  The spatial version of the fourth-order Dysthe equations describe the
evolution of weakly nonlinear narrowband wave trains in deep waters. For
unidirectional waves, the hidden Hamiltonian structure and new invariants are
unveiled by means of a gauge transformation to a new canonical form of the
evolution equations. A highly accurate Fourier-type spectral scheme is
developed to solve for the equations and validate the new conservation laws,
which are satisfied up to machine precision. Further, traveling waves are
numerically investigated using the Petviashvili method. It is found that their
collision appears inelastic, suggesting the non-integrability of the Dysthe
equations.
