Periodic orbits, localization in normal mode space, and the Fermi-Pasta-Ulam problem

The Fermi-Pasta-Ulam problem was one of the first computational experiments. It has stirred the physics community since, and resisted a simple solution for half a century. The combination of straightforward simulations, efficient computational schemes for finding periodic orbits, and analytical estimates allows us to achieve significant progress. Recent results on $q$-breathers, which are time-periodic solutions that are localized in the space of normal modes of a lattice and maximize the energy at a certain mode number, are discussed, together with their relation to the Fermi-Pasta-Ulam problem. The localization properties of a $q$-breather are characterized by intensive parameters, that is, energy densities and wave numbers. By using scaling arguments, $q$-breather solutions are constructed in systems of arbitrarily large size. Frequency resonances in certain regions of wave number space lead to the complete delocalization of $q$-breathers. The relation of these features to the Fermi-Pasta-Ulam problem are discussed.Comment: 19 pages, 9 figures, to appear in Am. J. Phy

Numerical And Experimental Study Of Multi-Point Forming Of Thick Double-Curvature Plates From Aluminum Alloy 7075

The paper describes various rod type work tools intended for forming parts and their design peculiarities and technological processes they are used in. We present the device for multi-point forming thick double-curvature plates with the use of reconfigurable core punch and die in large temperature and speed range. The results of finite element modeling of forming and machining process are demonstrated. It is revealed that heating the work piece results in pressing of the rod into the work piece in the areas of maximum pressure. The depth of pressing depends on mechanical behavior of the material at forming temperature and force to forming rods. The paper presents the results of experiments on developing of multi-point forming plates

q-breathers in finite two- and three-dimensional nonlinear acoustic lattices

Nonlinear interaction between normal modes dramatically affects energy equipartition, heat conduction and other fundamental processes in extended systems. In their celebrated experiment Fermi, Pasta and Ulam (FPU, 1955) observed that in simple one-dimensional nonlinear atomic chains the energy must not always be equally shared among the modes. Recently, it was shown that exact and stable time-periodic orbits, coined $q$-breathers (QBs), localize the mode energy in normal mode space in an exponential way, and account for many aspects of the FPU problem. Here we take the problem into more physically important cases of two- and three-dimensional acoustic lattices to find existence and principally different features of QBs. By use of perturbation theory and numerical calculations we obtain that the localization and stability of QBs is enhanced with increasing system size in higher lattice dimensions opposite to their one-dimensional analogues.Comment: 4 pages, 5 figure

q-breathers in Discrete Nonlinear Schroedinger lattices

$q$-breathers are exact time-periodic solutions of extended nonlinear systems continued from the normal modes of the corresponding linearized system. They are localized in the space of normal modes. The existence of these solutions in a weakly anharmonic atomic chain explained essential features of the Fermi-Pasta-Ulam (FPU) paradox. We study $q$-breathers in one- two- and three-dimensional discrete nonlinear Sch\"{o}dinger (DNLS) lattices -- theoretical playgrounds for light propagation in nonlinear optical waveguide networks, and the dynamics of cold atoms in optical lattices. We prove the existence of these solutions for weak nonlinearity. We find that the localization of $q$-breathers is controlled by a single parameter which depends on the norm density, nonlinearity strength and seed wave vector. At a critical value of that parameter $q$-breathers delocalize via resonances, signaling a breakdown of the normal mode picture and a transition into strong mode-mode interaction regime. In particular this breakdown takes place at one of the edges of the normal mode spectrum, and in a singular way also in the center of that spectrum. A stability analysis of $q$-breathers supplements these findings. For three-dimensional lattices, we find $q$-breather vortices, which violate time reversal symmetry and generate a vortex ring flow of energy in normal mode space.Comment: 19 pages, 9 figure

Stability of non-time-reversible phonobreathers

Non-time reversible phonobreathers are non-linear waves that can transport energy in coupled oscillator chains by means of a phase-torsion mechanism. In this paper, the stability properties of these structures have been considered. It has been performed an analytical study for low-coupling solutions based upon the so called {\em multibreather stability theorem} previously developed by some of the authors [Physica D {\bf 180} 235]. A numerical analysis confirms the analytical predictions and gives a detailed picture of the existence and stability properties for arbitrary frequency and coupling.Comment: J. Phys. A.:Math. and Theor. In Press (2010