SIAM Journal on Mathematical Analysis

Article	Citations
A Rigorous Derivation of a Boltzmann System for a Mixture of Hard-Sphere Gases	50
Stability Results for the Robin-Laplacian on Nonsmooth Domains	35
Global Solvability and Vanishing Shear Viscosity Limit for a Simplified Compressible Navier--Stokes System with Temperature-Dependent Viscosity	29
Local Existence of Smooth Solutions for the Semigeostrophic Equations on Curved Domains	27
Equivariant Pyragas Control of Discrete Waves	27
The Hasimoto Transformation for a Finite Length Vortex Filament and its Application	21
A Perturbative Approach to the Parabolic Optimal Transport Problem	17
Long-Time Limit of Nonlinearly Coupled Measure-Valued Equations that Model Many-Server Queues with Reneging	15
Global-in-Time Regular Unique Solutions with Positive Temperature to One-Dimensional Thermoelasticity	14
Modified Scattering of Small Data Solutions for the Vlasov–Poisson System with a Repulsive Potential	12
Discrete Approximation of the Griffith Functional by Adaptive Finite Elements	12
Incompressible Limit of Ideal Magnetohydrodynamic Equations with a Perfectly Conducting Wall Condition	12
Two New Functional Inequalities and Their Application to the Eventual Smoothness of Solutions to a Chemotaxis-Navier–Stokes System with Rotational Flux	12
Static Solutions to the Spherically Symmetric Einstein–Vlasov System: A Particle Number–Casimir Approach	11
Large Global Solutions for the Energy-Critical Nonlinear Schrödinger Equation	10
On the Ill-Posedness of the Triple Deck Model	9
Navier–Stokes Blowup Rates in Certain Besov Spaces Whose Regularity Exceeds the Critical Value by \({\boldsymbol{\epsilon \in [1,2]}}\)	9
Asymptotic Stability of KdV Solitons on the Half-Line: A Study in the Energy Space	9
Invisibility Cloaking and Transformation Optics for Three Dimensional Manifolds and Applications in Cosmology	9
Strict g-Convexity for Generated Jacobian Equations with Applications to Global Regularity	9
Entropic Regularization of NonGradient Systems	9
Existence and Stability of Localized Patterns in the Population Models with Large Advection and Strong Allee Effect	9
Local Existence for the Landau Equation with Hard Potentials	9
Effective Fronts of Polygon Shapes in Two Dimensions	8
Generalized Impedance Boundary Conditions with Vanishing or Sign-Changing Impedance	8

Stochastic Logarithmic Schrödinger Equations: Energy Regularized Approach	8
On One-Dimensional Bose Gases with Two-Body and (Critical) Attractive Three-Body Interactions	8
On the Diffusive Limits of Radiative Heat Transfer System I: Well-Prepared Initial and Boundary Conditions	8
On the Temperature Distribution of a Body Heated by Radiation	8
Existence of Solutions for a Class of One-Dimensional Models of Pedestrian Evacuations	8
Scattering for Nonradial 3D NLS with Combined Nonlinearities: The Interaction Morawetz Approach	8
Exponential Mixing for the White-Forced Complex Ginzburg–Landau Equation in the Whole Space	8
Weak-Strong Uniqueness for Maxwell--Stefan Systems	7
Weak-Strong Uniqueness for an Elastic Plate Interacting with the Navier--Stokes Equation	7
Conservation, Convergence, and Computation for Evolving Heterogeneous Elastic Wires	7
Global Strong Well-Posedness of the Stochastic Bidomain Equations with FitzHugh–Nagumo Transport	7
Renormalization and Existence of Finite-Time Blow-Up Solutions for a One-Dimensional Analogue of the Navier–Stokes Equations	7
Weighted Analytic Regularity for the Integral Fractional Laplacian in Polygons	7
$\Gamma$-Convergence for Functionals Depending on Vector Fields. II. Convergence of Minimizers	6
Spectral Instability of Peakons in the $b$-Family of the Camassa--Holm Equations	6
Leray's Backward Self-Similar Solutions to the 3D Navier--Stokes Equations in Morrey Spaces	6
\({\boldsymbol{{L}^{1}}}\)-Gradient Flow of Convex Functionals	6
On the Half-Space or Exterior Problems of the 3D Compressible Elastic Navier–Stokes–Poisson Equations	6
The Magnetic Liouville Equation as a Semiclassical Limit	6
Metric Entropy for Hamilton--Jacobi Equations with Uniformly Directionally Convex Hamiltonian	6
Novel Resolution Analysis for the Radon Transform in \(\mathbb R^2\) for Functions with Rough Edges	6
Higher Regularity for Entropy Solutions of Conservation Laws with Geometrically Constrained Discontinuous Flux	6
Newtonian Limits and Global Stability of Shock Solutions for Piston Problem to the Relativistic Full Euler System	6
Diffusion-Driven Blow-Up for a Nonlocal Fisher-KPP Type Model	6
Stability for Time-Domain Elastic Wave Equations	6
Global Regularity for a Nonlocal PDE Describing Evolution of Polynomial Roots Under Differentiation	6
The Complex-Scaled Half-Space Matching Method	6
Global in Time Weak Solutions to Singular Three-Dimensional Quasi-Geostrophic Systems	6
A Reduced Model for Plates Arising as Low-Energy \(\boldsymbol{\Gamma}\) -Limit in Nonlinear Magnetoelasticity	6
On Well-Posedness of Quantum Fluid Systems in the Class of Dissipative Solutions	6
Uniqueness of Nontrivial Solutions for Degenerate Monge–Ampere Equations	6
Global Existence of Weak Solutions to the Compressible Navier--Stokes Equations with Temperature-Depending Viscosity Coefficients	6
Bistable Wavefronts in the Delayed Belousov–Zhabotinsky Reaction	5
Localized Sensitivity Analysis at High-Curvature Boundary Points of Reconstructing Inclusions in Transmission Problems	5
Behavior of Spiral Wave Spectra with a Rank-Deficient Diffusion Matrix	5
Free and Harmonic Trapped Spin-1 Bose–Einstein Condensates in \({{\mathbb{R}}^{3}}\)	5
Long-Time Behavior and Stability for Quasilinear Doubly Degenerate Parabolic Equations of Higher Order	5
Global Solution to the Klein--Gordon--Zakharov Equations with Uniform Energy Bounds	5
Quasi-neutral Limit of Nernst–Planck–Navier–Stokes System	5
Sharp Nonuniqueness of Solutions to Stochastic Navier–Stokes Equations	5
Rigorous Envelope Approximation for Interface Wave Packets in Maxwell’s Equations with Two Dimensional Localization	5
Existence of Pulses for Monotone Reaction-Diffusion Systems	5
Dynamical Systems and Hamilton–Jacobi–Bellman Equations on the Wasserstein Space and their L2 Representations	5
On the Existence and Stability of Modified Maxwell Steklov Eigenvalues	4
Szegö--Weinberger Type Inequalities for Symmetric Domains with Holes	4
Time Analyticity for Nonlocal Parabolic Equations	4
A Bourgain–Brezis–Mironescu Formula Accounting for Nonlocal Antisymmetric Exchange Interactions	4
A New Approach to Space-Time Boundary Integral Equations for the Wave Equation	4
Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm	4
Inverse Boundary Problem for the Two Photon Absorption Transport Equation	4

The Bilaplacian with Robin Boundary Conditions	4
Asymptotics for Semidiscrete Entropic Optimal Transport	4
On the Stability of Radial Solutions to an Anisotropic Ginzburg--Landau Equation	4
Global Quasi-Neutral Limit for a Two-Fluid Euler–Poisson System in Several Space Dimensions	4
Asymptotics of the Hard Edge Pearcey Determinant	4
Fuglede-Type Arguments for Isoperimetric Problems and Applications to Stability Among Convex Shapes	4
An Inverse Problem for Hypersonic Flow Past a Curved Cone	4
Unique Continuation Properties from One Time for Hyperbolic Schrödinger Equations	4
Equilibrium Configurations of a Symmetric Body Immersed in a Stationary Navier–Stokes Flow in a Planar Channel	4
Almost Global Existence for Kirchhoff Equations Around Global Solutions	4
Subsonic Flows Past a Profile with a Vortex Line at the Trailing Edge	4
Inverse Scattering for the Biharmonic Wave Equation with a Random Potential	4
Diffusive Limit of the Vlasov–Poisson–Fokker–Planck Model: Quantitative and Strong Convergence Results	4
Local Well-Posedness for the Boltzmann Equation with Very Soft Potential and Polynomially Decaying Initial Data	4
On Asymptotic Stability on a Center Hypersurface at the Soliton for Even Solutions of the Nonlinear Klein–Gordon Equation When \(\boldsymbol{2 \ge p \gt \frac{5}{3}}\)	4
Formation and Construction of a Multidimensional Shock Wave for the First-Order Hyperbolic Conservation Law with Smooth Initial Data	4
Initial-Boundary Value Problems for Poiseuille Flow of Nematic Liquid Crystal via Full Ericksen–Leslie Model	4
Poisson Equation on Wasserstein Space and Diffusion Approximations for Multiscale McKean–Vlasov Equation	4
The Local Well-Posedness of the Relativistic Vlasov–Maxwell–Landau System with the Specular Reflection Boundary Condition	4
Existence and Asymptotic Behavior of Ground States for Rotating Bose–Einstein Condensates	4
Compressible Navier--Stokes System with Transport Noise	4
Smoluchowski Coagulation Equation with Velocity Dependence	4
Second-Order Homogenization of Periodic Schrödinger Operators with Highly Oscillating Potentials	4
Higher Order Airy and Painlevé Asymptotics for the mKdV Hierarchy	4
Cauchy Theory for General Kinetic Vicsek Models in Collective Dynamics and Mean-Field Limit Approximations	4
Large \(\boldsymbol{|k|}\) Behavior for the Reflection Coefficient for Davey-Stewartson II Equations	4
Rigorous Derivation of Michaelis–Menten Kinetics in the Presence of Slow Diffusion	3
On the Global Well-Posedness and Analyticity of Some Electrodiffusion Models in Ideal Fluids and Porous Media	3
Asymptotic Stability for Diffusion with Dynamic Boundary Reaction from Ginzburg–Landau Energy	3
Decompositions of High-Frequency Helmholtz Solutions via Functional Calculus, and Application to the Finite Element Method	3
Uniform Asymptotic Expansions for the Zeros of Bessel Functions	3
A Sharp Interface Limit of a Nonlocal Variational Model for Pattern Formation in Biomembranes	3
Quantitative Coarse-Graining of Markov Chains	3
Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions	3
Disjoint Data Inverse Problem on Manifolds with Quantum Chaos Bounds	3
Structural Stability of Radial Interior Subsonic Steady-States to n-D Euler-Poisson System of Semiconductor Models with Sonic Boundary	3
Nonlinear Inverse Optimal Transport: Identifiability of the Transport Cost from Its Marginals and Optimal Values	3
Optimal Decay Rates of the Compressible Euler Equations with Time-Dependent Damping in \({\mathbb {R}}^n\): (II) Overdamping Case	3
Transonic Steady-States of Euler--Poisson Equations for Semiconductor Models with Sonic Boundary	3
Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations	3
On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole \(\mathbb{R}^N\) via Lusternik-Schnirelmann Category	3
The Calderón Problem for the Fractional Wave Equation: Uniqueness and Optimal Stability	3
Differentiability With Respect to the Initial Condition for Hamilton--Jacobi Equations	3
Improved Concentration of Laguerre and Jacobi Ensembles	3
Monotonicity Properties of Limits of Solutions to the Semidiscrete Scheme for a Class of Perona–Malik Type Equations	3
Gradient Estimates for Stokes and Navier--Stokes Systems with Piecewise DMO Coefficients	3
A Bayesian Model for Dynamic Mass Reconstruction from PET Listmode Data	3
On Feasibility of Extrapolation of Completely Monotone Functions	3
Global Limit Theorem for Parabolic Equations with a Potential	3
Planar Viscous Shocks with Periodic Perturbations for Scalar Multidimensional Viscous Conservation Laws	3
Convergence of Boundary Layers of Chemotaxis Models with Physical Boundary Conditions I: Degenerate Initial Data	3
Convergence Rates in the Nonrelativistic Limit of the Cubic Klein–Gordon Equation	3
Longer Lifespan for Many Solutions of the Kirchhoff Equation	3
Almost Everywhere Nonuniqueness of Integral Curves for Divergence-Free Sobolev Vector Fields	3
Analysis of the Rolling Carpet Strategy to Eradicate an Invasive Species	3
Linear Stability of the Couette Flow in the 3D Isentropic Compressible Navier--Stokes Equations	3
Long Time Decay and Asymptotics for the Complex mKdV Equation	3
Stability of Traveling Oscillating Fronts in Complex Ginzburg Landau Equations	3
Mean-Field Limit Derivation of a Monokinetic Spray Model with Gyroscopic Effects	3
Theory of Weak Asymptotic Autonomy of Pullback Stochastic Weak Attractors and Its Applications to 2D Stochastic Euler Equations Driven by Multiplicative Noise	3
Local Regularity Near Boundary for the Stokes and Navier–Stokes Equations	3
A Representation Formula for Viscosity Solutions of Nonlocal Hamilton–Jacobi Equations and Applications	3
Global Well-Posedness and Scattering for Fourth-Order Schrödinger Equations on Waveguide Manifolds	3
Nonlocal Problems with Local Boundary Conditions II: Green’s Identities and Regularity of Solutions	3
On an Electrostatic Problem and a New Class of Exceptional Subdomains of \(\mathbb{R}^3\)	3
The Second Bogolyubov Theorem and Global Averaging Principle for SPDEs with Monotone Coefficients	3
Existence and Smoothing Effect of Measure Valued Solution to the Homogeneous Boltzmann Equation with Debye–Yukawa Potential	3
The Problem of Optimal Camouflaging	3
On Saint-Venant Compatibility and Stress Potentials in Manifolds with Boundary and Constant Sectional Curvature	3
A Strong Comparison Principle for the Generalized Dirichlet Problem for Monge–Ampère	3
Modulated Energy Estimates for Singular Kernels and their Applications to Asymptotic Analyses for Kinetic Equations	3
Conditional Regularity for the Compressible MHD System with Inhomogeneous Boundary Data	2
Global Solutions of Wave-Klein--Gordon Systems in 2+1 Dimensional Space-Time with Strong Couplings in Divergence Form	2
Weak Optimal Transport with Unnormalized Kernels	2
Optimal Regularity of Mixed Dirichlet-Conormal Boundary Value Problems for Parabolic Operators	2
On the Treatment of Exterior Domains for the Time-Harmonic Equations of Stellar Oscillations	2
Propagation of Moments for Large Data and Semiclassical Limit to the Relativistic Vlasov Equation	2
The Discrete Collision-Induced Breakage Equation with Mass Transfer: Well-Posedness and Stationary Solutions	2
Large-Time Behavior of Composite Waves of Viscous Shocks for the Barotropic Navier–Stokes Equations	2
Partial Data Inverse Problems for Magnetic Schrödinger Operators with Potentials of Low Regularity	2

On Echo Chains in the Linearized Boussinesq Equations Around Traveling Waves	2
Elliptic Equations with Degenerate Weights	2
Weakly Nonlinear Multiphase Geometric Optics for Hyperbolic Quasilinear Boundary Value Problems: Construction of a Leading Profile	2
A Regularity Theory for Static Schrödinger Equations on \(\boldsymbol{\mathbb{R}}\)d in Spectral Barron Spaces	2
Approximation of Calogero–Moser Lattices by Benjamin–Ono Equations	2
Infinite-Dimensional Hamilton–Jacobi Equations for Statistical Inference on Sparse Graphs	2
Corotational Hookean Models of Dilute Polymeric Fluids: Existence of Global Weak Solutions, Weak-Strong Uniqueness, Equilibration, and Macroscopic Closure	2
Mild Ill-Posedness for 2D Magneto-Micropolar Fluid Equations Near a Background Magnetic Field	2
Asymptotic Theory for a General Class of Short-Range Interaction Functionals	2
The Scattering Resonances for Schrödinger-Type Operators with Unbounded Potentials	2
Unconditional Regularity and Trace Results for the Isentropic Euler Equations with \({\boldsymbol{\gamma = 3}}\)	2
Gradient Flows for Probabilistic Frame Potentials in the Wasserstein Space	2
The Peskin Problem with \({\boldsymbol{\dot B^1_{\infty,\infty }}}\) Initial Data	2
Multiple Solutions to Cylindrically Symmetric Curl-Curl Problems and Related Schrödinger Equations with Singular Potentials	2
Analysis of Time-Harmonic Maxwell Impedance Problems in Anisotropic Media	2
Partial Data Inverse Problem for Hyperbolic Equation with Time-Dependent Damping Coefficient and Potential	2
Exceptional Points in Parity--Time-Symmetric Subwavelength Metamaterials	2
Dynamics of Threshold Solutions for Energy Critical NLS with Inverse Square Potential	2
Linearized Optimal Transport on Manifolds	2
Inverse Resonance Problems for Energy-Dependent Potentials on the Half-Line	2
Generalized Maslov Indices for Non-Hamiltonian Systems	2
Plasma Echoes Near Stable Penrose Data	2
On the Placement of an Obstacle so as to Optimize the Dirichlet Heat Content	2
On the Convergence from Boltzmann to Navier–Stokes–Fourier for General Initial Data	2
Enhanced Dissipation and Transition Threshold for the Poiseuille Flow in a Periodic Strip	2
Unique Reconstruction of the Heat-Reflection Indices at Solid Interfaces	2
Upper Bound for the Grand Canonical Free Energy of the Bose Gas in the Gross–Pitaevskii Limit	2
Ramified Optimal Transportation with Payoff on the Boundary	2
Global Regular Solutions for one-dimensional Degenerate Compressible Navier--Stokes Equations with Large Data and Far Field Vacuum	2
A McKean--Vlasov SDE and Particle System with Interaction from Reflecting Boundaries	2
Existence, Nonexistence, and Degeneracy of Limit Solutions to \({{p}}\)-Laplace Problems Involving Hardy Potentials as \({{p}\to 1^+}\)	2
Nonconvergence of the Rotating Stratified Flows Toward the Quasi-Geostrophic Dynamics	2
The Anisotropic Calderón Problem for High Fixed Frequency	2
Immediate Blowup of Entropy-Bounded Classical Solutions to the Vacuum Free Boundary Problem of Nonisentropic Compressible Navier–Stokes Equations	2
Strichartz and Local Smoothing Estimates for Stochastic Dispersive Equations with Linear Multiplicative Noise	2
A Dynamical Approach to Lower Gradient Estimates for Viscosity Solutions of Hamilton–Jacobi Equations	2
The Calderón Problem with Finitely Many Unknowns is Equivalent to Convex Semidefinite Optimization	2
Pinning in the Extended Lugiato–Lefever Equation	2
Traceability of Water Pollution: An Inversion Scheme via Dynamic Complex Geometrical Optics Solutions	2
NonLocal Optimized Schwarz Method for the Helmholtz Equation with Physical Boundaries	2
Persistence, Extinction, and Spreading Properties of Noncooperative Fisher–KPP Systems in Space-Time Periodic Media	2
Inverse Elastic Scattering for a Random Potential	2
Gradient Regularity for a Class of Widely Degenerate Parabolic Systems	2
On Fractional Benney Type Systems	2
Vanishing Shear Viscosity Limit for the Navier–Stokes Equations with Cylindrical Symmetry: Boundary Layer and Optimal Convergence Rate	2
Propagating Terrace in a Two-Tubes Model of Gravitational Fingering	2
The Monge–Ampère System: Convex Integration in Arbitrary Dimension and Codimension	2
Global Subsonic Jet with Strong Transonic Shock over a Convex Cornered Wedge for the Two-Dimensional Steady Full Euler Equations	2
Two-Layer Neural Networks with Values in a Banach Space	2
Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift	2
Planar Stationary Solution to Outflow Problem for Compressible Heat Conducting Gas in \(\mathbb{R}^3_+\): Stability and Convergence Rate	2
Distributional Extension and Invertibility of the \(\boldsymbol{k}\)-Plane Transform and Its Dual	2
An Inverse Problem for the Porous Medium Equation with Partial Data and a Possibly Singular Absorption Term	2
Linear Stability Analysis for a Free Boundary Problem Modeling Multilayer Tumor Growth with Time Delay	2
On a Phase Field Model for RNA-Protein Dynamics	2
Ground States of Two-Component Bose–Einstein Condensates with Josephson Junction: The Endpoint Cases	2
Global Stability and Nonvanishing Vacuum States of 3D Compressible Navier–Stokes Equations	2
Parabolic Approximation of Quasilinear Wave Equations with Applications in Nonlinear Acoustics	2
Nonlinear Fokker–Planck Equations with Time-Dependent Coefficients	2
On an Obstacle Problem for the Brakke Flow with a Generalized Right-Angle Boundary Condition	2
Stress Blowup Analysis When a Suspending Rigid Particle Approaches the Boundary in Stokes Flow: 2-Dimensional Case	2
A Simplified Voltage-Conductance Kinetic Model for Interacting Neurons and Its Asymptotic Limit	2
Nontrivial Equilibrium Solutions and General Stability for Stochastic Evolution Equations with Pantograph Delay and Tempered Fractional Noise	2
Vanishing Capillarity Limit of the Navier--Stokes--Korteweg System in One Dimension with Degenerate Viscosity Coefficient and Discontinuous Initial Density	1
Identification of Nonlinear Beam-Hardening Effects in X-ray Tomography	1
Asymptotically Autonomous Robustness in Probability of Random Attractors for Stochastic Navier-Stokes Equations on Unbounded Poincaré Domains	1
Erratum: Lipschitz Stability for Backward Heat Equation with Application to Fluorescence Microscopy	1
From the Vlasov-Poisson-Boltzmann System to the Vlasov-Poisson-Landau System with Coulomb Potential	1
Large Deviations for Stochastic Generalized Porous Media Equations Driven by Lévy Noise	1
Sobolev Regularity for Optimal Transport Maps of Nonconvex Planar Domains	1
Asymptotic Behavior of Solutions to the Cauchy Problem for 1D \({\boldsymbol{p}}\)-System with Space-Dependent Damping	1
Terrace Solutions for Non-Lipschitz Multistable Nonlinearities	1
Critical Threshold for Global Regularity of the Euler--Monge--Ampère System with Radial Symmetry	1
Weak Solutions for a Bifluid Model for a Mixture of Two Compressible Noninteracting Fluids with General Boundary Data	1
Global Regularity for a Radiation Hydrodynamics Model with Viscosity and Thermal Conductivity	1
Normalized Solutions for Lower Critical Choquard Equations with Critical Sobolev Perturbation	1
The Calderón Problem for Space-Time Fractional Parabolic Operators with Variable Coefficients	1
Symmetry of Periodic Traveling Waves for Nonlocal Dispersive Equations	1
Isothermal Limit of Entropy Solutions of the Euler Equations for Isentropic Gas Dynamics	1
Interface Propagation Properties for a Nonlocal Thin-Film Equation	1
Homogenization for Locally Periodic Elliptic Problems on a Domain	1
A Regularity Theory for Parabolic Equations with Anisotropic Nonlocal Operators in \(\boldsymbol{L_{{q}}(L_{{p}})}\) Spaces	1
On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators	1
Global Existence and Optimal Time-Decay Rates of the Compressible Navier–Stokes–Euler System	1
Zero-Inertia Limit: From Particle Swarm Optimization to Consensus-Based Optimization	1
Supersonic Reacting Jet Flows from a Three-Dimensional Divergent Conical Nozzle	1
On a Supersonic-Sonic Patch in the Three-Dimensional Steady Axisymmetric Transonic Flows	1
Approximations of Interface Topological Invariants	1
Curvature Effects in Pattern Formation: Well-Posedness and Optimal Control of a Sixth-Order Cahn–Hilliard Equation	1
Viscosity Solutions for Nonlocal Equations with Space-Dependent Operators	1
Polarized High-frequency Wave Propagation Beyond the Nonlinear Schrödinger Approximation	1
Periodic Perturbations of a Composite Wave of Two Viscous Shocks for 1-d Full Compressible Navier--Stokes Equations	1
Theory of Invariant Manifolds for Infinite-Dimensional Nonautonomous Dynamical Systems and Applications	1
A Mathematical Study of a Hyperbolic Metamaterial in Free Space	1
A Geometric Proof of the Quasi-Linearity of the Water Waves System	1