PDE Characteristics
1 PDE Characteristics
Characteristics are special curves (or surfaces in higher dimensions) in the domain of a [[partial-differential-equations|PDE]] along which information travels and along which the PDE simplifies. They serve two purposes:
- For first-order PDEs they reduce the problem to a family of ODEs, which is the Method of Characteristics.
- For second-order PDEs they determine the type of the equation (hyperbolic, parabolic or elliptic), and with it the qualitative behaviour of solutions.
The guiding picture is the Transport Equation \(u_t+cu_x=0\): the solution is constant along the lines \(x-ct=\text{const}\), so these lines carry the information, and all that matters is to follow them back to the initial data.
2 First-Order PDEs
2.1 Characteristic Curves
Consider the quasilinear first-order PDE for \(u(x,y)\) \[ a(x,y,u)\,u_x+b(x,y,u)\,u_y=c(x,y,u). \] The left-hand side is a derivative of \(u\) in the direction \((a,b)\) (see directional derivative), so the PDE only involves the behaviour of \(u\) along that direction field. Follow a curve \((x(s),y(s))\) whose tangent is \((a,b)\) and set \(U(s)=u(x(s),y(s))\). By the chain rule \[ \frac{dU}{ds}=u_x\frac{dx}{ds}+u_y\frac{dy}{ds}=au_x+bu_y=c. \]
The characteristic equations of the PDE above are the ODE system \[ \frac{dx}{ds}=a(x,y,u),\qquad\frac{dy}{ds}=b(x,y,u),\qquad\frac{du}{ds}=c(x,y,u). \] Their solution curves \((x(s),y(s),u(s))\) are the characteristic curves, and their projections \((x(s),y(s))\) to the \((x,y)\)-plane are the characteristics (or base characteristics).
Note: In the linear case \(a,b\) do not depend on \(u\), so the base characteristics can be found first from \(dy/dx=b/a\) and the equation for \(u\) is then solved along them. In the quasilinear case the three equations are coupled and must be solved together.
2.2 Examples
For \(u_t+cu_x=0\): \(\dot t=1\), \(\dot x=c\), \(\dot u=0\). The characteristics are the lines \(x-ct=x_0\) and \(u\) is constant on each, so \(u=\varphi(x-ct)\) for any function \(\varphi\).
For \(yu_x-xu_y=0\): \(\dot x=y\), \(\dot y=-x\), \(\dot u=0\). Since \(\frac{d}{ds}(x^2+y^2)=2xy-2xy=0\), the characteristics are circles \(x^2+y^2=r^2\) about the origin and \(u\) is constant on each circle. The general solution is \[ u(x,y)=\varphi(x^2+y^2), \] a radially symmetric function, as we expect since the operator \(y\partial_x-x\partial_y\) is the generator of rotations.
For \(u_t+uu_x=0\) the speed depends on \(u\): \(\dot t=1\), \(\dot x=u\), \(\dot u=0\). So \(u\) is constant on each characteristic, which are therefore straight lines \[ x=x_0+u_0(x_0)\,t, \] with slope determined by the initial data, and the solution is \(u(t,x)=u_0(x_0)\) where \(x_0\) solves this equation. Take \(u_0(x)=-x\). Then \(x=x_0(1-t)\), so \[ u(t,x)=-\frac{x}{1-t}. \] All characteristics meet at the origin at \(t=1\), where the solution blows up. Beyond that time the classical solution breaks down (a shock forms). For \(u_0\) with \(u_0'<0\) somewhere the breaking time is \(t^*=-1/\min u_0'\).
2.3 Data on a Curve
To determine a solution we prescribe \(u\) on an initial curve \(\Gamma:(x_0(r),y_0(r))\), \(u=u_0(r)\), and take the characteristic through each point of \(\Gamma\) as the starting point of the ODE system.
If \(a,b,c\) are smooth and the data curve \(\Gamma\) is transversal to the characteristics, \[ a\,y_0'(r)-b\,x_0'(r)\neq0, \] then there is a unique local solution through \(\Gamma\). If \(\Gamma\) is itself a characteristic, then either no solution exists or infinitely many do.
Proof. The map \((r,s)\mapsto(x(r,s),y(r,s))\) obtained by solving the characteristic equations from \(\Gamma\) has Jacobian at \(s=0\) \[ \det\begin{pmatrix}x_r&x_s\\y_r&y_s\end{pmatrix}=x_0'\,b-y_0'\,a\neq0, \] so by the inverse function theorem it is locally invertible, and \(u(x,y)=U(r(x,y),s(x,y))\) is a well-defined solution. If \(\Gamma\) is characteristic, \(a y_0'-bx_0'=0\) and the map degenerates.
For \(u_x+u_y=0\) the solutions are \(u=\varphi(x-y)\) and the characteristics are the lines \(y=x+\text{const}\). If we prescribe \(u(x,x)=g(x)\) on the characteristic \(y=x\) then \(\varphi(0)=g(x)\) for all \(x\), so there is no solution unless \(g\) is a constant and then there are infinitely many (\(\varphi\) is arbitrary away from \(0\)). Prescribing instead \(u(x,0)=g(x)\) on the transversal line \(y=0\) gives the unique solution \(u=g(x-y)\).
3 Second-Order PDEs
3.1 Characteristic Curves
Consider the second-order PDE in two variables \[ Au_{xx}+2Bu_{xy}+Cu_{yy}+(\text{lower order terms})=0, \] with coefficients \(A,B,C\) functions of \((x,y)\). Seek new coordinates \(\xi=\xi(x,y)\) for which \(u_{\xi\xi}\) does not appear. Substituting \(u(x,y)=v(\xi(x,y),\eta(x,y))\), the coefficient of \(v_{\xi\xi}\) is \[ A\xi_x^2+2B\xi_x\xi_y+C\xi_y^2, \] so we need \(\xi\) to satisfy the first-order PDE \(A\xi_x^2+2B\xi_x\xi_y+C\xi_y^2=0\). Level sets \(\xi=\text{const}\) are curves \(y=y(x)\) with slope \(dy/dx=-\xi_x/\xi_y\), and substituting \(\xi_x/\xi_y=-dy/dx\) gives:
The characteristic curves of \(Au_{xx}+2Bu_{xy}+Cu_{yy}+\dots=0\) are the curves \(y(x)\) solving \[ A\left(\frac{dy}{dx}\right)^2-2B\frac{dy}{dx}+C=0 \quad\Longleftrightarrow\quad \frac{dy}{dx}=\frac{B\pm\sqrt{B^2-AC}}{A}. \]
The number of real characteristic families is determined by the sign of the discriminant \(D=B^2-AC\).
3.2 Classification
At a point the equation is
- hyperbolic if \(B^2-AC>0\): two real families of characteristics,
- parabolic if \(B^2-AC=0\): one real family,
- elliptic if \(B^2-AC<0\): no real characteristics.
Using the characteristics as coordinates reduces the equation to a canonical form:
| Type | Characteristic coordinates | Canonical form | Model |
|---|---|---|---|
| Hyperbolic | \(\xi,\eta\) the two families | \(v_{\xi\eta}+\dots=0\) | Wave Equation |
| Parabolic | \(\eta\) the single family, \(\xi\) any independent | \(v_{\xi\xi}+\dots=0\) | Heat Equation |
| Elliptic | \(\xi\pm i\eta\) complex family | \(v_{\xi\xi}+v_{\eta\eta}+\dots=0\) | Laplace Equation |
- Wave equation \(u_{tt}-c^2u_{xx}=0\). In the variables \((t,x)\) we have \(A=1,B=0,C=-c^2\), so \(D=c^2>0\) and \(dx/dt=\pm c\): the characteristics are \(x\pm ct=\text{const}\). Taking these as coordinates gives \(u_{\xi\eta}=0\), as in d’Alembert’s solution.
- Heat equation \(u_t=ku_{xx}\). In \((x,t)\), \(A=k\) and \(B=C=0\) so \(D=0\): parabolic, with the single family \(dt/dx=0\), i.e. the lines \(t=\text{const}\).
- Laplace equation \(u_{xx}+u_{yy}=0\). \(A=C=1,B=0\) so \(D=-1<0\): elliptic, with no real characteristics (\(dy/dx=\pm i\)).
Consider \(u_{xx}+2u_{xy}-3u_{yy}=0\). Here \(A=1,B=1,C=-3\) and \(D=4>0\), so it is hyperbolic, with \[ \frac{dy}{dx}=1\pm2\in\{3,-1\}\quad\Longrightarrow\quad \xi=y-3x,\quad\eta=y+x. \] By the chain rule, \(u_x=-3v_\xi+v_\eta\) and \(u_y=v_\xi+v_\eta\), so \[ \begin{aligned} u_{xx}&=9v_{\xi\xi}-6v_{\xi\eta}+v_{\eta\eta},\\ u_{xy}&=-3v_{\xi\xi}-2v_{\xi\eta}+v_{\eta\eta},\\ u_{yy}&=v_{\xi\xi}+2v_{\xi\eta}+v_{\eta\eta}, \end{aligned} \] and the equation becomes \(-16v_{\xi\eta}=0\). Hence \(v=F(\xi)+G(\eta)\) and \[ u(x,y)=F(y-3x)+G(y+x). \]
3.3 Mixed Type
When the coefficients depend on position, the type can change from point to point. The Tricomi equation \(u_{xx}+xu_{yy}=0\) has \(D=-x\), so it is elliptic for \(x>0\), parabolic on the line \(x=0\) and hyperbolic for \(x<0\). This occurs for example in transonic flow, where the flow changes from subsonic (elliptic) to supersonic (hyperbolic).
3.4 Several Variables
For \(\sum_{i,j}a_{ij}u_{x_ix_j}+\dots=0\) a surface \(\{\phi=\text{const}\}\) is characteristic if \[ \sum_{i,j}a_{ij}\,\phi_{x_i}\phi_{x_j}=0. \] For the wave equation this reads \(\phi_t^2=c^2|\nabla\phi|^2\) (the eikonal equation), whose solutions include the light cones \(|x-x_0|=c(t-t_0)\). The type is read from the signs of the eigenvalues of \((a_{ij})\): all the same sign is elliptic, one sign differing from the rest is hyperbolic, and a zero eigenvalue is parabolic.
4 Interpretation
For a hyperbolic equation:
- A discontinuity in the data (or in derivatives of the solution) is propagated only along the characteristics.
- The solution at \((x_0,t_0)\) depends only on the data in the region bounded by the characteristics through that point (the domain of dependence), and influences only the region between the characteristics going forward (the domain of influence).
| Type | Real characteristics | Information speed | Singularities | Example |
|---|---|---|---|---|
| Hyperbolic | two families | finite | travel along characteristics | wave |
| Parabolic | one family | infinite | smoothed out instantly | heat |
| Elliptic | none | no propagation | solutions smooth in the interior | Laplace |
Elliptic equations have no characteristics, so no direction is special: the solution at a point depends on the data on the whole boundary and is as smooth as the coefficients allow. This is why elliptic problems are posed as boundary value problems and hyperbolic and parabolic ones as initial value problems.
Note: An initial curve being characteristic is exactly the degenerate case in which the Cauchy problem fails. For the wave equation the data \(u,u_t\) on \(t=0\) is fine, since \(t=0\) is never a characteristic for \(c<\infty\), but data on the light line \(x=ct\) does not determine the solution.