On the Integrability of the Averaged KdV and Benney Equations

Abstract

The general theory of hamiltonian systems of hydrodynamic type was developed by B.A.Dubrovin and S.P.Novikov [7],[8]. Here we study only one-dimensional hamiltonian systems possessing a complete set of Riemann invariants arising in the theory of multiphase averaging of completely integrable equations (see [2], [4], [13], [20], [32]), for example the Whitham equations (the averaged 1-phase KdV equation):

$$\begin{array}{*{20}{c}} {u_{t}^{1} = {{v}_{1}}\left( u \right)\cdot u_{x}^{1},{{v}_{1}}\left( u \right) = \frac{{{{u}^{1}} + {{u}^{2}} + {{u}^{3}}}}{3} - \frac{{2\cdot \left( {{{u}^{2}} - {{u}^{1}}} \right)\cdot K\left( s \right)}}{{3\cdot \left( {K\left( s \right) - E\left( s \right)} \right)}},} \\ {u_{t}^{2} = {{v}_{2}}\left( u \right)\cdot u_{x}^{2},{{v}_{2}}\left( u \right) = \frac{{{{u}^{1}} + {{u}^{2}} + {{u}^{3}}}}{3} - \frac{{2\cdot \left( {{{u}^{2}} - {{u}^{1}}} \right)\cdot \left( {1 - {{s}^{2}}} \right)\cdot K\left( s \right)}}{{3\cdot \left( {E\left( s \right) - \left( {1 - {{s}^{2}}} \right)\cdot K\left( s \right)} \right)}},} \\ {u_{t}^{3} = {{v}_{3}}\left( u \right)\cdot u_{x}^{3},{{v}_{3}}\left( u \right) = \frac{{{{u}^{1}} + {{u}^{2}} + {{u}^{3}}}}{3} + \frac{{2\cdot \left( {{{u}^{3}} - {{u}^{1}}} \right)\cdot \left( {1 - {{s}^{2}}} \right)\cdot K\left( s \right)}}{{3\cdot E\left( s \right)}},} \\ \end{array}$$

(1)

(here s ²=(u ²-u ¹)/(u ³-u ¹)); E(s), K(s) are the complete ellipticintegrals and Benney equations:

$$\begin{array}{*{20}{c}} {h_{t}^{i} + {{{\left( {{{q}^{i}}\cdot {{h}^{i}}} \right)}}_{x}} = 0,i = 1, \ldots ,N,} \\ {q_{t}^{i} + {{q}^{i}}\cdot q_{x}^{i} + {{S}_{x}} = 0,S = {{h}^{1}} + + {{h}^{N}}} \\ \end{array}$$

(2)