A boundary integral formulation for particle trajectories in Stokes waves

Recently important theorems have been established presenting qualitative results for particle trajectories below a Stokes wave.A diversity of orbit patterns were described,including the case of a closed orbit when a Stokes wave propagates in the presence of an adverse current. In this work these results are revisited in aquantitative fashion through a boundary integral formulation which leads to very accurate numerical simulations of particle trajectories. The boundaryintegral formulation allows the accurate evaluation of the vector field of the (particle's) dynamical system, without resorting to a series expansion and a smallparameter. Accurate trajectories are benchmarked against well known expansions for weakly nonlinear waves. Simulations are then performedbeyond this regime. Closed orbits are found in the presence of an adverse current, as well as non-smooth trajectories that have not been reported.These occur for both adverse and favorable currents.