On the validity of mean-field amplitude equations for counterpropagating wavetrains

We rigorously establish the validity of the equations describing the evolution of one-dimensional long wavelength modulations of counterpropagating wavetrains for a hyperbolic model equation, namely the sine-Gordon equation. We consider both periodic amplitude functions and localized wavepackets. For the localized case, the wavetrains are completely decoupled at leading order, while in the periodic case the amplitude equations take the form of mean-field (nonlocal) Schr\"odinger equations rather than locally coupled partial differential equations. The origin of this weakened coupling is traced to a hidden translation symmetry in the linear problem, which is related to the existence of a characteristic frame traveling at the group velocity of each wavetrain. It is proved that solutions to the amplitude equations dominate the dynamics of the governing equations on asymptotically long time scales. While the details of the discussion are restricted to the class of model equations having a leading cubic nonlinearity, the results strongly indicate that mean-field evolution equations are generic for bimodal disturbances in dispersive systems with \O(1) group velocity.Comment: 16 pages, uuencoded, tar-compressed Postscript fil

Valorization of a biomass: phytochemicals in oilseed by-products

A huge amount of residue is released every year in agricultural and food industries. If this waste is valorised properly, it would be very beneficial from both economic and environmental aspects. Recently, seeds of oil crops have been getting great interest due to their waste rich in a great variety of lipophilic and hydrophilic phytochemicals. On the other hand, oilseed processing by-products comprise approximately 35 millions of tons seed for oil in European Union. Therefore, this remarkable amount of residue should be considered as a resource from waste to health. The purpose of the present research is to introduce the types of oilseed by-product and their high added value phytochemicals by summarizing the recent studies on the valorization of different wastes of several oilseed crops. This research also supplies an overview of the oilseed sources with their botanical names, production regions and target phytochemical ingredients. Moreover, further remarks with some certain concerns are discussed