In this paper, a symmetry classification of a -nonlinear wave equation where is a smooth function on , using Lie group method, is given. The basic infinitesimal method for calculating symmetry groups is presented, and used to determine the general symmetry group of this $…
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48 results for “nonlinear wave equation”
Constructs periodic solutions for wave equations, including Einstein's, with negative cosmological constant.
problem Finding periodic solutions for nonlinear wave equations, especially Einstein's equations with negative cosmological constant.
method Analytic continuation to construct periodic solutions.
result Infinite-dimensional families of time-periodic solutions for vacuum and Einstein-Maxwell-dilaton-scalar fields.
CNNs reconstruct medium properties from wave probing responses.
problem Determining medium properties from wave responses.
method Deep convolutional neural networks (CNNs) for nonlinear wave equations.
result Quantitative dependence of network depth and units on medium complexity.
Using the scalar curvature of the product manifold S^{2}X R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M. T. Mustafa [H. Azad and M. T. Mustafa, Symmetry an…
Adaptive Wave Models for Option Pricing Evolution: Nonlinear and Quantum Schrödinger Approachesq-fin.PR
Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black--Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equatio…
The paper establishes scattering theory for wave equations on Schwarzschild spacetime.
problem Defocusing semilinear wave equations on Schwarzschild spacetime.
method Combining energy and pointwise decay results with Sobolev embedding, constructing scattering operator.
result Construction of a scattering operator mapping past to future scattering data.
The moving coframe method is applied to solve the local equivalence problem for the class of nonlinear wave equations in two independent variables under an action of the pseudo-group of contact transformations. The structure equations and the complete sets of differential invariants for symmetry groups are found. The s…
Measuring wave sources uniquely identifies manifold properties.
problem Determining Riemannian manifold structure from wave observations.
method Semilinear wave equation measurements at a single point.
result Topological, differential, and geometric structure can be inferred.
In this paper, we study the schrodinger equation and wave equation with the Dirichlet boundary condition on a connected finite graph. The explicit expressions for solutions are given and the energy conservations are derived. Applications to the corresponding nonlinear problems are indicated.
A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing 'controlled Brownian behavior' of financial markets, is formally defined by adaptive nonlinear Schrödinger (NLS) equations, defining the option-pricing wave function in terms of the stock…
The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct…
New method finds precise late-time behavior of wave equations.
problem Analyzing late-time behavior of wave equations with inverse-square potentials.
method Physical-space-based method for deriving late-time asymptotics.
result Sharp, uniform decay estimates in time for asymptotic late-time tails.
The paper constructs and analyzes self-similar blowup solutions for a wave map equation.
problem Existence and stability of self-similar blowup solutions for a wave map equation.
method Construction of self-similar solutions, detailed nonlinear stability analysis, spectral analysis of linearized operators.
result Sharp semigroup bounds and nonlinear stability of all discretely self-similar profiles in all dimensions.
Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese websmath-ph
Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web…
Schwarzschild black hole stability proven using a new gauge.
problem Linear stability of Schwarzschild black holes to gravitational perturbations.
method Proved stability by imposing a generalised wave gauge.
result Uniform boundedness and polynomial decay of solutions.
We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of t…
In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is int…
This paper solves a complex equation using Lie symmetry approach to find solitary wave and multiple soliton solutions.
problem Solving a (3 + 1)-dimensional KdV type equation.
method Lie group of transformation method to find infinitesimal generators and commutator table.
result Exact solutions of KdV type equation in explicit form.
Second paper in series solves Einstein vacuum equations for three impulsive waves.
problem Solving local Cauchy problem for impulsive gravitational waves.
method Geometric commutators for energy estimates, fractional-derivative regularity, anisotropic Sobolev embedding.
result Scalar field becomes everywhere Lipschitz and away from singular region.
New approach simplifies proof of wave equations on black holes.
problem Global existence and decay for semilinear wave equations on extremal Reissner-Nordström black holes.
method Develops a new approach based on weaker estimates, avoiding near-horizon sharp estimates.
result Simpler and more streamlined proof without requiring near-horizon sharp estimates.
Paper proves stability of multi-dimensional rarefaction waves in gas dynamics.
problem Challenges in constructing multi-dimensional rarefaction waves in gas dynamics.
method Geometric Weighted Energy Method (GWEM) to overcome derivative losses.
result Established nonlinear stability of multi-dimensional rarefaction waves for compressible Euler equations.
Physics-informed neural networks solve physics problems using neural nets.
problem Discovering nonlinear PDEs from data.
method Two classes of algorithms: continuous time and discrete time models.
result Demonstrated effectiveness on various physics problems.
Recently, a novel adaptive wave model for financial option pricing has been proposed in the form of adaptive nonlinear Schrödinger (NLS) equation [Ivancevic a], as a high-complexity alternative to the linear Black-Scholes-Merton model [Black-Scholes-Merton]. Its quantum-mechanical basis has been elaborated in [Ivancevi…
Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.
problem Interaction of three impulsive gravitational waves in Einstein vacuum equations.
method Geometric estimates and wave estimates to prove local solution and continuity.
result Local solution to Einstein vacuum equations with three impulsive gravitational waves, Lipschitz continuity.
Study Dirichlet symbols related to univalent functions and nonlinear wave equations.
problem Characterize Dirichlet symbols associated with univalent functions and their properties.
method Local analysis of Dirichlet symbols near the diagonal on the bidisk, using the Grunsky operator and Schwarzian derivative.
result Find a new concept of Schwarzian asymptotic variance and bound its effective average amplitude.
Novel approach to wave equations near null infinity in flat spacetimes.
problem Analyzing regularity and decay of wave equations near null infinity in asymptotically flat spacetimes.
method Microlocal analysis in a compactified spacetime with corners, focusing on edge-type wave operators.
result Microlocal regularity propagates across null infinity via radial sets, leading to new estimates for wave equations.
In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave equations satisfied by the curvatures are derived. The relation of hypergeometric flo…
This paper addresses parameter estimation for wave equations with Markovian switching.
problem Parameter estimation for wave equations with abrupt changes.
method Bayesian statistical framework using discrete sparse Bayesian learning.
result Strong performance in parameter estimation for variable coefficient PDEs.
Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
Authors develop a new theory to smooth spacetime connections and remove singularities in GR shock waves.
problem Singularities in General Relativity shock waves and optimal regularity of spacetime connections.
method Established a general multi-dimensional existence theory for Reintjes-Temple equations using elliptic regularity in spaces.
result Regularities of GR shock waves can always be removed by coordinate transformations, extending Uhlenbeck compactness to Lorentzian geometry.
In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations are strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with …
Algorithm samples constrained stochastic differential equations.
problem Sampling stochastic differential equations with complex constraints.
method Pathspace Metropolis-adjusted manifold sampling.
result Demonstrated effectiveness in various constrained conditions.
Generalizes Hasimoto transformation to arbitrary flows on space curves.
problem Modeling and analyzing wave motions on space curves.
method Develops a mapping between curve evolution and scalar equations, transforming general vector fields in the Frenet frame.
result Binormal flows are generally length preserving but bending energy is fragile.
Gravitational waves are predicted by the general theory of relativity. In [6] D. Christodoulou showed that gravitational waves have a nonlinear memory. We proved in [3] that the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In the present paper, we study this …
Survey on stability of Minkowski spacetime in relativity.
problem Nonlinear stability of Minkowski spacetime in general relativity.
method Decay assumptions, geometric foliations, energy identities, and gauge choices.
result Understanding of decay, dispersion, and geometry-analysis interplay.
Financial rogue wavesq-fin.PR
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.
Energy methods solve Dirac-type equations in 2D Minkowski space.
problem Solving Dirac-type equations in 2D Minkowski space.
method Energy methods for linear and nonlinear equations.
result Existence results for Dirac-type equations.
Online algorithm identifies PDEs from noisy data snapshots.
problem Identifying PDEs from sequential solution snapshots.
method Combines weak-form discretization with online proximal gradient descent.
result Efficiently identifies and tracks systems with time-varying coefficients.
Financial Rogue Waves Appearing in the Coupled Nonlinear Volatility and Option Pricing Modelq-fin.PR
The coupled nonlinear volatility and option pricing model presented recently by Ivancevic is investigated, which generates a leverage effect, i.e., stock volatility is (negatively) correlated to stock returns, and can be regarded as a coupled nonlinear wave alternative of the Black-Scholes option pricing model. In this…
We investigate bi-Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non-stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric-induced nonlinear connection (N-connection) structure. On spacetim…
Construction of type II blowup solutions for the 1-corotational energy supercritical wave mapsmath.AP
Constructs blowup solutions for wave maps with specific symmetry.
problem Energy supercritical wave maps with 1-corotational symmetry.
method Reduction to 1D semilinear wave equation, concentration of universal profile, modulation techniques, finite-dimensional problem solving, Brouwer fixed point theorem.
result Construction of blowup solutions for wave maps.
Study on solitary waves on noncompact manifolds, proving existence and stability, with instability in most cases.
problem Existence and stability of solitary waves on noncompact manifolds.
method Existence via perturbation, stability via spectral analysis of linearized operator, Vakhitov-Kolokolov criterion for instability.
result Existence and instability of solitary waves on most noncompact manifolds, with numerical evidence.
We prove that for spacetimes solving the Einstein-Maxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In [5] D. Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a perma…
Researchers solve a complex equation to embed graphs with negative curvature.
problem Embedding graphs in with negative Gauss curvature.
method Solving a fully nonlinear Monge-Ampère equation using energy estimates and Nash-Moser iteration.
result Local solvability of the fully nonlinear equation for negative curvature.
Paper explores non-uniqueness and uniqueness class for wave equations on graphs.
problem Non-uniqueness of solutions to wave equations on infinite graphs.
method Analyticity of solutions in the uniqueness class, extension to a wide class of linear evolution equations.
result Sharp uniqueness class for solutions of wave equations on graphs.
Proves global existence for quasilinear wave equations with weak null condition.
problem Global existence for quasilinear wave equations with weak null condition.
method p-weighted energy method, hierarchical structure in semilinear terms, robust methods.
result Proves global existence for a larger class of quasilinear wave equations.
Rotational elasticitymath-ph
We consider an infinite 3-dimensional elastic continuum whose material points experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which give…
Stable blowup profile identified for wave maps in all dimensions.
problem Stability of blowup solutions for wave maps in supercritical energy.
method Novel stability analysis using similarity variables on the whole space.
result Global nonlinear stability of the corotational self-similar blowup profile.