Instability of solitary waves on Euler's elastica

Stability of solitary waves in a thin inextensible and unshearable rod of infinite length is studied. Solitary-wave profile ofthe elastica of such a rod without torsion has the form of a planar loop and its speed depends on a tension in the rod. The linear instability of a solitary-wave profile subject to perturbations escaping from the plane of the loop is established for a certain range of solitary-wave speeds. It is done using the properties of the Evans function, an analytic function on the right complex half-plane, that has zeroes if and only if there exist the unstable modes of the linearization around a solitary-wave solution. The result follows from comparison of the behaviour of the Evans function in some neighbourhood of the origin with its asymptotic at infinity. The explicit computation of the leading coefficient of the Taylor series of the Evans function near the origin is performed by means of the symbolic computer language.Comment: 19 pages, 2 figure

Cooling a magnetic resonance force microscope via the dynamical back-action of nuclear spins

We analyze the back-action influence of nuclear spins on the motion of the cantilever of a magnetic force resonance microscope. We calculate the contribution of nuclear spins to the damping and frequency shift of the cantilever. We show that, at the Rabi frequency, the energy exchange between the cantilever and the spin system cools or heats the cantilever depending on the sign of the high-frequency detuning. We also show that the spin noise leads to a significant damping of the cantilever motion.Comment: 15 pqges, 11 figures. The last part of Section IV.A and Section IV B were rewritten. We added three new graphs: Figs. 5, 7, 9 and all graphs in this subsection were recalculated for T=600 mK as in the experiment. Several new references were adde