The spectral stability problem for periodic traveling water waves on a two-dimensional fluid of infinite depth is investigated via a perturbative approach, computing the spectrum as a function of the wave amplitude beginning with a flat surface. We generalize our previous results by considering the crucially important situation of eigenvalues with multiplicity greater than one (focusing on the generic case of multiplicity two) in the case of flat water. We use this extended method of Transformed Field Expansions (which accounts for resonant spectrum) to numerically simulate the evolution of the eigenvalues as the wave amplitude is increased. From this we conclude that all of these configurations are spectrally stable (for waveheight sufficiently small), though our analysis currently excludes the well-known Benjamin-Feir instability. We complement the numerical results with an explicit calculation of the first nonzero correction to the linear spectrum of resonant deep water waves. Two countably infinite families of collisions of eigenvalues with opposite Krein signature which do not lead to instability are presented.

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A model equation for gravity-capillary waves in deep water is proposed. This model is a quadratic approximation of the deep water potential flow equations and has wavepacket-type solitary wave solutions. The model equation supports line solitary waves which are spatially localized in the direction of propagation and constant in the transverse direction, and lump solitary waves which are spatially localized in both directions. Branches of both line and lump solitary waves are computed via a numerical continuation method. The stability of each type of wave is examined. The transverse instability of line solitary waves is predicted by a similar instability of line solitary waves in the nonlinear Schrödinger equation.The spectral stability of lumps is predicted using the waves' speed energy relation. The role of wave collapse in the stability of these waves is also examined. Numerical time evolution is used to confirm stability predictions and observe dynamics, including instabilities and solitary wave collisions.

A model equation governing the primitive dynamics of wave packets near an extremum of the linear dispersion relation at finite wavenumber is derived. In two spatial dimensions, we include the effects of weak variation of the wave in the direction transverse to the direction of propagation. The resulting equation is contrasted with the Kadomtsev-Petviashvilli and Nonlinear Schrödinger (NLS) equations. The model is derived as an approximation to the equations for deep water gravity-capillary waves, but has wider applications. Both line solitary waves and solitary waves which decay in both the transverse and propagating directions-lump solitary waves-are computed. The stability of these waves is investigated and their dynamics are studied via numerical time evolution of the equation.