On the Initial Value Problem of the Whitham Averaged System

Abstract

The initial value problem in question is concerning the Whitham averaged system:

$${{\beta }_{{it}}} + {{\lambda }_{i}}\left( {{{\beta }_{1}},{{\beta }_{2}},{{\beta }_{3}}} \right){{\beta }_{{ix}}} = 0i = 1,2,3$$

(1)

where

$$ {{\lambda }_{1}}\left( {{{\beta }_{1}},{{\beta }_{2}},{{\beta }_{3}}} \right) = 2\left( {{{\beta }_{1}} + {{\beta }_{2}} + {{\beta }_{3}}} \right) + 4\left( {{{\beta }_{1}} - {{\beta }_{2}}} \right)\frac{{K\left( s \right)}}{{E\left( s \right)}} $$

(2)

$$ {{\lambda }_{2}}\left( {{{\beta }_{1}},{{\beta }_{2}},{{\beta }_{3}}} \right) = 2\left( {{{\beta }_{1}} + {{\beta }_{2}} + {{\beta }_{3}}} \right) + 4\left( {{{\beta }_{2}} - {{\beta }_{1}}} \right)\frac{{{{s}^{2}}K\left( s \right)}}{{E\left( s \right) - \left( {1 - {{s}^{2}}} \right)K\left( s \right)}} $$

(3)

$$ \begin{array}{*{20}{c}} {{{\lambda }_{3}}\left( {{{\beta }_{1}},{{\beta }_{2}},{{\beta }_{3}}} \right) = 2\left( {{{\beta }_{1}} + {{\beta }_{2}} + {{\beta }_{3}}} \right) + 4\left( {{{\beta }_{2}} - {{\beta }_{3}}} \right)\frac{{K\left( s \right)}}{{E\left( s \right) - K\left( s \right)}}} \\ {{{s}^{2}} = \frac{{{{\beta }_{2}} - {{\beta }_{3}}}}{{{{\beta }_{1}} - {{\beta }_{3}}}}} \\ \end{array} $$

(4)

K(s) and E(s) are complete elliptic integrals of first and second kind.