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.

We have investigated shallow water flows through a channel with a contraction by experimental and theoretical means. The horizontal channel consists of a sluice gate and an upstream channel of constant width b 0 ending in a linear contraction of minimum width b c . Experimentally, we observe upstream steady and moving bores/shocks, and oblique waves in the contraction, as single and multiple ͑steady͒ states, as well as a steady reservoir with a complex hydraulic jump in the contraction occurring in a small section of the b c / b 0 and Froude number parameter plane. One-dimensional hydraulic theory provides a comprehensive leading-order approximation, in which a turbulent frictional parametrization is used to achieve quantitative agreement. An analytical and numerical analysis is given for two-dimensional supercritical shallow water flows. It shows that the one-dimensional hydraulic analysis for inviscid flows away from hydraulic jumps holds surprisingly well, even though the two-dimensional oblique hydraulic jump patterns can show large variations across the contraction channel.

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.

Abstract. We study the motion of an interface between two irrotational, incompressible fluids, with elastic bending forces present; this is the hydroelastic wave problem. We prove a global bifurcation theorem for the existence of families of spatially periodic traveling waves on infinite depth. Our traveling wave formulation uses a parameterized curve, in which the waves are able to have multi-valued height. This formulation and the presence of the elastic bending terms allows for the application of an abstract global bifurcation theorem of "identity plus compact" type. We furthermore perform numerical computations of these families of traveling waves, finding that, depending on the choice of parameters, the curves of traveling waves can either be unbounded, reconnect to trivial solutions, or end with a wave which has a self-intersection. Our analytical and computational methods are able to treat in a unified way the cases of positive or zero mass density along the sheet, the cases of single-valued or multi-valued height, and the cases of single-fluid or interfacial waves.

Crapper waves are a family of exact periodic travelling wave solutions of the free-surface irrotational incompressible Euler equations; these are pure capillary waves, meaning that surface tension is accounted for, but gravity is neglected. For certain parameter values, Crapper waves are known to have multi-valued height. Using the implicit function theorem, we prove that any of the Crapper waves can be perturbed by the effect of gravity, yielding the existence of gravity-capillary waves nearby to the Crapper waves. This result implies the existence of travelling gravity-capillary waves with multi-valued height. The solutions we prove to exist include waves with both positive and negative values of the gravity coefficient. We also compute these gravity perturbed Crapper waves by means of a quasi-Newton iterative scheme (again, using both positive and negative values of the gravity coefficient). A phase diagram is generated, which depicts the existence of singlevalued and multi-valued travelling waves in the gravity-amplitude plane. A new largest water wave is computed, which is composed of a string of bubbles at the interface.