Towards a sufficient criterion for collapse in 3D Euler equations

A sufficient integral criterion for a blow-up solution of the Hopf equations (the Euler equations with zero pressure) is found. This criterion shows that a certain positive integral quantity blows up in a finite time under specific initial conditions. Blow-up of this quantity means that solution of the Hopf equation in 3D can not be continued in the Sobolev space $H^2({\cal R}^3)$ for infinite time

High energy neutrino absorption by W production in a strong magnetic field

An influence of a strong external magnetic field on the neutrino self-energy operator is investigated. The width of the neutrino decay into the electron and W boson, and the mean free path of an ultra-high energy neutrino in a strong magnetic field are calculated. A kind of energy cutoff for neutrinos propagating in a strong field is defined.Comment: 13 pages, LaTeX, 1 EPS figure, submitted to Physics Letters

Backlund transformations for finite-dimensional integrable systems: a geometric approach

We present a geometric construction of Backlund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Backlund transformations, which are naturally parametrized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel-Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Backlund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Backlund transformations that was recently introduced. Moreover, we recover Backlund transformations and discretizations which have up to now been constructed by ad-hoc methods, and we find Backlund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables.Comment: 37 pages, Late