Spike Solutions Research Articles

On the existence of some new positive interior spike solutions to a semilinear Neumann problem

In this paper we are concerned with the following Neumann problem { ε 2 Δ u − u + f ( u ) = 0 , u > 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where ε is a small positive parameter, f is a superlinear and subcritical nonlinearity, Ω is a smooth and bounded domain in R N . Solutions with multiple boundary peaks have been established for this problem. It has also been proved that for any integer k there exists an interior k-peak solution which concentrates, as ε → 0 + , at k sphere packing points in Ω. In this paper we prove the existence of a second interior k-peak solution provided that k is large enough, and we conjecture that its peaks are located along a straight line. Moreover, when Ω is a two-dimensional strictly convex domain, we also construct a third interior k-peak solution provided that k is large enough, whose peaks are aligned on a closed curve near ∂ Ω.

The normal field instability under side-wall effects: comparison of experiments and computations

We consider a single spike of ferrofluid, arising in a small cylindrical container, when a vertically oriented magnetic field is applied. The height of the spike as well as the surface topography is measured experimentally by two different technologies and calculated numerically using the finite element method. As a consequence of the finite size of the container, the numerics uncovers an imperfect bifurcation to a single spike solution, which is forward. This is in contrast to the standard transcritical bifurcation to hexagons, common for rotational symmetric systems with broken up-down symmetry. The numerical findings are corroborated in the experiments. The small hysteresis observed is explained in terms of a hysteretic wetting of the side wall.

Electrocardiogram of the Mixmaster universe

The Mixmaster dynamics is revisited in a new light as revealing a series of transitions in the complex scale invariant scalar invariant of the Weyl curvature tensor best represented by the speciality index , which gives a four-dimensional measure of the evolution of the spacetime independent of all the three-dimensional gauge-dependent variables except for the time used to parametrize it. Its graph versus time characterized by correlated isolated pulses in its real and imaginary parts corresponding to curvature wall collisions serves as a sort of electrocardiogram of the Mixmaster universe, with each such pulse pair arising from a single circuit or ‘complex pulse’ around the origin in the complex plane. These pulses in the speciality index and their limiting points on the real axis seem to invariantly characterize some of the so-called spike solutions in inhomogeneous cosmology and should play an important role as a gauge-invariant lens through which to view current investigations of inhomogeneous Mixmaster dynamics.

High accuracy and precision isotope dilution mass spectrometry: An application to the determination of Mo in seawater

Two isotope dilution calibration approaches, a standard two-step and a curve method, were examined in an effort to achieve highest accuracy and precision for determination of Mo in seawater. On-line matrix separation using immobilized 8-hydroxyquinoline permitted transient measurements to be made by both sector field sequential (SF-ICP-MS) and multicollector (MC-ICP-MS) inductively coupled plasma mass spectrometry. Concentrations of 9.42 ± 0.22 and 9.396 ± 0.053 ng g−1Mo (U, 95% confidence interval, k = 3.18, n = 4) were obtained in NASS-5 seawater using the standard two-step ID method with SF-ICP-MS and MC-ICP-MS, respectively, in good agreement with the certified value of 9.4 ± 1.0 ng g−1 (U, 95% confidence interval, k = 2; concentration converted from ng ml−1 to ng g−1). A 4-fold enhancement in the precision of determination was evident using MC-ICP-MS. Performance of the standard two-step ID and the ID curve method was also compared following an off-line column separation which permitted steady-state solution nebulization with MC-ICP-MS detection. Undertaking mass bias drift correction during a measurement sequence generated identical concentrations of 9.383 ± 0.016 and 9.383 ± 0.014 ng g−1Mo (U, 95% confidence interval, k = 3.18, n = 4) in NASS-5 using the two-step ID and the ID curve method, respectively, despite significant blank correction arising from the off-line column separation which amounted to 4–5% of the total Mo in the sample. Although the ID curve method uses an additional set of reverse ID calibration solutions, it benefits from the advantages of not needing to employ a certified isotopic reference material for mass bias correction or to know isotope abundances of the analyte in the sample, the standard or the enriched spike. An analysis of the sources of uncertainty suggests that the contribution from determination of the ratio is no longer dominant. Further improvements below 0.1% could be achieved by optimizing other sources of uncertainties, such as increasing the masses of enriched spike and natural abundance Mo standard solutions as well as the mass of high purity Mo standard used for its preparation, assuming similar measurement precision for the ratios and the same number of replicate samples measured.

Quantification of selenium species in petroleum refinery wastewaters using ion chromatography coupled to post-column isotope dilution analysis ICP-MS

A post-column Isotope Dilution Analysis (IDA) methodology has been used to carry out the quantitative speciation of selenium (Se) in industrial wastewaters by ion chromatography coupled to ICP-MS with an octapole reaction system (ORS). The interfering argon dimers on the 78Se and 80Se isotopes were suppressed by pressurizing the octapole chamber with 4 mL min-1 of hydrogen. The inorganic Se species selenite [Se (IV)], selenate [Se (VI)] and selenocyanate [SeCN-] were separated by ion chromatography (IC) using isocratic elution with 20 mmol L-1 Na2CO3 / 2 mmol L-1 NaHCO3 as eluent. After the separation, the eluent carbonates were converted to CO2 on-line by a membrane self-regenerating suppressor, in order to avoid clogging problems in the nebulizer or in the cones of the ICP-MS interface. Detection limits (as 3σ of the baseline noise) of 0.18, 0.13 and 1.0 ng g-1 were achieved for Se (IV), Se (VI) and SeCN- respectively. Quantification of selenium in each separated species was performed using post-column isotope dilution analysis by continuous mixing of an enriched 77Se spike solution with the effluent from the suppressor column. Finally, the 78Se/77Se or 80Se/77Se isotope ratios were monitored and the concentration of the corresponding selenium species was evaluated. The proposed methodology has been successfully applied to study the quantitative speciation of selenium in petroleum refinery wastewaters.

Spiky strings inAdS4×CP3with Neveu-Schwarz flux

We study general rotating string solution in the AdS4 × CP3 background with a BNS holonomy turned on over CP1 ⊂ CP3. We find the giant magnon and single spike solutions for the string moving in this background corresponding to open spin chain. We calculate the corresponding dispersion relation among various conserved charges for both the cases. We further study the finite size effect on both the giant magnon and single spike solutions.

Giant magnon and spike solutions with two spins inAdS4×CP3

In the string theory in AdS_4 \times CP^3 we construct the giant magnon and spike solutions with two spins in two kinds of subspaces of R_t \times CP^3 and derive the dispersion relations for them. For the single giant magnon solution in one subspace we show that its dispersion relation is associated with that of the big one-spin giant magnon solution in the RP^2 subspace. For the single giant magnon solution in the other complementary subspace its dispersion relation is similar to that of the one-spin giant magnon solution living in the S^2 subspace but has one additional spin dependence.

Spiky strings on AdS4×CP3

We study a giant magnon and a spike solution for the string rotating on AdS(4) X CP**3 geometry. We consider rigid rotating fundamental string in the SU(2) X SU(2) sector inside the CP**3 and find out the general form of all the conserved charges. We find out the dispersion relation corresponding to both the known giant magnon and the new spike solutions. We further study the finite size correction in both cases.

Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction

We consider the following elliptic system: \[ \left \{\begin {array}{l} \varepsilon ^2 \Delta u-\lambda _1 u+\mu _1 u^3 + \beta u v^2 =0 \ \mbox {in} \ \Omega , \varepsilon ^2 \Delta v-\lambda _2 v+\mu _2 v^3 + \beta u^2 v =0 \ \mbox {in} \ \Omega , u,v >0 \ \mbox {in} \ \Omega , \ u=v=0 \ \mbox {on} \ \partial \Omega , \end {array} \right . \] where $\Omega \subset \mathbb {R}^N (N\leq 3)$ is a smooth and bounded domain, $\varepsilon >0$ is a small parameter, $\lambda _1, \lambda _2, \mu _1, \mu _2 >0$ are positive constants and $\beta \ne 0$ is a coupling constant. We show that there exists an interval $I=[a_0, b_0]$ and a sequence of numbers $0<\beta _1 <\beta _2 <...<\beta _n <...$ such that for any $\beta \in (0, +\infty ) \backslash (I \cup \{ \beta _1,..., \beta _n, ...\})$, the above problem has a solution such that both $u$ and $v$ develop a spike layer at the innermost part of the domain. Central to our analysis is the nondegeneracy of radial solutions in $\mathbb {R}^N$.

Neumann-Rosochatius integrable system for strings onAdS4× ℂℙ3

We use the reduction of the string dynamics onAdS4 × 3 to the Neumann-Rosochatius integrable system. All constraints can be expressed simply in terms of a few parameters. We analyze the giant magnon and single spike solutions on Rt × 3 with two angular momenta in detail and find the energy-charge relations. The finite-size effects of the giant magnon and single spike solutions are analyzed.

Vibrating giant spikes and the large-winding sector

The single spike is a rigidly rotating classical string configuration closely related to the giant magnon. We calculate bosonic and fermionic modes of this solution, from which we see that it is not supersymmetric. It can be viewed as an excitation above a hoop of string wound around the equator, in the same sense that the magnon is an excitation above an orbiting point particle. We find the operator which plays the role of the Hamiltonian for this sector, which compared to the magnon's E-J has the angular momentum replaced by a winding charge. The single spike solution is unstable, and we use the modes to attempt a semi-classical computation of its lifetime.

On the giant magnon and spike solutions for strings onAdS3×S3

We study solutions for the rotating strings on the sphere with a background NS-NS field and on the Anti-de-Sitter spacetime. We show the existence of magnon and single spike solutions on R$\times$S$^2$ in the presence of constant magnetic field as two limiting cases. We also study the solution for strings on AdS$_3\times$ S$^3$ with Melvin deformation. The dispersion relations among various conserved charges are shown to receive finite corrections due to the deformation parameter. We further study the rotating string on AdS$_3 \times$ S$^3$ geometry with two conserved angular momenta on S$^3$ and one spin along the AdS$_3$. We show that there exists two kinds of solutions: a circular string solution and a helical string. We find out the dispersion relation among various charges and give physical interpretation of these solutions.

New explicit spike solutions—non-local component of the generalized Mixmaster attractor

By applying a standard solution-generating transformation to an arbitrary vacuum Bianchi type II solution, one generates a new solution with spikes commonly observed in numerical simulations. It is conjectured that the spike solutions are part of the generalized Mixmaster attractor.

Scattering of single spikes

We apply the dressing method to a string solution given by a static string wrapped around the equator of a three-sphere and find that the result is the single spike solution recently discussed in the literature. Further application of the method allows the construction of solutions with multiple spikes. In particular we construct the solution describing the scattering of two single spikes and compute the scattering phase shift. As a function of the dressing parameters, the result is exactly the same as the one for the giant magnon, up to non-logarithmic terms. This suggests that the single spikes should be described by an integrable spin chain closely related to the one associated to the giant magnons. The field theory interpretation of such spin chain however is still unclear.

Reaction-Difusion Model of Early Carcinogenesis: The Effects of Influx of Mutated Cells

In this paper we explore a new model of field carcinogenesis, inspired by lung cancer precursor lesions, which includes dynamics of a spatially distributed population of pre-cancerous cells c(t, x), constantly supplied by an influx μ of mutated normal cells. Cell proliferation is controlled by growth factor molecules bound to cells, b(t, x). Free growth factor molecules g(t, x) are produced by precancerous cells and may diffuse before they become bound to other cells. The purpose of modelling is to investigate the existence of solutions, which correspond to formation of multiple spatially isolated lesions of pre-cancerous cells or, mathematically, to stable spike solutions. These multiple lesions are consistent with the field theory of carcinogenesis. In a previous model published by these authors, the influx of mutated cells was equal to zero, μ = 0, which corresponded to a single pre-malignant colony of cells. In that model, stable patterns appeared only if some of the growth factor was supplied from outside, arguably, a biologically tenuous hypothesis. In the present model, when μ > 0, that hypothesis is no more required, which makes this model more realistic. We present a range of results, both mathematical and computational, which taken together allow understanding the dynamics of this model. The equilibrium solutions in the current model result from the balance between new premalignant colonies being initiated and the old ones dying out.

Precise measurement of 228Ra/226Ra for 226Ra determination employing total integration and simultaneous 228Th correction by multicollector ICP-MS using multiple ion counters

A precise method of measuring 228Ra/226Ra for 226Ra determination used in U-series disequilibrium studies has been developed using the total integration method with simultaneous 228Th isobaric interference correction by multicollector ICP-MS with multiple ion counters (MIC). A sample solution of 0.2 ml was separately taken into a test tube, and all 226Ra+ and 228Ra+ signals as well as 229Th+ and 228Th16O+ from beginning to end were integrated simultaneously using four channeltrons, IC1, IC3, IC4 and IC5, respectively. Gains between IC1 and IC3 and the mass discrimination factor for 228Ra/226Ra determination were canceled out by measurement of a Ra spike solution with an accurately known Ra isotope ratio determined by TIMS, with which samples were bracketed. The isobaric interferences of 228Th+ were simultaneously corrected from the signal of 228Th16O+ and the oxide forming ratio of 228Th, which was separately determined by simultaneous measurements of 229Th+ and 229Th16O+ by IC3 and IC5. The gains between IC3 and IC5 for the Th correction were also canceled out. The ion yield of Ra in MIC-ICP-MS was ∼1.5%. The intermediate precision using 6.6, 3.3, 1.7, 0.83 and 0.41 fg of 226Ra with 228Ra/226Ra = ∼5 was 0.33, 0.93, 0.76, 1.6 and 1.5%, respectively. The 228Ra/226Ra ratio showed no systematic change dependent on 228Th/228Ra up to 0.25, indicating the validity of the 228Th correction. At similar amounts of Ra, our method gives intermediate precision values similar to or better than previous MC-ICP-MS and TIMS studies. The precision of the method was verified using another Ra spike with 228Ra/226Ra = ∼1 determined by TIMS. The analytical performance of the method was further investigated using the silicate samples JB-2 and JB-3 basalts issued by GSJ.

Application of the isotope dilution technique for 93Zr determination in an irradiated cladding material by multiple collector-inductively coupled plasma mass spectrometry

The determination of 93Zr concentration, a long-lived radionuclide present in spent nuclear fuel and in the structural components of nuclear reactors, is a major issue for nuclear waste disposal purpose and to validate neutronic calculation codes. To measure 93Zr concentration in irradiated cladding material with a high precision, an analytical method based on the use of multiple collector-inductively coupled plasma mass spectrometer (MC-ICPMS) combined to isotope dilution technique was developed. First a radiochemical separation of zirconium from a zircaloy sample (a zirconium alloy used as a cladding material for nuclear fuel elements), has allowed to obtain a very pure zirconium fraction with no potential isobaric interferences for mass spectrometric measurements. Then as the determination of all zirconium isotope ratios in the sample is necessary for the isotope dilution method, a MC-ICPMS procedure was developed to perform these precise measurements. Finally, the determination of 93Zr concentration in the same sample was performed, after preparation and calibration of a 96Zr spike solution. The uncertainties obtained on isotope ratios of zirconium by MC-ICPMS were in the order of 0.1%. The final uncertainty obtained on the 93Zr concentration in the nuclear material used and after chemical purification was lower than 0.6%.

Single spike solutions for strings onS2andS3

We study solutions for rigidly rotating strings on a two-sphere. Among them we find two limiting cases that have a particular interest, one is the already known giant magnon and the other we call the single spike solution. The limiting behavior of this last solution is a string infinitely wrapped around the equator. It differs from that solution by the existence of a single spike of height 0 that points toward the north pole. We study its properties and compute its energy E and angular momentum J as a function of 0. We further generalize the solution by adding one angular momentum to obtain a solution on S 3 . We find a spin chain interpretation of these results in terms of free fermions and the Hubbard model but the exact relation with the same models derived from the field theory is not clear.

D1-brane in β-deformed background

We study various configurations of rotating and wound D1-brane in AdS5 × S5 background and in its β deformed version. We find giant magnon and spike solutions on the world-volume of D1-brane in AdS5 × S5 background. We also analyse the equations of motion of D1-brane in β-deformed background. We show that in the limit of large electric flux on world-volume of D1-brane they reduce to the equations that describe collection of large number of fundamental strings. We also construct rotating and wound D1-brane solution that has two equal spins on S5γ.

On the multi-spin magnon and spike solutions from membranes

Recently important classes of solitonic string solutions were obtained—giant magnons and single spikes. In previous study we showed the existence of giant magnon-like membrane solutions and studied their properties. In this paper we investigate classical rotating membranes representing analog of a specific class of string spiky solutions. Using the reduction to the Neumann–Rosochatius integrable system we find analog of the string single spike solutions. In contrast to the magnon-like solutions, this case is characterized with finite difference of energy and “winding number” and finite spins as well.