The Hubbard model on a complete graph: Exact Analytical results

We derive the analytical expression of the ground state of the Hubbard model with unconstrained hopping at half filling and for arbitrary lattice sites.Comment: Email:[email protected]

Quantum Lattice Solitons

The number state method is used to study soliton bands for three anharmonic quantum lattices: i) The discrete nonlinear Schr\"{o}dinger equation, ii) The Ablowitz-Ladik system, and iii) A fermionic polaron model. Each of these systems is assumed to have $f$-fold translational symmetry in one spatial dimension, where $f$ is the number of freedoms (lattice points). At the second quantum level $(n=2)$ we calculate exact eigenfunctions and energies of pure quantum states, from which we determine binding energy $(E_{\rm b})$, effective mass $(m^{*})$ and maximum group velocity $(V_{\rm m})$ of the soliton bands as functions of the anharmonicity in the limit $f \to \infty$. For arbitrary values of $n$ we have asymptotic expressions for $E_{\rm b}$, $m^{*}$, and $V_{\rm m}$ as functions of the anharmonicity in the limits of large and small anharmonicity. Using these expressions we discuss and describe wave packets of pure eigenstates that correspond to classical solitons.Comment: 21 pages, 1 figur

Quantum signatures of breather-breather interactions

The spectrum of the Quantum Discrete Nonlinear Schr\"odinger equation on a periodic 1D lattice shows some interesting detailed band structure which may be interpreted as the quantum signature of a two-breather interaction in the classical case. We show that this fine structure can be interpreted using degenerate perturbation theory.Comment: 4 pages, 4 fig

Lagrangian Formalism in Perturbed Nonlinear Klein-Gordon Equations

We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term we are able to write the Lagrangian density and then, calculate the Lagrangian as a function of collective variables. We use the Lagrangian formalism together with the Rice {\it Ansatz} to derive the equations of motion of the collective coordinates (CCs) for the perturbed sine-Gordon (sG) and $\phi^{4}$ systems. For the $N$ collective coordinates, regardless of the {\it Ansatz} used, we show that, for the nonlinear Klein-Gordon equations, this approach is equivalent to the {\it Generalized Traveling Wave Ansatz} ({\it GTWA})Comment: 9 page

Discrete Nonlinear Schrodinger Equations with arbitrarily high order nonlinearities

A class of discrete nonlinear Schrodinger equations with arbitrarily high order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrodinger equation and the Ablowitz-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers and moving solutions, are investigated

Table Top Dome Tester

The Erichsen Cupping Test was used as a basis to design a dome tester. The intention of a dome tester is to test sheet metal material properties in all directions. This was done by clamping a piece of sheet metal and using a piston and hydraulic press to punch through the material. The force used to punch the material and the height of the deforming material can be gathered and the sheet metal properties can then be calculated. At the end of the project the team was able to successfully design and manufacture a hydraulic dome tester. However, due to unforeseen circumstances, the final assembly was unable to be tested. The machine should be capable of loading samples up to 20 tons with a stroke length of 2.0 inches. The dome tester currently resides in the manufacturing lab at the University of Akron where students will perform the stampability test themselves. They can apply real life engineering theory and study material properties of sheet metal during their coursework

Exact Localized Solutions of Quintic Discrete Nonlinear Schr\"odinger Equation

We study a new quintic discrete nonlinear Schr\"odinger (QDNLS) equation which reduces naturally to an interesting symmetric difference equation of the form $\phi_{n+1}+\phi_{n-1}=F(\phi_n)$. Integrability of the symmetric mapping is checked by singularity confinement criteria and growth properties. Some new exact localized solutions for integrable cases are presented for certain sets of parameters. Although these exact localized solutions represent only a small subset of the large variety of possible solutions admitted by the QDNLS equation, those solutions presented here are the first example of exact localized solutions of the QDNLS equation. We also find chaotic behavior for certain parameters of nonintegrable case.Comment: 12 pages,4 figures(eps files),revised,Physics Letters A, In pres

Mobile Localization in nonlinear Schrodinger lattices

Using continuation methods from the integrable Ablowitz-Ladik lattice, we have studied the structure of numerically exact mobile discrete breathers in the standard Discrete Nonlinear Schrodinger equation. We show that, away from that integrable limit, the mobile pulse is dressed by a background of resonant plane waves with wavevectors given by a certain selection rule. This background is seen to be essential for supporting mobile localization in the absence of integrability. We show how the variations of the localized pulse energy during its motion are balanced by the interaction with this background, allowing the localization mobility along the lattice.Comment: 10 pages, 11 figure

Discrete Breathers in Two-Dimensional Anisotropic Nonlinear Schrodinger lattices

We study the structure and stability of discrete breathers (both pinned and mobile) in two-dimensional nonlinear anisotropic Schrodinger lattices. Starting from a set of identical one-dimensional systems we develop the continuation of the localized pulses from the weakly coupled regime (strongly anisotropic) to the homogeneous one (isotropic). Mobile discrete breathers are seen to be a superposition of a localized mobile core and an extended background of two-dimensional nonlinear plane waves. This structure is in agreement with previous results on onedimensional breather mobility. The study of the stability of both pinned and mobile solutions is performed using standard Floquet analysis. Regimes of quasi-collapse are found for both types of solutions, while another kind of instability (responsible for the discrete breather fission) is found for mobile solutions. The development of such instabilities is studied, examining typical trajectories on the unstable nonlinear manifold.Comment: 13 pages, 9 figure