Method of characteristics

Characteristics of first-order partial differential equation

For a first-order PDE, the method of characteristics discovers so called characteristic curves along which the PDE becomes an ODE. Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.

Two-dimensional quasilinear PDE

For the sake of simplicity, we initially direct our attention to the case of a function of two independent variables x and y. Consider a quasilinear PDE of the form

For a differentiable function ${\displaystyle (x,y)\mapsto u(x,y)}$, consider the graph of u, which is the set $${\displaystyle \mathrm{gph}(u)=\{(x,y,z)\in {\mathbb{R}}^3\mid z=u(x,y)\}}$$ A normal vector to ${\displaystyle \mathrm{gph}(u)}$ is given by

$${\displaystyle n(x,y)=\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}(x,y),\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x,y),-1\right).}$$

Consider the vector field

The dot product of the vector field (2) with the normal vector to ${\displaystyle \mathrm{gph}(u)}$ at each ${\displaystyle (x,y,u(x,y))\in \mathrm{gph}(u)}$ is$${\displaystyle \left[\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}(x,y)\\ {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}(x,y)\\ -1\end{array}\right]\cdot \left[\begin{array}{c}a(x,y,u(x,y))\\ b(x,y,u(x,y))\\ c(x,y,u(x,y))\end{array}\right]=a(x,y,u(x,y))\frac{\mathrm{\partial }u}{\mathrm{\partial }x}(x,y)+b(x,y,u(x,y))\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x,y)-c(x,y,u(x,y)).}$$

Comparing the right-hand side of the above equation with (1), it is evident the following statements are equivalent:

• the right-hand side of the above equation is zero;
• ${\displaystyle u}$ is a solution to (1);
• the vector field (2) is orthogonal to the normal vectors of ${\displaystyle \mathrm{gph}(u)}$ at every point ${\displaystyle (x,y,z)\in \mathrm{gph}(u)}$;
• the vector field (2) is tangent to the surface ${\displaystyle \mathrm{gph}(u)}$ at every point ${\displaystyle (x,y,z)\in \mathrm{gph}(u)}$;

In other words, the graph of the solution to (1) is the union of integral curves of the vector field (2). Each integral curve is called a characteristic curve of the PDE (1) equation and follow as the solutions of the characteristic equations:

$${\displaystyle \{\begin{array}{cc}{\displaystyle \frac{dx}{dt}}& =a(x,y,z),\\ {\displaystyle \frac{dy}{dt}}& =b(x,y,z),\\ {\displaystyle \frac{dz}{dt}}& =c(x,y,z).\end{array}}$$

A parametrization invariant form of the Lagrange–Charpit equations is:

${\displaystyle \frac{dx}{a(x,y,z)}=\frac{dy}{b(x,y,z)}=\frac{dz}{c(x,y,z)}.}$

N-dimensional linear and quasilinear PDE

Consider now a PDE of the form

${\displaystyle \sum _{i=1}^n{a}_i(x_1,\dots ,x_n,u)\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}=c(x_1,\dots ,x_n,u).}$

For this PDE to be linear, the coefficients a_i may be functions of the spatial variables only, and independent of u. For it to be quasilinear, a_i may also depend on the value of the function, but not on any derivatives. The distinction between these two cases is inessential for the discussion here.

For a linear or quasilinear PDE, the characteristic curves are given parametrically by

${\displaystyle (x_1,\dots ,x_n,u)=(X_1(s),\dots ,X_n(s),U(s))}$

${\displaystyle u(\mathbf{X}(s))=U(s)}$

for some univariate functions ${\displaystyle s\mapsto (X_i(s){)}_i,U(s)}$ of one real variable ${\displaystyle s}$ satisfying the following system of ordinary differential equations

Equations (4) and (5) give the characteristics of the PDE.

Fully nonlinear PDE

Consider the partial differential equation

where the variables p_i are shorthand for the partial derivatives

${\displaystyle p_i=\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}.}$

Let ${\displaystyle s\mapsto (x_1(s),\dots ,x_n(s),u(s),p_1(s),\dots ,p_n(s))}$ be a curve in R²ⁿ⁺¹. Suppose that u is any solution, and that

${\displaystyle u(s)=u(x_1(s),\dots ,x_n(s)).}$

The derivatives with respect to ${\displaystyle s}$ of ${\displaystyle x_i,}$ ${\displaystyle u,}$ and ${\displaystyle p_i}$ are written as ${\displaystyle {\dot{x}}_i,}$, ${\displaystyle \dot{u},}$ and ${\displaystyle {\dot{p}}_i,}$ respectively. Along a solution, differentiating (6) with respect to s gives

${\displaystyle \sum _i(F_{x_i}+F_u{p}_i){\dot{x}}_i+\sum _i{F}_{p_i}{\dot{p}}_i=0}$

${\displaystyle \dot{u}-\sum _i{p}_i{\dot{x}}_i=0}$

${\displaystyle \sum _i({\dot{x}}_{i}d{p}_i-{\dot{p}}_{i}d{x}_i)=0.}$

The second equation follows from applying the chain rule to a solution u, and the third follows by taking an exterior derivative of the relation ${\displaystyle du-\sum _i{p}_{i}d{x}_i=0}$. Manipulating these equations gives

$${\displaystyle \{\begin{array}{cc}{\dot{x}}_i& =\lambda F_{p_i},\\ {\dot{p}}_i& =-\lambda (F_{x_i}+F_u{p}_i),\\ \dot{u}& =\lambda \sum _i{p}_i{F}_{p_i}\end{array}}$$

where λ is a constant. Writing these equations more symmetrically, one obtains the Lagrange–Charpit equations for the characteristic

${\displaystyle \frac{{\dot{x}}_i}{F_{p_i}}=-\frac{{\dot{p}}_i}{F_{x_i}+F_u{p}_i}=\frac{\dot{u}}{\sum p_i{F}_{p_i}}.}$

Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the Monge cone of the differential equation should everywhere be tangent to the graph of the solution.