A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides

We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. Our main result provides an explicit condition for uniqueness which takes into account the physically significant components, corresponding to guided and non-guided waves; this condition reduces to the classical Sommerfeld-Rellich condition in the relevant cases. Finally, we also show that our condition is satisfied by a solution, already present in literature, of the problem under consideration

An unexpected nitrate distribution in the tropical North Atlantic at 18°N, 30°W—implications for new production

During a R.V. Meteor JGOFS-NABE cruise to a tropical site in the northeast Atlantic in spring 1989, three different vertical regimes with respect to nitrate distribution and availability within the euphotic zone were observed. Besides dramatic variations in the depth of the nitracline, a previously undescribed nose-like nitrate maximum within the euphotic zone was the most prominent feature during this study. Both the vertical structure of phytoplankton biomass and the degree of absolute and relative new production were related to the depth of the nitracline, which in turn was dependent on the occurrence/non-occurrence of the subsurface subtropical salinity maximum (S(max)). The mesoscale variability of the nitracline depth, as indicated from a pre-survey grid, and published data on the frequent occurrence of the S(max) in tropical waters suggest higher variability of new production and F-ratio than usually expected for oligotrophic oceans. The importance of salt fingering and double diffusion for nitrate transport into the euphotic zone is discussed

The linearization method in hydrodynamical stability theory

This book presents the theory of the linearization method as applied to the problem of steady-state and periodic motions of continuous media. The author proves infinite-dimensional analogues of Lyapunov's theorems on stability, instability, and conditional stability for a large class of continuous media. In addition, semigroup properties for the linearized Navier-Stokes equations in the case of an incompressible fluid are studied, and coercivity inequalities and completeness of a system of small oscillations are proved

Variational principles for nonpotential operators

This book develops a variational method for solving linear equations with B-symmetric and B-positive operators and generalizes the method to nonlinear equations with nonpotential operators. The author carries out a constructive extension of the variational method to "nonvariational" equations (including parabolic equations) in classes of functionals which differ from the Euler-Lagrange functionals. In this connection, some new functions spaces are considered. Intended for mathematicians working in the areas of functional analysis and differential equations, this book would also prove useful for researchers in other areas and students in advanced courses who use variational methods in solving linear and nonlinear boundary value problems in continuum mechanics and theoretical physics