Published: April 22, 2014

Toward a general solution of the three-wave resonant interaction equations

Ruth A. Martin

Applied MathematicsUniversity of Colorado Boulder

Date and time: 

Tuesday, April 22, 2014 - 10:00am

Location: 

ECOT 831

Abstract: 

The resonant interaction of three wavetrains is one of the simplest forms of nonlinear interaction for
dispersive waves of small amplitude. This behavior arises frequently in applications ranging from nonlinear
optics to internal waves through the study of the weakly nonlinear limit of a dispersive system. The slowly
varying amplitudes of the three waves satisfy a set of integrable nonlinear partial dierential equations known
as the three-wave equations. So far, these universally occurring equations have been solved in only a limited
number of congurations. For example, Zakharov and Manakov (1973, 1976) and Kaup (1976) used inverse
scattering to solve the three-wave equations in one spatial dimension on the real line. Similarly, solutions in
two or three spatial dimensions on the whole space were worked out by Zakharov (1976), Kaup (1980), and
others. The known methods of analytic solution fail in the case of periodic boundary conditions, although
numerical simulations of the problem typically impose these conditions.
We hope to nd a general solution to the three-wave equations, which has the advantage of being compat-
ible with a wide variety of boundary conditions and any number of spatial dimensions. To nd the general
solution of an nth order system of ordinary dierential equations, it is sucient to nd a function that
satises the ODEs and has n constants of integration. The general solution of a PDE, however, is not well
dened and is usually dicult, if not impossible, to attain. In fact, there is only a small number of PDEs with
known general solutions. We present a method to construct the general solution of the three-wave equations
using a Painleve-type analysis. For now, we consider a convergent Laurent series solution (in time), which
contains ve real-valued functions (in space) that are arbitrary except for some dierentiability constraints.
A full general solution of the problem would involve six such functions.

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