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The method of characteristics transforms a PDE into a system of ODEs along special curves called characteristics, reducing the problem to something solvable with ODE techniques. For a first-order PDE, characteristics are curves along which the solution is constant or satisfies a simple ODE. For second-order hyperbolic PDEs, characteristics are the curves along which information propagates. This method reveals the geometric structure underlying wave propagation and transport phenomena, and provides explicit solutions to important classes of PDEs.
The method of characteristics is based on a beautiful geometric insight: a first-order PDE can be interpreted as a statement about directional derivatives. The transport equation u_t + cu_x = 0 says that the directional derivative of u in the direction (1, c) in the (t, x)-plane is zero. This means u is constant along lines with slope dx/dt = c, called characteristics. The general solution is u(x,t) = f(x - ct) for any function f determined by initial data.
For a general first-order quasilinear PDE of the form a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u), the method produces a system of characteristic ODEs: dx/ds = a, dy/ds = b, du/ds = c, where s parameterizes the characteristic curves. Along these curves, the PDE reduces to an ODE for u, which can often be solved explicitly. The solution surface in (x, y, u)-space is swept out by these characteristic curves, each originating from a point on the initial curve.
The real power and subtlety of the method emerges with nonlinear equations. In Burgers' equation u_t + uu_x = 0, the characteristic speed depends on the solution itself—faster parts of the wave overtake slower parts, causing characteristics to converge and eventually cross. At that moment, a smooth solution ceases to exist and shock waves form. This breakdown is not a defect of the method but a genuine physical phenomenon: it explains the formation of sonic booms, breaking ocean waves, and traffic jams.
For second-order hyperbolic PDEs like the wave equation u_tt - c²u_xx = 0, the characteristics come in two families: x - ct = constant and x + ct = constant. D'Alembert's formula u(x,t) = ½[f(x-ct) + f(x+ct)] + (1/2c)∫g(s)ds is a direct consequence of the characteristic structure, decomposing the solution into right-traveling and left-traveling waves. The domain of dependence and domain of influence for any point are determined by the characteristic cone through that point.
The method of characteristics extends to systems of first-order PDEs and to higher dimensions, where characteristic surfaces replace characteristic curves. In gas dynamics, the characteristic structure of the Euler equations determines the propagation of sound waves, contact discontinuities, and shock waves. Understanding characteristics is essential for designing numerical schemes (upwind methods, Godunov schemes) that respect the underlying physics of wave propagation.