Wave Maps Research Articles

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

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.

Mode Stability of Self-Similar Wave Maps in Higher Dimensions

We consider co-rotational wave maps from Minkowski space in $d+1$ dimensions to the $d$-sphere. Recently, Bizo\'n and Biernat found explicit self-similar solutions for each dimension $d\geq 4$. We give a rigorous proof for the mode stability of these self-similar wave maps.

Quantization of Time-Like Energy for Wave Maps into Spheres

In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru (Commun. Math. Phys. 298(1):139–230, 2010; Commun. Math. Phys. 298(1):231–264, 2010), but more specifically into the unit Euclidean sphere {mathbb{S}^{n-1}subsetmathbb{R}^{n}} with {ngeq2}, and study further the dynamics of the sequence of wave maps that are obtained in Sterbenz and Tataru (Commun. Math. Phys. 298(1):231–264, 2010) at the final rescaling for a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space. Similar and stronger results have been recently obtained in the equivariant setting by several authors (Côte, Commun. Pure Appl. Math. 68(11):1946–2004, 2015; Côte, Commun. Pure Appl. Math. 69(4):609–612, 2016; Côte, Am. J. Math. 137(1):139–207, 2015; Côte et al., Am. J. Math. 137(1):209–250, 2015; Krieger, Commun. Math. Phys. 250(3):507–580, 2004), where better control on the dispersive term concentrating on the null boundary of the cone is provided, and in some cases the asymptotic decomposition is shown to hold for all time. Here, however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Rivière (Ann. Math. 149(2):785–829, 1999; Duke Math. J. 111:177–193, 2002), which we make work here in the hyperbolic context of Sterbenz and Tataru (Commun. Math. Phys. 298(1), 231–264, 2010).

Regularity criterion for the wave map in a bounded domain

We prove a regularity criterion for the wave map in a bounded domain when the space dimension n = 2 or 3 . We also prove the global well-posedness of strong solutions for the wave map in a bounded domain when the space dimension n = 1 .

Demonstration of optical rogue waves using a laser diode emitting at 980 nm and a fiber Bragg grating.

Rogue waves are observed for the first time, to the best of our knowledge, in a 980 nm laser diode subject to filtered optical feedback via a fiber Bragg grating. By counting the number of rogue waves in a fixed time window, a rogue wave map is established experimentally as a function of both the optical feedback ratio and the laser current. The comparison with low frequency fluctuations (LFFs) reveals that the rogue waves observed in our system are, in fact, LFF jump-ups.

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 [9], 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.

A proof for the mode stability of a self-similar wave map

We study the fundamental self-similar solution to the SU(2) sigma model, found by Shatah and Turok–Spergel. We give a rigorous proof for its mode stability. Based on earlier results by the second author, the present paper constitutes the last building block for a completely rigorous proof of the nonlinear stability of the Shatah–Turok–Spergel wave map.

A class of large global solutions for the wave-map equation

In this paper we consider the equation for equivariant wave maps from R3+1 to S-3 and we prove global in forward time existence of certain C-infinity-smooth solutions which have infinite critical Sobolev norm (H) overdot(3/4) (R-3) x (H) overdot(1/2) (R-3). Our construction provides solutions which can moreover satisfy the additional size condition parallel to u(0, .)parallel to L-infinity(vertical bar-vertical bar >= 1) > M for arbitrarily chosen M > 0. These solutions are also stable under suitable perturbations. Our method, strongly inspired by work of Krieger and Schlag, is based on a perturbative approach around suitably constructed approximate self similar solutions.

QUest for the Arrhythmogenic Substrate of Atrial fibRillation in Patients Undergoing Cardiac Surgery (QUASAR Study): Rationale and Design.

The heterogeneous presentation and progression of atrial fibrillation (AF) implicate the existence of different pathophysiological processes. Individualized diagnosis and therapy of the arrhythmogenic substrate underlying AF may be required to improve treatment outcomes. Therefore, this single-center study aims to identify the arrhythmogenic areas underlying AF by intra-operative, high-resolution, multi-site epicardial mapping in 600 patients with different heart diseases. Participants are divided into 12 groups according to the underlying heart diseases and presence of prior AF episodes. Mapping is performed with a 192-electrode array for 5–10 s during sinus rhythm and (induced) AF of the entire atrial surface. Local activation times are converted into activation and wave maps from which various electrophysiological parameters are derived. Postoperative cardiac rhythm registrations and a 5-year follow-up will show the incidence of postoperative and persistent AF. This project provides the first step in the development of a tool for individual AF diagnosis and treatment.Electronic supplementary materialThe online version of this article (doi:10.1007/s12265-016-9685-1) contains supplementary material, which is available to authorized users.

Stability of stationary wave maps from a curved background to a sphere

We study time and space equivariant wave maps from $M\times\mathbb{R}\rightarrow S^2,$ where $M$ is diffeomorphic to a two dimensional sphere and admits an action of $SO(2)$ by isometries. We assume that metric on $M$ can be written as $dr^2+f^2(r)d\theta^2$ away from the two fixed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared with the case where $M$ is isometric to the standard sphere (considered by Shatah and Tahvildar-Zadeh [34]), is the the use of triangle comparison theorems to obtain pointwise bounds on the fundamental solution on a curved background.

Numerical solutions of Einstein’s equations for cosmological spacetimes with spatial topologyS3and symmetry group U(1)

In this paper we consider the single patch pseudo-spectral scheme for tensorial and spinorial evolution problems on the 2-sphere presented in [3,4] which is based on the spin-weighted spherical harmonics transform. We apply and extend this method to Einstein's equations and certain classes of spherical cosmological spacetimes. More specifically, we use the hyperbolic reductions of Einstein's equations obtained in the generalized wave map gauge formalism combined with Geroch's symmetry reduction, and focus on cosmological spacetimes with spatial S3-topologies and symmetry groups U(1) or U(1) x U(1). We discuss analytical and numerical issues related to our implementation. We test our code by reproducing the exact inhomogeneous cosmological solutions of the vacuum Einstein field equations obtained in [7].

MultiSymplectic Discretization of Wave Map Equations

We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint and can be seen as a generalisation of the Shake algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions.

Curie point depth, heat-flow and radiogenic heat production deduced from the spectral analysis of the aeromagnetic data for geothermal investigation on the Menderes Massif and the Aegean Region, western Turkey

In this study, the Curie point depth (CPD), heat-flow and radiogenic heat production maps of the Menderes Massif (MM) and the Aegean Region-Turkey were prepared for the thermal crustal structure from the aeromagnetic data by means of a spectral analysis method. The estimated CPDs in the study area vary from 6.21 to 12.41km. The heat-flow values were calculated for two (2) different thermal conductivities of 2.5–2.7Wm−1K−1. The radiogenic heat production was also calculated between 0.38 and 0.80μWm−3 on the high-grade metamorphic units of the study area. The estimated heat-flow and radiogenic heat production maps indicate that a high temperature in the upper crust and thin lithosphere in the observation area are probably resulted from the crustal extension, active deformation and magmatism. All of the produced maps in this study were correlated with the analytic signal (AS) map which indicates the locations of the bodies causing magnetic anomalies. In addition, maxspots of the horizontal gradients from the magnetic anomalies were also emplaced onto the produced maps to indicate the boundaries of critical magmatic units in the study area. These correlations allow reasonable and meaningful interpretation for the reasons of high geothermal gradient and high heat-flow. In addition, a longitudinal seismic wave (Vp velocity) map was presented from the relationship between seismic velocity and radiogenic heat production in the western Anatolia, for the first time and they were correlated with the previous investigations. The estimated Vp velocities range from about 6 to 6.4km/sn and low Vp values may be related to the partial melting of subducted structures in the western Anatolia. As a result, two new potential geothermal areas were determined by the integration of different methods used in this study.

Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation

We prove global regularity, scattering and a priori bounds for the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge on (1+4)-dimensional Minkowski space. The proof is based upon a modified Bahouri-Gerard profile decomposition [1] and a concentration compactness/rigidity argument by Kenig-Merle [5], following the method developed by the first author and Schlag [10] in the context of critical wave maps. This is a preliminary version of the final paper.

A Refined Threshold Theorem for (1 + 2)-Dimensional Wave Maps into Surfaces

The recently established threshold theorem for energy critical wave maps states that wave maps with energy less than that of the ground state (i.e., a minimal energy nontrivial harmonic map) are globally regular and scatter on $${\mathbb{R}^{1+2}}$$ . In this note we give a refinement of this theorem when the target is a closed orientable surface, by taking into account an additional invariant of the problem, namely the topological degree. We show that the sharp energy threshold for global regularity and scattering is in fact twice the energy of the ground state for wave maps with degree zero, whereas wave maps with nonzero degree necessarily have at least the energy of the ground state. We also give a discussion on the formulation of a refined threshold conjecture for the energy critical SU(2) Yang–Mills equation on $${\mathbb{R}^{1+4}}$$ .

On the Global Stability of the Wave-map Equation in Kerr Spaces with Small Angular Momentum

This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map $$\Phi $$ defined from a fixed Kerr solution $${\mathcal K}(M,a)$$ , $$0\le a \le M $$ , with values in the two dimensional hyperbolic space $${\mathbb H}^2$$ . A particular such wave map is given by the complex Ernst potential associated to the axial Killing vector-field $$\mathbf{Z}$$ of $${\mathcal K}(M,a)$$ . We conjecture that this stationary solution is stable, under small axially symmetric perturbations, in the domain of outer communication (DOC) of $${\mathcal K}(M,a)$$ , for all $$0\le a<M$$ and we provide preliminary support for its validity, by deriving convincing stability estimates for the linearized system.

Stable soliton resolution for exterior wave maps in all equivariance classes

In this paper we consider finite energy ℓ-equivariant wave maps from Rt,x1+3\\(R×B(0,1))→S3 with a Dirichlet boundary condition at r=1, and for all ℓ∈N. Each such ℓ-equivariant wave map has a fixed integer-valued topological degree, and in each degree class there is a unique harmonic map, which minimizes the energy for maps of the same degree. We prove that an arbitrary ℓ-equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class, i.e., soliton resolution. This extends the recent results of the first, second, and fourth authors on the 1-equivariant equation to higher equivariance classes, and thus completely resolves a conjecture of Bizoń, Chmaj and Maliborski, who observed this asymptotic behavior numerically. The proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d=2ℓ+3, which are established in the companion paper [13].

Shear wave mapping of skeletal muscle using shear wave wavefront reconstruction based on ultrasound color flow imaging

We have proposed a quantitative shear wave imaging technique for continuous shear wave excitation. Shear wave wavefront is observed directly by color flow imaging using a general-purpose ultrasonic imaging system. In this study, the proposed method is applied to experiments in vivo, and shear wave maps, namely, the shear wave phase map, which shows the shear wave propagation inside the medium, and the shear wave velocity map, are observed for the skeletal muscle in the shoulder. To excite the shear wave inside the skeletal muscle of the shoulder, a hybrid ultrasonic wave transducer, which combines a small vibrator with an ultrasonic wave probe, is adopted. The shear wave velocity of supraspinatus muscle, which is measured by the proposed method, is 4.11 ± 0.06 m/s (N = 4). This value is consistent with those obtained by the acoustic radiation force impulse method.

Generic Self-Similar Blowup for Equivariant Wave Maps and Yang–Mills Fields in Higher Dimensions

We consider equivariant wave maps from the $(d+1)$--dimensional Minkowski spacetime into the $d$-sphere for $d\geq 4$. We find a new explicit stable self-similar solution and give numerical evidence that it plays the role of a universal attractor for generic blowup. An analogous result is obtained for the $SO(d)$ symmetric Yang-Mills field for $d\geq 6$.