Landau vortices on the hyperbolic plane, we develop the distorted Fourier transform for a general class of radial non-self-adjoint matrix Schrödinger operators on the hyperbolic plane. This applies in particular to the operator obtained by linearizing the equivariant Ginzburg-Landau equation on the hyperbolic plane around the degree one vortex. We systematically construct the distorted Fourier transform by writing the Stone formula for complex energies and taking the limit as the energy tends to the spectrum of the operator on the real line. This approach entails a careful analysis of the resolvent for complex energies in a neighborhood of the real line. It is the analogue of the approaches used in \cite{KS,ES2, LSS25}, where the limiting operator as $r\to\infty$ is not self-adjoint and which we carry out for all energies. Our analysis serves as the starting point for the study of the stability of the Ginzburg-Landau vortex under equivariant perturbations.

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arXiv2025

Equivariant stability of vortices in Manton's Chern-Simons-Schrödinger system on the hyperbolic plane

In this work we study magnetic vortices on the hyperbolic plane for a Chern-Simons-Schrödinger system introduced by Manton. The model can be thought of as the Schrödinger analogue of the Abalian-Higgs model. It consists of a system of partial differential equations, where the complex Higgs field $Φ$ evolves according to a nonlinear Schrödinger equation coupled to an electromagnetic field $A$. We restrict attention to the self-dual (Bogomolny) case under equivariance symmetry. For each $m\geq 1$ we prove the asymptotic stability of the equivariant vortex of degree $m$. The main novelties are unraveling the favorable structure of the equations after a nonlinear Darboux transform, and the analysis of the elliptic operator relating the original and the transformed variables.

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arXiv2025

On the Gross-Pitaevskii evolution linearized around the degree-one vortex

We study the evolution of the Gross-Pitaevskii equation linearized around the Ginzburg-Landau vortex of degree one under equivariant symmetry. Among the main results of this work, we determine the spectrum of the linearized operator, uncover a remarkable $L^2$-norm growth phenomenon related to a zero-energy resonance, and provide a complete construction of the distorted Fourier transform at small energies. The latter hinges upon a meticulous analysis of the behavior of the resolvent in the upper and lower half-planes in a small disk around zero-energy.

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arXiv2025

Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry

We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the $n$-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for $n \geq 5$. The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution.

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arXiv2024

Stability of the 3-dimensional catenoid for the hyperbolic vanishing mean curvature equation

We prove that the $3$-dimensional catenoid is asymptotically stable as a solution to the hyperbolic vanishing mean curvature equation in Minkowski space, modulo suitable translation and boost (i.e., modulation) and with respect to a codimension one set of initial data perturbations. The modulation and the codimension one restriction on the initial data are necessary (and optimal) in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. The $3$-dimensional problem is more challenging than the higher (specifically, $5$ and higher) dimensional case addressed in the previous work of the authors with J.~Lührmann, due to slower temporal decay of waves and slower spatial decay of the catenoid. To overcome these issues, we introduce several innovations, such as a proof of Morawetz- (or local-energy-decay-) estimates for the linearized operator with slowly decaying kernel elements based on the Darboux transform, a new method to obtain Price's-law-type bounds for waves on a moving catenoid, as well as a refined profile construction designed to capture a crucial cancellation in the wave-catenoid interaction. In conjunction with our previous work on the higher dimensional case, this paper outlines a systematic approach for studying other soliton stability problems for $(3+1)$-dimensional quasilinear wave equations.

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arXiv2021

Well-posedness for the free boundary hard phase model in general relativity

The hard phase model describes a relativistic barotropic and irrotational fluid with sound speed equal to the speed of light. In the framework of general relativity, the fluid, as a matter field, affects the geometry of the background spacetime. Therefore the motion of the fluid must be coupled to the Einstein equations which describe the structure of the underlying spacetime. In this work we prove a priori estimates and well-posedness in Sobolev spaces for this model with free boundary. Estimates for the curvature are derived using the Bianchi equations in a frame that is parallel transported by the fluid velocity. The fluid velocity is also decomposed with respect to this parallel frame, and its components are estimated using a coupled interior-boundary system of wave equations.

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arXiv2020

Well-posedness of the free boundary hard phase fluids in Minkowski background and its Newtonian limit

The hard phase model describes a relativistic barotropic irrotational fluid with sound speed equal to the speed of light. In this paper, we prove the local well-posedness for this model in the Minkowski background with free boundary. Moreover, we show that as the speed of light tends to infinity, the solution of this model converges to the solution of the corresponding Newtonian free boundary problem for incompressible fluids. In the appendix we explain how to extend our proof to the general barotropic fluid free boundary problem.

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arXiv2019

Local smoothing estimates for Schrödinger equations on hyperbolic space

We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space $\mathbb{H}^d$. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood-Paley machinery on $\mathbb{H}^d$ based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on $\mathbb{H}^{d}$. Specifically, some of the estimates established in this paper play a crucial role in the authors' proof of the nonlinear asymptotic stability of harmonic maps under the Schrödinger maps evolution on the hyperbolic plane; see [29]. As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-dependent perturbations, including polynomially decaying second order ones, and small lower order nonsymmetric perturbations.

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arXiv2019

Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane Under the Schrödinger Maps Evolution

We consider the Cauchy problem for the Schrödinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map $Q$ under the Schrödinger maps evolution with respect to non-equivariant perturbations, provided $Q$ obeys a suitable linearized stability condition. This condition is known to hold for all equivariant harmonic maps with values in the hyperbolic plane and for a subset of those maps taking values in the sphere. One of the main technical ingredients in the paper is a global-in-time local smoothing and Strichartz estimate for the operator obtained by linearization around a harmonic map, proved in the companion paper [36].

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arXiv2017

On tidal energy in Newtonian two-body motion

In this work, which is based on an essential linear analysis carried out by Christodoulou, we study the evolution of tidal energy for the motion of two gravitating incompressible fluid balls with free boundaries obeying the Euler-Poisson equations. The orbital energy is defined as the mechanical energy of the two bodies' center of mass. According to the classical analysis of Kepler and Newton, when the fluids are replaced by point masses, the conic curve describing the trajectories of the masses is a hyperbola when the orbital energy is positive and an ellipse when the orbital energy is negative. The orbital energy is conserved in the case of point masses. If the point masses are initially very far, then the orbital energy is positive, corresponding to hyperbolic motion. However, in the motion of fluid bodies the orbital energy is no longer conserved because part of the conserved energy is used in deforming the boundaries of the bodies. In this case the total energy $\tilde{\mathcal{E}}$ can be decomposed into a sum $\tilde{\mathcal{E}}:=\widetilde{\mathcal{E}_{\mathrm{orbital}}}+\widetilde{\mathcal{E}_{\mathrm{tidal}}}$, with $\widetilde{\mathcal{E}_{\mathrm{tidal}}}$ measuring the energy used in deforming the boundaries, such that if $\widetilde{\mathcal{E}_{\mathrm{orbital}}}<-c<0$ for some absolute constant $c>0$, then the orbit of the bodies must be bounded. In this work we prove that under appropriate conditions on the initial configuration of the system, the fluid boundaries and velocity remain regular up to the point of the first closest approach in the orbit, and that the tidal energy $\widetilde{\mathcal{E}_{\mathrm{tidal}}}$ can be made arbitrarily large relative to the total energy $\tilde{\mathcal{E}}$. In particular under these conditions $\widetilde{\mathcal{E}_{\mathrm{orbital}}}$, which is initially positive, becomes negative before the point of the first closest approach.

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arXiv2015

On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary and Constant Vorticity: An Appendix

In a recent work [1] the authors studied the dynamics of the interface separating a vacuum from an inviscid incompressible fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid is additionally assumed to be irrotational, and we proved that for data which are size $ε$ perturbations of an equilibrium state, the lifespan $T$ of solutions satisfies $T \gtrsim ε^{-2}$. The key to the proof is to find a nonlinear transformation of the unknown function and a coordinate change, such that the equation for the new unknown in the new coordinate system has no quadratic nonlinear terms. For the related irrotational gravity water wave equation with constant gravity the analogous transformation was carried out by the last author in [3]. While our approach is inspired by the last author's work [3], the self-gravity in the present problem is a new nonlinearity which needs separate investigation. Upon completing [1] we learned of the work of Ifrim and Tataru [2] where the gravity water wave equation with constant gravity and constant vorticity is studied and a similar estimate on the lifespan of the solution is obtained. In this short note we demonstrate that our transformations in [1] can be easily modified to allow for nonzero constant vorticity, and a similar energy method as in [1] gives an estimate $T\gtrsimε^{-2}$ for the lifespan $T$ of solutions with data which are size $ε$ perturbations of the equilibrium. In particular, the effect of the constant vorticity is an extra linear term with constant coefficient in the transformed equation, which can be further transformed away by a bounded linear transformation. This note serves as an appendix to the aforementioned work of the authors.

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arXiv2015

On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary

We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler-Poission system in moving bounded simply connected domains. A family of equilibrium solutions of the system are the perfect balls moving at constant velocity. We show that for smooth data which are small perturbations of size $ε$ of these static states, measured in appropriate Sobolev spaces, the solution exists and remains of size $ε$ on a time interval of length at least $cε^{-2},$ where $c$ is a constant independent of $ε.$ This should be compared with the lifespan $O(ε^{-1})$ provided by local well-posdness. The key ingredient of our proof is finding a nonlinear transformation which removes quadratic terms from the nonlinearity. An important difference with the related gravity water waves problem is that unlike the constant gravity for water waves, the self-gravity in the Euler-Poisson system is nonlinear. As a first step in our analysis we also show that the Taylor sign condition always holds and establish local well-posedness for this system.

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arXiv2015

The Cauchy problem for wave maps on hyperbolic space in dimensions $d \geq 4$

We establish global well-posedness and scattering for wave maps from $d$-dimensional hyperbolic space into Riemannian manifolds of bounded geometry for initial data that is small in the critical Sobolev space for $d \geq 4$. The main theorem is proved using the moving frame approach introduced by Shatah and Struwe. However, rather than imposing the Coulomb gauge we formulate the wave maps problem in Tao's caloric gauge, which is constructed using the harmonic map heat flow. In this setting the caloric gauge has the remarkable property that the main `gauged' dynamic equations reduce to a system of nonlinear scalar wave equations on $\mathbb{H}^{d}$ that are amenable to Strichartz estimates rather than tensorial wave equations (which arise in other gauges such as the Coulomb gauge) for which useful dispersive estimates are not known. This last point makes the heat flow approach crucial in the context of wave maps on curved domains.

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arXiv2015

Equivariant Wave Maps on the Hyperbolic Plane with Large Energy

In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space into surfaces of revolution that was initiated in [13, 14]. When the target is the hyperbolic plane we proved in [13] the existence and asymptotic stability of a 1-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps

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arXiv2015

Gap Eigenvalues and Asymptotic Dynamics of Geometric Wave Equations on Hyperbolic Space

In this paper we study $k$-equivariant wave maps from the hyperbolic plane into the $2$-sphere as well as the energy critical equivariant $SU(2)$ Yang-Mills problem on $4$-dimensional hyperbolic space. The latter problem bears many similarities to a $2$-equivariant wave map into a surface of revolution. As in the case of $1$-equivariant wave maps considered in~\cite{LOS1}, both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schrödinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.

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