1.2. EXAMPLES 5

b. Nonlinear equations.

1. Eikonal equation

|Du| = 1.

2. Nonlinear Poisson equation

−Δu = f(u).

3. p-Laplacian equation

div(|Du|p−2Du)

= 0.

4. Minimal surface equation

div

Du

(1 +

|Du|2)1/2

= 0.

5. Monge–Amp` ere equation

det(D2u)

= f.

6. Hamilton–Jacobi equation

ut + H(Du, x) = 0.

7. Scalar conservation law

ut + div F(u) = 0.

8. Inviscid Burgers’ equation

ut + uux = 0.

9. Scalar reaction-diﬀusion equation

ut − Δu = f(u).

10. Porous medium equation

ut −

Δ(uγ)

= 0.

11. Nonlinear wave equation

utt − Δu + f(u) = 0.

12. Korteweg–deVries (KdV) equation

ut + uux + uxxx = 0.

13. Nonlinear Schr¨ odinger equation

iut + Δu =

f(|u|2)u.