Airy equation

The (homogeneous) Airy equation is given by

${\displaystyle \,y_{xx}-xy=0.\!}$

This equation can be solved by power series expansion or Laplace transform techniques to give the Airy functions. The solutions to a wide class of problems may be expressed in terms of these functions. One such problem is the reduced (or linearized) Korteweg-de Vries equation

${\displaystyle \,u_{t}+u_{xxx}=0,\!}$

which is the linear component of many equations of KdV type. The relationship between the two equations is that if y(x) solves the Airy equation, then

${\displaystyle \,u(t,x):=t^{-1/3}y(x/(3t)^{1/3})\!}$

solves the linearized KdV equation (and for the correct choice of y, can in fact be used as the fundamental solution for this equation).

For applications to nonlinear perturbation problems, it is often important to study the more general inhomogeneous linear component of the KdV equation

${\displaystyle \,u_{t}+u_{xxx}=F\!}$

for various forcing terms F. Of course, the inhomogeneous and homogeneous equations are related by Duhamel's formula. A large number of linear, bilinear, trilinear, and multilinear estimates for this equation are known; see the page on Airy estimates for more details.