Bi-Hamiltonian structure and (non) integrability of nonlinear wave equations

Parallel propagation of magnetohydrodynamic waves in relativistic electron-positron plasmas is described by a vector version of the modified Korteweg-de Vries equation. This is shown to be nonintegrable by the bi-Hamiltonian structure method, because it fails to provide the recursion operator needed to generate an infinite sequence of conserved densities. Only three of these conserved densities had been found explicitly by direct investigations based on the symmetry properties of the new equation.