Characteristics and transport

Consider u_t+a(x,t)u_x=b(x,t,u). Along a path X(t), the chain rule gives d u(X(t),t)/dt=u_t+X'(t)u_x. Choosing X'=a converts the PDE into the scalar ODE U'=b(X,t,U). Initial values for X and U come from a curve carrying the prescribed data. If the coefficient a is smooth enough to give a unique flow and the source is locally Lipschitz in U, this characteristic system supplies a local construction. Its existence interval and regularity still require checking.

For constant a=c and zero source, X(t)=ξ+ct and U(t)=f(ξ). Solving ξ=x−ct gives u(x,t)=f(x−ct). The minus sign describes a profile moving right when c is positive: the peak initially at ξ arrives at ξ+ct. No stretching or smoothing occurs because every point moves at the same speed. On the whole line, translation preserves every Lp norm for which the initial profile belongs to Lp, and transports nonsmooth profiles as well when the equation is interpreted distributionally.

On a finite interval with positive speed, characteristics enter through the left endpoint and leave through the right. Prescribing the left boundary history is therefore natural; prescribing an independent right boundary history can overdetermine the problem. At a given interior point, trace the characteristic backward. It either reaches the initial line first or reaches the inflow boundary first. That encounter determines which datum to use. At the corner, compatible values are needed if a continuous classical solution is required.

A general data curve must also be noncharacteristic: its tangent cannot coincide with the direction along which the PDE already specifies change. Giving values along a characteristic may fail to determine neighboring characteristics, or may contradict the ODE along that same curve. Geometrically, solving the problem uses the initial-curve parameter and elapsed characteristic time as local coordinates. A nonzero Jacobian allows inversion. Checking this condition turns a tempting parametric formula into a legitimate local solution rather than a collection of unrelated trajectories.

For inviscid Burgers transport u_t+u u_x=0, the value remains constant along a characteristic but now sets its speed. Starting from ξ, the path is X=ξ+t f(ξ). Its derivative with respect to ξ is 1+t f'(ξ). If an initial slope is negative, faster material on the left catches slower material on the right, and this derivative can vanish. Before that time, the inverse map gives a classical solution; at crossing, assigning both carried values to one point is impossible.

A shock is not obtained by continuing the multivalued classical formula as though it were a function. One instead returns to the conservation law and an admissibility condition, studied later in this module. In the waves lab, set amplitude ratio to zero and play the remaining sin(2x−t) component. Its phase travels at speed one half, an exact constant-speed transport example. Bringing back the second component changes interference, not speed through the local value. The lab does not implement Burgers characteristics, boundary inflow, or shock formation.