Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems

B{\'e}zout 's theorem states that dense generic systems of n multivariate quadratic equations in n variables have 2 n solutions over algebraically closed fields. When only a small subset M of monomials appear in the equations (fewnomial systems), the number of solutions may decrease dramatically. We focus in this work on subsets of quadratic monomials M such that generic systems with support M do not admit any solution at all. For these systems, Hilbert's Nullstellensatz ensures the existence of algebraic certificates of inconsistency. However, up to our knowledge all known bounds on the sizes of such certificates -including those which take into account the Newton polytopes of the polynomials- are exponential in n. Our main results show that if the inequality 2|M| -- 2n $\le$ \sqrt 1 + 8{\nu} -- 1 holds for a quadratic fewnomial system -- where {\nu} is the matching number of a graph associated with M, and |M| is the cardinality of M -- then there exists generically a certificate of inconsistency of linear size (measured as the number of coefficients in the ground field K). Moreover this certificate can be computed within a polynomial number of arithmetic operations. Next, we evaluate how often this inequality holds, and we give evidence that the probability that the inequality is satisfied depends strongly on the number of squares. More precisely, we show that if M is picked uniformly at random among the subsets of n + k + 1 quadratic monomials containing at least $\Omega$(n 1/2+$\epsilon$) squares, then the probability that the inequality holds tends to 1 as n grows. Interestingly, this phenomenon is related with the matching number of random graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results showing that certificates in inconsistency can be computed for systems with more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201

Unstaggered-staggered solitons in two-component discrete nonlinear Schr\"{o}dinger lattices

We present stable bright solitons built of coupled unstaggered and staggered components in a symmetric system of two discrete nonlinear Schr\"{o}dinger (DNLS) equations with the attractive self-phase-modulation (SPM) nonlinearity, coupled by the repulsive cross-phase-modulation (XPM) interaction. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. The results are obtained in an analytical form, using the variational and Thomas-Fermi approximations (VA and TFA), and the generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the stability. The analytical predictions are verified against numerical results. Almost all the symbiotic solitons are predicted by the VA quite accurately, and are stable. Close to a boundary of the existence region of the solitons (which may feature several connected branches), there are broad solitons which are not well approximated by the VA, and are unstable

Stable Control of Pulse Speed in Parametric Three-Wave Solitons

We analyze the control of the propagation speed of three wave packets interacting in a medium with quadratic nonlinearity and dispersion. We found analytical expressions for mutually trapped pulses with a common velocity in the form of a three-parameter family of solutions of the three-wave resonant interaction. The stability of these novel parametric solitons is simply related to the value of their common group velocity

Quantum shock waves in the Heisenberg XY model

We show the existence of quantum states of the Heisenberg XY chain which closely follow the motion of the corresponding semi-classical ones, and whose evolution resemble the propagation of a shock wave in a fluid. These states are exact solutions of the Schroedinger equation of the XY model and their classical counterpart are simply domain walls or soliton-like solutions.Comment: 15 pages,6 figure

Sinking and floating rates of natural phytoplankton assemblages in Lake Erken

Sinking rates of the <120 mu m size phytoplankton fraction of water from Lake Erken were determined during the summer 1992 by following the increase of chlorophyll a in the 10 ml-bottom layer in replicate 100 ml settling cylinders. Changes in chlorophyll a concentrations as a function of incubation time allowed two fractions to be separated. Fast sinking rates varied between values of 1.9 m/day when pennate and centric diatoms and coccal cyanobacteria were dominant tin cell concentration) and values of 0.5 m/day when cryptophytes and chrysophytes dominated the <120 mu m size fraction. Slow sinking rates decreased from 0.04 m/day at the beginning of July to 0.02 m/day in late July. Photosynthesis-Irradiance parameters (P-max(B) light saturated photosynthesis and #alpha#(B), light limited photosynthesis) were lower in the fast sinking fraction (P-max(B) = 1.3 - 2.4 mu gC/mu gChl/h and #alpha#(B) = 0.01 - 0.04 mu gC/mu gChl/h/(mu E/m(2)/s) than in the slow or non-sinking one (P-max(B) = 3.9 - 6.4 mu gC/mu gChl/h and #alpha#(B) = 0.03 - 0.08 mu gC/mu gChl/h/(mu E/m(2)/s). P-max(B) and #alpha#B of the planktonic Gloeotrichia echinulata, a colonial broom-forming cyanobacterium, were similar to those found in the fast sinking fraction. Mean floating rates of G. echinulata were around 43 m/d from 15 to 27 July and increased by a factor of two afterwards. G. echinulata colonies migrating upwards from sediments and captured in inverted traps showed a mean floating rate of 104 m/d

Quantized representation of some nonlinear integrable evolution equations on the soliton sector

The Hirota algorithm for solving several integrable nonlinear evolution equations is suggestive of a simple quantized representation of these equations and their soliton solutions over a Fock space of bosons or of fermions. The classical nonlinear wave equation becomes a nonlinear equation for an operator. The solution of this equation is constructed through the operator analog of the Hirota transformation. The classical N-solitons solution is the expectation value of the solution operator in an N-particle state in the Fock space.Comment: 12 page

Controlled Generation of Dark Solitons with Phase Imprinting

The generation of dark solitons in Bose-Einstein condensates with phase imprinting is studied by mapping it into the classic problem of a damped driven pendulum. We provide simple but powerful schemes of designing the phase imprint for various desired outcomes. We derive a formula for the number of dark solitons generated by a given phase step, and also obtain results which explain experimental observations.Comment: 4pages, 4 figure

The Davey Stewartson system and the B\"{a}cklund Transformations

We consider the (coupled) Davey-Stewartson (DS) system and its B\"{a}cklund transformations (BT). Relations among the DS system, the double Kadomtsev-Petviashvili (KP) system and the Ablowitz-Ladik hierarchy (ALH) are established. The DS hierarchy and the double KP system are equivalent. The ALH is the BT of the DS system in a certain reduction. {From} the BT of coupled DS system we can obtain new coupled derivative nonlinear Schr\"{o}dinger equations.Comment: 13 pages, LaTe

Completely integrable models of non-linear optics

The models of the non-linear optics in which solitons were appeared are considered. These models are of paramount importance in studies of non-linear wave phenomena. The classical examples of phenomena of this kind are the self-focusing, self-induced transparency, and parametric interaction of three waves. At the present time there are a number of the theories based on completely integrable systems of equations, which are both generations of the original known models and new ones. The modified Korteweg-de Vries equation, the non- linear Schrodinger equation, the derivative non-linear Schrodinger equation, Sine-Gordon equation, the reduced Maxwell-Bloch equation, Hirota equation, the principal chiral field equations, and the equations of massive Thirring model are gradually putting together a list of soliton equations, which are usually to be found in non-linear optics theory.Comment: Latex, 17 pages, no figures, submitted to Pramana