Airy equation: Difference between revisions

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(The equation discussed here was not the Airy equation)
(explicit relationship between Airy and linear KdV)
Line 3: Line 3:
:<math> y_{xx} - x y = 0. </math>
:<math> y_{xx} - x y = 0. </math>


This equation can be solved by power series expansion 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 [[Korteweg-de Vries equation]]
This equation can be solved by power series expansion 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]]


:<math> u_t + u_{xxx} = 0, </math>
:<math> u_t + u_{xxx} = 0, </math>


which is the linear component of many [[KdV equations|equations of KdV type]]. For applications to nonlinear perturbation problems, it is often important to study the more general inhomogeneous linear component of the KdV equation
which is the linear component of many [[KdV equations|equations of KdV type]]. The relationship between the two equations is that if y(x) solves the Airy equation, then
 
:<math>u(t,x) := t^{-1/3} y( x / (3t)^{1/3} )</math>
 
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


:<math> u_t + u_{xxx} = F </math>
:<math> u_t + u_{xxx} = F </math>

Revision as of 06:07, 17 April 2009

The (homogeneous) Airy equation is given by

This equation can be solved by power series expansion 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

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

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

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.