Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice

We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus $k$. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential ($k\to 0$) our solutions model a quasi-one dimensional quantum degenerate Bose-Fermi mixture trapped in optical lattice. In the limit $k\to 1$ the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On Soliton Interactions for a Hierarchy of Generalized Heisenberg Ferromagnetic Models on SU(3)/S(U(1) ×\times U(2)) Symmetric Space

We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In particular, the one-soliton solutions for NLEEs with even dispersion laws are not traveling waves; their velocities and their amplitudes are time dependent. Calculating the asymptotics of the N-soliton solutions for t \rightarrow \pm \infty we analyze the interactions of quadruplet solitons

Rational Bundles and Recursion Operators for Integrable Equations on A.III-type Symmetric Spaces

We analyze and compare the methods of construction of the recursion operators for a special class of integrable nonlinear differential equations related to A.III-type symmetric spaces in Cartan's classification and having additional reductions.Comment: 13 pages, 1 figur

Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces

A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the 'squared solutions' (generalized exponentials) are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analyzed

On integrable wave interactions and Lax pairs on symmetric spaces

Multi-component generalizations of derivative nonlinear Schrödinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z2-graded Lie algebras and A.III symmetric spaces are studied. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed. The latter lead to multi-component integrable equations with CPT-symmetry. Furthermore, the fundamental analytic solutions (FAS) are constructed and the spectral properties of the associated Lax operators are briefly discussed. The Riemann–Hilbert problem (RHP) for the multi-component generalizations of DNLS equation of Kaup–Newell (KN) and Gerdjikov–Ivanov (GI) types is derived. A modification of the dressing method is presented allowing the explicit derivation of the soliton solutions for the multi-component GI equation with both local and nonlocal reductions. It is shown that for specific choices of the reduction these solutions can have regular behavior for all finite x and t. The fundamental properties of the multi-component GI equations are briefly discussed at the end

On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources

Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.Comment: 16 pages, no figures, LaTe

POSITIVE AND NEGATIVE EFFECTS OF SIGNING A TRANSANTLANTIC TRADE AND INVESTMENT PARTNERSHIP ON THE USA AND EU ECONOMIES

В настоящата разработка е представена същността на трансатлантическо търговско и инвестиционно партньорство (ТТИП). Целта и е да докаже, че ТТИП носи повече позитиви отколкото негативи за икономиките на САЩ и ЕС. По време на преговорите, в резултат на различия в законодателната уредба на САЩ и ЕС, са поставени множество спорни въпроси, касаещи опазването на околната среда, здравето и безопасността на населението, уреждане на съдебните спорове, правила за защита на инвеститорите и др. За изясняване на спорните моменти в настоящата разработка е направен анализ на позитивите и негативите ефекти за икономиките на ЕС и САЩ от евентуалното подписване на ТТИП. На база анализа, може да се изведе заключението, че подписването на ТТИП трябва да се осъществи. Това ще се отрази позитивно на икономиките на САЩ и ЕС и ще допринесе за по-лесно преодоляване на икономическата криза, в която се намират те. This paper presents the essentials of the concept of transatlantic trade and investment partnership (TTIP). Its aim is to prove that TTIP brings more positives than negatives for the US and EU. During the negotiations, the differences in the legislation of the US and the EU brought forward many controversial issues concerning the environment, health and safety of the population, settlement of disputes, rules to protect investors and others. To clarify the controversial points, this paper analyses the positive and negative effects from signing a TTIP on the economies of the EU and the US. Based on the analysis, we can conclude that it is desirable a TTIP to be signed. This will have a positive impact on the economies of the US and EU and will contribute to an easier overcoming of the economic crisis they are currently in

Zakharov-Shabat System with Constant Boundary Conditions. Reflectionless Potentials and End Point Singularities

We consider scalar defocusing nonlinear Schroedinger equation with constant boundary conditions. We aim here to provide a self contained pedagogical exposition of the most important facts regarding integrability of that classical evolution equation. It comprises the following topics: direct and inverse scattering problem and the dressing method

N-Wave Equations with Orthogonal Algebras: Z_2 and Z_2 × Z_2 Reductions and Soliton Solutions

We consider N-wave type equations related to the orthogonal algebras obtained from the generic ones via additional reductions. The first Z_2-reduction is the canonical one. We impose a second Z_2-reduction and consider also the combined action of both reductions. For all three types of N-wave equations we construct the soliton solutions by appropriately modifying the Zakharov-Shabat dressing method. We also briefly discuss the different types of one-soliton solutions. Especially rich are the types of one-soliton solutions in the case when both reductions are applied. This is due to the fact that we have two different configurations of eigenvalues for the Lax operator L: doublets, which consist of pairs of purely imaginary eigenvalues, and quadruplets. Such situation is analogous to the one encountered in the sine-Gordon case, which allows two types of solitons: kinks and breathers. A new physical system, describing Stokes-anti Stokes Raman scattering is obtained. It is represented by a 4-wave equation related to the B_2 algebra with a canonical Z_2 reduction