arXiv research

Study finds all 4D Lie groups with harmonic curvature.

Develops methods to construct harmonic and wave maps into variable-curvature surfaces.

Study on $pp$-waves in isotropic quasi-Einstein manifolds.

Solves wave equation on non-flat harmonic manifolds using Abel transform and Fourier analysis.

A new mathematical approach detects frequency-based alterations in brain networks.

Wave maps connect to constant curvature surfaces, studying singularities and bifurcations.

Wave maps can have multiple bubbling solutions at blow-up points.

We present both, theory and an algorithm for solving time-harmonic wave problems in a general setting. The time-harmonic solutions will be achieved by computing time-periodic solutions of the original wave equations. Thus, an exact controllability technique is proposed to solve the time-dependent wave equations. We dis…

Initial data with zero mass must be in pp-wave spacetimes.

Develops methods to model and forecast inter-sectoral balance dynamics.

We consider the energy-critical half-wave maps equation $\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterizat…

Identifies conjugate points in spherical harmonics solutions of quasi-geostrophic equations.

In recent work, we have proven uniform decay bounds for solutions of the wave equation $\Box_gφ=0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $|φ|\le Cv_+^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_+=\ma…

Proves global existence for quasilinear wave equations with weak null condition.

Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${\mathbb R}^2$ with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potent…

In this paper, we introduce a new notion named as Schrödinger soliton. So-called Schrödinger solitons are defined as a class of special solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold $M$ into a Kähler manifold $N$. If the target manifold $N$ admits a Killing potential, th…

Survey and new results link hydrodynamics, molecular physics, and financial engineering.

The paper analyzes the spectra of compact quotients of the oscillator group.

In this article, we classify the set of asymptotic mass-like invariants for asymptotically hyperbolic metrics. It turns out that the standard mass is just one example (but probably the most important one) among the two families of invariants we find. These invariants are attached to finite-dimensional representations o…

In this paper, we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, and obtain the local well-posedness result in $H^s, s>2$. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstei…

Stability of Minkowski space-time in higher dimensions proven for arbitrary small perturbations.

These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spa…

Rigidity theorem shows massless hyperboloidal data embeds into Minkowski space.

This work proves Kerr black holes are dynamically stable under certain perturbations.

Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …

Eigenfunction value distribution shows unimodal density with maximum at zero.

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-spac…

Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.

New Galilean spacetimes found as pp-wave reductions.

The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler-Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy …

Analyzes Gerstner's trochoidal waves and their geometric properties.

Wave fronts on certain surfaces become dense.

Analyzes properties of stiffness tensors for elastic wave imaging.

Deep learning model predicts wind-wave relationship.

Stability of Minkowski space-time in Einstein-Yang-Mills system proven.

Impulsive waves contradict a 1962 conjecture about pp-waves.

Extends Penrose limit to Finsler spacetimes.

New findings on plane waves in 3D spacetimes, showing non-unimodular elliptic plane waves are unique.

Classifies solutions to vacuum weighted Einstein equations on pr-waves.

High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton e…

Study compact plane waves, showing they are essentially standard.

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…

New method finds precise late-time behavior of wave equations.

Paper explores non-uniqueness and uniqueness class for wave equations on graphs.

We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.

The financial rogue waves are reported analytically in the nonlinear option pricing model due to Ivancevic, which is nonlinear wave alternative of the Black-Scholes model. These solutions may be used to describe the possible physical mechanisms for rogue wave phenomenon in financial markets and related fields.

New metrics link Kähler and pp-wave spacetimes.

New method detects spike-and-wave epileptiform discharges using Kendall's Tau-b.