A Dirac Sea and thermodynamic equilibrium for the quantized three-wave interaction

The classical version of the three wave interaction models the creation and destruction of waves; the quantized version models the creation and destruction of particles. The quantum three wave interaction is described and the Bethe Ansatz for the eigenfunctions is given in closed form. The Bethe equations are derived in a rigorous fashion and are shown to have a thermodynamic limit. The Dirac sea of negative energy states is obtained as the infinite density limit. Finite particle/hole excitations are determined and the asymptotic relation of energy and momentum is obtained. The Yang-Yang functional for the relative free energy of finite density excitations is constructed and is shown to be convex and bounded below. The equations of thermal equilibrium are obtained

Action-Angle variables for the Gel'fand-Dikii flows

Using the scattering transform for $n^{th}$ order linear scalar operators, the Poisson bracket found by Gel'fand and Dikii, which generalizes the Gardner Poisson bracket for the KdV hierarchy, is computed on the scattering side. Action-angle variables are then constructed. Using this, complete integrability is demonstrated in the strong sense. Real action-angle variables are constructed in the self-adjoint case

A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator

\We consider an inverse scattering problem for Schr\"odinger operators with energy dependent potentials. The inverse problem is formulated as a Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for two distinct symmetry classes. As an application we prove global existence theorems for the two distinct systems of partial differential equations $u_t+(u^2/2+w)_x=0, w_t\pm u_{xxx}+(uw)_x=0$ for suitably restricted, complementary classes of initial data

Topological phase for entangled two-qubit states and the representation of the SO(3)group

We discuss the representation of the $SO(3)$ group by two-qubit maximally entangled states (MES). We analyze the correspondence between $SO(3)$ and the set of two-qubit MES which are experimentally realizable. As a result, we offer a new interpretation of some recently proposed experiments based on MES. Employing the tools of quantum optics we treat in terms of two-qubit MES some classical experiments in neutron interferometry, which showed the $\pi$-phase accrued by a spin-$1/2$ particle precessing in a magnetic field. By so doing, we can analyze the extent to which the recently proposed experiments - and future ones of the same sort - would involve essentially new physical aspects as compared with those performed in the past. We argue that the proposed experiments do extend the possibilities for displaying the double connectedness of $SO(3)$, although for that to be the case it results necessary to map elements of $SU(2)$ onto physical operations acting on two-level systems.Comment: 25 pages, 9 figure

Calculation of mechanical and thermal influences during coiling of hot strip

Coiled steel strip is the final product from flat hot rolling processes. With increasing demand for higher quality of hot rolled strips, especially the evolution of strip flatness during and after coiling becomes a crucial aspect. The main impacts on the flatness properties of hot rolled strips result from residual stresses and “eigen-strains” induced by the last hot rolling passes, by strip cooling at the run-out table, and finally, by the mechanical and thermal conditions during and after the coiling process itself. In this paper, a mathematical model is presented, which takes into account the mechanical and thermal effects on hot rolled strip during and after the coiling process. To improve the prediction quality of the underlying process, a customized self-developed 3D finite-element model has been developed and programmed in C++, leading to a software prototype, which is highly superior to commercial FEM-packages with respect to calculation time and storage capacities. The model is based on a dynamic implicit total Lagrangian formulation. All relevant devices directly interacting with the strip, such as pinch rolls, coiler rolls and mandrel are incorporated in the calculation model. Well known and established methods in the solid-shell theory, like the EAS- and ANS-method, were applied to prevent the occurrence of locking phenomena resulting from low order interpolation functions. Selected benchmark tests were performed to evaluate the accuracy of these novel solid-shell elements in comparison to the results attained by the FEM- package ABAQUS©. The results obtained so far agree very satisfactorily. A further important topic is the contact and friction algorithm, where Coulomb’s friction law is applied. The accurate and reliable determination of the contact between strip and interacting devices as well as the aspect of self-contact was treated by applying a sophisticated two dimensional contact search algorithm, leading to a significantly reduced calculation time. The highly non-linear time-dependent system of equations is integrated by utilizing the (implicit) Newmark time-integration scheme. The developed calculation model serves as an effective tool to predict the interesting stress-distributions and local plastic deformations inside the strip, which induce residual stresses and strip unflatness (latent or even manifest waviness). Furthermore, this tool p ovides the basis for further improvements and investigations on hot rolling production lines

Variational Principles for Water Waves

We describe the Hamiltonian structures, including the Poisson brackets and Hamiltonians, for free boundary problems for incompressible fluid flows with vorticity. The Hamiltonian structure is used to obtain variational principles for stationary gravity waves both for irrotational flows as well as flows with vorticity.Comment: 20 page

On Soliton-type Solutions of Equations Associated with N-component Systems

The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.Comment: 43 pages, 16 figure

Little groups of irreps of O(3), SO(3), and the infinite axial subgroups

Little groups are enumerated for the irreps and their components in any basis of O(3) and SO(3) up to rank 9, and for all irreps of C$_{\infty}$, C$_{\infty h}$, C$_{\infty v}$, D$_{\infty}$ and D$_{\infty h}$. The results are obtained by a new chain criterion, which distinguishes massive (rotationally inequivalent) irrep basis functions and allows for multiple branching paths, and are verified by inspection. These results are relevant to the determination of the symmetry of a material from its linear and nonlinear optical properties and to the choices of order parameters for symmetry breaking in liquid crystals.Comment: 28 pages and 3 figure