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Using the inverse spectral theory of the nonlinear Schrödinger (NLS) equation we correlate the development of rogue waves in oceanic sea states characterized by the JONSWAP spectrum with the proximity to homoclinic solutions of the NLS equation. We find in numerical simulations of the NLS equation that rogue waves develop for JONSWAP initial data that is "near" NLS homoclinic data, while rogue waves do not occur for JONSWAP data that is "far" from NLS homoclinic data. We show the nonlinear spectral decomposition provides a simple criterium for predicting the occurrence and strength of rogue waves (PACS: 92.10. Hm, 47.20.Ky, 47.35+i).

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Analytical solutions are developed for the Richards' equation following the analysis of Broadbridge and White. Included here is the solution for drainage and redistribution of a partially or deeply wetted profile. Additionally, infiltration for various initial conditions is examined as well as evaporation at the upper boundary. In all cases the surface flux is constant, whether it be zero for drainage, positive for infiltration, or negative for evaporation. The solutions assume specific forms for the soil water diffusivity and hydraulic conductivity functions: a(b − θ)−2 and β + γ(b − θ) + λ/[2(b − θ)], respectively. Here θ is the water content and a, b, β, γ, and λ are constants.

Several recently developed multisymplectic schemes for Hamiltonian PDEs have been shown to preserve associated local conservation laws and constraints very well in long time numerical simulations. Backward error analysis for PDEs, or the method of modified equations, is a useful technique for studying the qualitative behavior of a discretization and provides insight into the preservation properties of the scheme. In this paper we initiate a backward error analysis for PDE discretizations, in particular of multisymplectic box schemes for the nonlinear Schrodinger equation. We show that the associated modified differential equations are also multisymplectic and derive the modified conservation laws which are satisfied to higher order by the numerical solution. Higher order preservation of the modified local conservation laws is verified numerically.