Hyperbolic partial differential equation

A partial differential equation whose principal part has real characteristic directions and supports finite-speed propagation. Wave equations are standard examples.

Characteristic curve: A curve along which a partial differential equation reduces to an ordinary differential equation or its solution carries information. Hyperbolic equations transmit data along these distinguished paths.
Principal symbol: The polynomial or matrix-valued expression formed from the highest-order derivative terms of a differential operator. Its algebraic structure determines whether a differential operator is hyperbolic.
Wave equation: A second-order partial differential equation describing propagation of waves at a specified speed. It is the canonical linear example of a hyperbolic partial differential equation.
Elliptic partial differential equation: A partial differential equation whose principal symbol has elliptic structure, as in Laplace's equation. Elliptic equations typically impose global constraints rather than finite-speed wave propagation.
Domain of dependence: The region of initial or boundary data that can influence a solution at a specified point. Characteristic directions determine which data can affect each point.
Characteristic variety: The set of covectors where the principal symbol of a differential operator becomes singular. Its real geometry encodes characteristic directions for hyperbolic equations.
Maxwell's equations: A system of equations describing how electric and magnetic fields are generated and evolve. In suitable formulations, electromagnetic fields obey hyperbolic evolution equations.
Parabolic partial differential equation: A partial differential equation with parabolic principal structure, commonly modeling diffusion. The heat equation's instantaneous spreading contrasts with hyperbolic finite-speed propagation.
Finite propagation speed: The property that disturbances influence a system only within a bounded region over finite time. It distinguishes hyperbolic propagation from instantaneous influence in many other equations.
Initial data: Values or derivatives prescribed on an initial surface to determine an evolution equation's solution. Their placement relative to characteristic surfaces affects whether evolution is possible.

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