Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

Integrable evolution equations in time-dependent domains

We discuss the implementation of a method of solving initial boundary value problems in the case of integrable evolution equations in a time-dependent domain. This method is applied to a dispersive linear evolution equation with spatial derivatives of arbitrary order and to the defocusing nonlinear Schrödinger equation, in the domain l(t)

Surprisingly slow reaction of dimethylsilylene with dimethylgermane: time-resolved kinetic studies and related quantum chemical calculations

Time-resolved studies of silylene, SiH2, and dimethylsilylene, SiMe2, generated by the 193 nm laser flash photolysis of appropriate precursor molecules have been carried out to obtain rate constants for their bimolecular reactions with dimethylgermane, Me2GeH2, in the gas phase. SiMe2 + Me2GeH2 was studied at five temperatures in the range 299-555 K. Problems of substrate UV absorption at 193 nm at temperatures above 400 K meant that only three temperatures could be used reliably for rate constant measurement. These rate constants gave the Arrhenius parameters log(A/cm(3) molecule(-1) s(-1)) = -13.25 +/- 0.16 and E-a = -(5.01 +/- 1.01) kJ mol(-1). Only room temperature studies of SiH2 were carried out. These gave values of (4.05 +/- 0.06) x 10(-10) cm(3) molecule(-1) s(-1) (SiH2 + Me2GeH2 at 295 K) and also (4.41 +/- 0.07) x 10(-10) cm(3) molecule(-1) s(-1) (SiH2 + MeGeH3 at 296 K). Rate constant comparisons show the surprising result that SiMe2 reacts 12.5 times slower with Me2GeH2 than with Me2SiH2. Quantum chemical calculations (G2(MP2,SVP)//B3LYP level) of the model Si-H and Ge-H insertion processes of SiMe2 with SiH4/MeSiH3 and GeH4/MeGeH3 support these findings and show that the lower reactivity of SiMe2 with Ge-H bonds is caused by a higher secondary barrier for rearrangement of the initially formed complexes. Full details of the structures of intermediate complexes and the discussion of their stabilities are given in the paper. Other, related, comparisons of silylene reactivity are also presented.

The addition reaction between silylene and ethyne: further isotope studies, pressure dependence studies, and quantum chemical calculations

Time-resolved kinetic studies of the reaction of dideutero-silylene, SiD2, generated by laser flash photolysis of phenylsilane-d(3), have been carried out to obtain rate constants for its bimolecular reaction with C2H2. The reaction was studied in the gas phase over the pressure range 1-100 Torr in SF6 bath gas, at five temperatures in the range 297-600 K. The second-order rate constants obtained by extrapolation to the high-pressure limits at each temperature fitted the Arrhenius equation log(k(infinity)/cm(3) molecule(-1) s(-1)) = (-10.05 +/- 0.05) + (3.43 +/- 0.36 kJ mol(-1))/RT ln 10. The rate constants were used to obtain a comprehensive set of isotope effects by comparison with earlier obtained rate constants for the reactions of SiH2 with C2H2 and C2D2. Additionally, pressure-dependent rate constants for the reaction of SiH2 with C2H2 in the presence of He (1-100 Tort) were obtained at 300, 399, and 613 K. Quantum chemical (ab initio) calculations of the SiC2H4 reaction system at the G3 level support the initial formation of silirene, which rapidly isomerizes to ethynylsilane as the major pathway. Reversible formation of vinylsilylene is also an important process. The calculations also indicate the involvement of several other intermediates, not previously suggested in the mechanism. RRKM calculations are in semiquantitative agreement with the pressure dependences and isotope effects suggested by the ab initio calculations, but residual discrepancies suggest the possible involvement of the minor reaction channel, SiH2 + C2H2 - SWPO + C2H4. The results are compared and contrasted with previous studies of this reaction system.

Investigation of the prototype silylene reaction, SiH2+H2O(and D2O): time-resolved gas-phase kinetic studies, isotope effects, RRKM calculations, and quantum chemical calculations of the reaction energy surface

Time-resolved kinetic studies of the reaction of silylene, SiH2, with H2O and with D2O have been carried out in the gas phase at 296 and at 339 K, using laser flash photolysis to generate and monitor SiH2. The reaction was studied over the pressure range 10-200 Torr with SF6 as bath gas. The second-order rate constants obtained were pressure dependent, indicating that the reaction is a third-body assisted association process. Rate constants at 339 K were about half those at 296 K. Isotope effects, k(H)/k(D), were small averaging 1.076 0.080, suggesting no involvement of H- (or D-) atom transfer in the rate determining step. RRKM modeling was undertaken based on a transition state appropriate to formation of the expected zwitterionic donoracceptor complex, H2Si...OH2. Because the reaction is close to the low pressure (third order) region, it is difficult to be definitive about the activated complex structure. Various structures were tried, both with and without the incorporation of rotational modes, leading to values for the high-pressure limiting (i.e., true secondorder) rate constant in the range 9.5 x 10(-11) to 5 x 10(-10) cm(3) molecule' s(-1). The RRKM modeling and mechanistic interpretation is supported by ab initio quantum calculations carried out at the G2 and G3 levels. The results are compared and contrasted with the previous studies.

Experimental and theoretical evidence for homogeneous catalysis in the gas-phase reaction of SiH2 with H2O (and D2O): a combined kinetic and quantum chemical study

Time-resolved kinetic studies of the reaction of silylene, SiH2, with H2O and with D2O have been carried out in the gas phase at 297 K and at 345 K, using laser flash photolysis to generate and monitor SiH2. The reaction was studied independently as a function of H2O (or D2O) and SF6 (bath gas) pressures. At a fixed pressure of SF6 (5 Torr), [SiH2] decay constants, k(obs), showed a quadratic dependence on [H2O] or [D2O]. At a fixed pressure of H2O or D2O, k(obs) Values were strongly dependent on [SF6]. The combined rate expression is consistent with a mechanism involving the reversible formation of a vibrationally excited zwitterionic donor-acceptor complex, H2Si...OH2 (or H2Si...OD2). This complex can then either be stabilized by SF6 or it reacts with a further molecule of H2O (or D2O) in the rate-determining step. Isotope effects are in the range 1.0-1.5 and are broadly consistent with this mechanism. The mechanism is further supported by RRKM theory, which shows the association reaction to be close to its third-order region of pressure (SF6) dependence. Ab initio quantum calculations, carried out at the G3 level, support the existence of a hydrated zwitterion H2Si...(OH2)(2), which can rearrange to hydrated silanol, with an energy barrier below the reaction energy threshold. This is the first example of a gas-phase-catalyzed silylene reaction.

Quantum mechanics studies of the tautomers of dlhydrodiacetylformoin, an important Maillard intermediate and the synthesis of furaneol

Conformational analyses have been carried out on the acyclic and cyclic forms of dihydrodiacetylformoin, an important Maillard intermediate and precursor for furaneol. For the acyclic forms, the 2,5-dicarbonyl isomers have the lowest energy, while for the cyclic forms, the 3-carbonyl are favoured over the 4-carbonyl isomers. The likely path for cyclisation is investigated and it is shown that the favoured path is dependent upon the relative chiralities of the carbon atoms and in particular that the reaction proceeds more readily if C2 and C3 have different chiralities. After cyclisation, the reaction path to produce furaneol proceeds via the loss of a water molecule. This reaction has been studied with a model including two water molecules and a hydroxide anion and shows relatively low-energy barriers. (C) 2008 Published by Elsevier B.V.

Modelling metal cutting using modern ductile fracture mechanics: Quantitative explanations for some longstanding problems

The assumption that negligible work is involved in the formation of new surfaces in the machining of ductile metals, is re-examined in the light of both current Finite Element Method (FEM) simulations of cutting and modern ductile fracture mechanics. The work associated with separation criteria in FEM models is shown to be in the kJ/m2 range rather than the few J/m2 of the surface energy (surface tension) employed by Shaw in his pioneering study of 1954 following which consideration of surface work has been omitted from analyses of metal cutting. The much greater values of surface specific work are not surprising in terms of ductile fracture mechanics where kJ/m2 values of fracture toughness are typical of the ductile metals involved in machining studies. This paper shows that when even the simple Ernst–Merchant analysis is generalised to include significant surface work, many of the experimental observations for which traditional ‘plasticity and friction only’ analyses seem to have no quantitative explanation, are now given meaning. In particular, the primary shear plane angle φ becomes material-dependent. The experimental increase of φ up to a saturated level, as the uncut chip thickness is increased, is predicted. The positive intercepts found in plots of cutting force vs. depth of cut, and in plots of force resolved along the primary shear plane vs. area of shear plane, are shown to be measures of the specific surface work. It is demonstrated that neglect of these intercepts in cutting analyses is the reason why anomalously high values of shear yield stress are derived at those very small uncut chip thicknesses at which the so-called size effect becomes evident. The material toughness/strength ratio, combined with the depth of cut to form a non-dimensional parameter, is shown to control ductile cutting mechanics. The toughness/strength ratio of a given material will change with rate, temperature, and thermomechanical treatment and the influence of such changes, together with changes in depth of cut, on the character of machining is discussed. Strength or hardness alone is insufficient to describe machining. The failure of the Ernst–Merchant theory seems less to do with problems of uniqueness and the validity of minimum work, and more to do with the problem not being properly posed. The new analysis compares favourably and consistently with the wide body of experimental results available in the literature. Why considerable progress in the understanding of metal cutting has been achieved without reference to significant surface work is also discussed.

Solving optimal control problems with state constraints using nonlinear programming and simulation tools

This paper illustrates how nonlinear programming and simulation tools, which are available in packages such as MATLAB and SIMULINK, can easily be used to solve optimal control problems with state- and/or input-dependent inequality constraints. The method presented is illustrated with a model of a single-link manipulator. The method is suitable to be taught to advanced undergraduate and Master's level students in control engineering.

Wigner Transport models of the electron-phonon kinetics in quantum wires

Two quantum-kinetic models of ultrafast electron transport in quantum wires are derived from the generalized electron-phonon Wigner equation. The various assumptions and approximations allowing one to find closed equations for the reduced electron Wigner function are discussed with an emphasis on their physical relevance. The models correspond to the Levinson and Barker-Ferry equations, now generalized to account for a space-dependent evolution. They are applied to study the quantum effects in the dynamics of an initial packet of highly nonequilibrium carriers, locally generated in the wire. The properties of the two model equations are compared and analyzed.

The gas-phase reaction between silylene and 2-butyne: kinetics, isotope studies, pressure dependence studies and quantum chemical calculations

Time-resolved kinetic studies of the reactions of silylene, SiH2, and dideutero-silylene, SiD2, generated by laser. ash photolysis of phenylsilane and phenylsilane-d(3), respectively, have been carried out to obtain rate coefficients for their bimolecular reactions with 2-butyne, CH3C CCH3. The reactions were studied in the gas phase over the pressure range 1-100 Torr in SF6 bath gas at five temperatures in the range 294-612 K. The second-order rate coefficients, obtained by extrapolation to the high pressure limits at each temperature, fitted the Arrhenius equations where the error limits are single standard deviations: log(k(H)(infinity)/cm(3) molecule(-1) s(-1)) = (-9.67 +/- 0.04) + (1.71 +/- 0.33) kJ mol(-1)/RTln10 log(k(D)(infinity)/cm(3) molecule(-1) s(-1)) = (-9.65 +/- 0.01) + (1.92 +/- 0.13) kJ mol(-1)/RTln10 Additionally, pressure-dependent rate coefficients for the reaction of SiH2 with 2-butyne in the presence of He (1-100 Torr) were obtained at 301, 429 and 613 K. Quantum chemical (ab initio) calculations of the SiC4H8 reaction system at the G3 level support the formation of 2,3-dimethylsilirene [cyclo-SiH2C(CH3)=C(CH3)-] as the sole end product. However, reversible formation of 2,3-dimethylvinylsilylene [CH3CH=C(CH3)SiH] is also an important process. The calculations also indicate the probable involvement of several other intermediates, and possible products. RRKM calculations are in reasonable agreement with the pressure dependences at an enthalpy value for 2,3-dimethylsilirene fairly close to that suggested by the ab initio calculations. The experimental isotope effects deviate significantly from those predicted by RRKM theory. The differences can be explained by an isotopic scrambling mechanism, involving H - D exchange between the hydrogens of the methyl groups and the D-atoms in the ring in 2,3-dimethylsilirene-1,1-d(2). A detailed mechanism involving several intermediate species, which is consistent with the G3 energy surface, is proposed to account for this.

On the quantum mechanical scattering statistics of many particles

The probability of a quantum particle being detected in a given solid angle is determined by the S-matrix. The explanation of this fact in time-dependent scattering theory is often linked to the quantum flux, since the quantum flux integrated against a (detector-) surface and over a time interval can be viewed as the probability that the particle crosses this surface within the given time interval. Regarding many particle scattering, however, this argument is no longer valid, as each particle arrives at the detector at its own random time. While various treatments of this problem can be envisaged, here we present a straightforward Bohmian analysis of many particle potential scattering from which the S-matrix probability emerges in the limit of large distances.

On the time-dependent analysis of Gamow decay

Gamow's explanation of the exponential decay law uses complex 'eigenvalues' and exponentially growing 'eigenfunctions'. This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wavefunctions. Observing that the time evolution of any wavefunction is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover, the presentation can well be used in physics lectures to students.

The reaction between silylene and ammonia: some gas-phase kinetic and quantum chemical studies

Time-resolved kinetic studies of the reaction of silylene, SiH2, generated by 193 nm laser flash photolysis of silacyclopent-3-ene, have been carried out in the presence of ammonia, NH3. Second order kinetics were observed. The reaction was studied in the gas phase at 10 Torr total pressure in SF6 bath gas at each of the three temperatures, 299, 340 and 400 K. The second order rate constants (laser pulse energy of 60 mJ/pulse) fitted the Arrhenius equation: log(k/cm3 molecule-1 s-1) = (-10.37 ± 0.17) + (0.36 ± 1.12 kJ mol-1)/RTln10 Experiments at other pressures showed that these rate constants were unaffected by pressure in the range 10-100 Torr, but showed small decreases in value at 3 and 1 Torr. There was also a weak intensity dependence, with rate constants decreasing at laser pulse energies of 30 mJ/pulse. Ab initio calculations at the G3 level of theory, show that SiH2 + NH3 should form an initial adduct (donor-acceptor complex), but that energy barriers are too great for further reaction of the adduct. This implies that SiH2 + NH3 should be a pressure dependent association reaction. The experimental data are inconsistent with this and we conclude that SiH2 decays are better explained by reaction of SiH2 with the amino radical, NH2, formed by photodissociation of NH3 at 193 nm. The mechanism of this previously unstudied reaction is discussed.

Double jeopardy: the potential for problems when bats interact with breathable roofing membranes in the United Kingdom

In order to reduce environmental impacts and achieve sustainability, it is important to balance the interactions between the built and natural environment. The construction industry is becoming more aware of ecological concerns and the importance that biodiversity and maintenance ecosystem services has for sustainability. Bats constitute an important component of urban biodiversity and several species in the UK are highly dependent on buildings, making them particularly vulnerable to anthropogenic and environmental changes. Many buildings suitable for use as bat roosts often require re-roofing as they age and traditional bituminous roofing felts are frequently being replaced with breathable roofing membranes (BRMs). In the UK new building regulations and modern materials may substantially reduce the viability of existing roosts, yet at thesame time building regulations require that materials be fit for purpose. Reports suggest that both bats and BRMs may experience problems when the two interact. Such information makes it important to understand how house dwelling bats and BRMs may be affected. This paper considers the possible ways in which bats and BRMs may interact, how this could affect existing bat roosts within buildings and the implications for BRM service life predictions and warranties. Keywords –Breathable Roofing Membranes, Bats in Buildings, Material Deterioration, Sustainability, Conservation, Biodiversit