Symmetry reductions and exact solutions of shallow water wave equations

In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} - u_{xt} - u_{xx} = 0,$$ whereα andβ are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by settingu _x =U, have been discussed in the literature. The caseα=2β was discussed by Ablowitz, Kaup, Newell and Segur (Stud. Appl. Math.,53 (1974), 249), who showed that this case was solvable by inverse scattering through a second-order linear problem. This case and the caseα=β were studied by Hirota and Satsuma (J. Phys. Soc. Japan,40 (1976), 611) using Hirota's bi-linear technique. Further, the caseα=β is solvable by inverse scattering through a third-order linear problem. In this paper, a catalogue of symmetry reductions is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole (J. Math. Mech,18 (1969), 1025). The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth Painlevé transcendents and Weierstrass elliptic functions. The nonclassical method yields a plethora of exact solutions of (1) withα=β which possess a rich variety of qualitative behaviours. These solutions all like a two-soliton solution fort < 0 but differ radically fort > 0 and may be viewed as a nonlinear superposition of two solitons, one travelling to the left with arbitrary speed and the other to the right with equal and opposite speed. These families of solutions have important implications with regard to the numerical analysis of SWW and suggests that solving (1) numerically could pose some fundamental difficulties. In particular, one would not be able to distinguish the solutions in an initial-value problem since an exponentially small change in the initial conditions can result in completely different qualitative behaviours. We compare the two-soliton solutions obtained using the nonclassical method to those obtained using the singular manifold method and Hirota's bi-linear method. Further, we show that there is an analogous nonlinear superposition of solutions for two (2+1)dimensional generalisations of the SWW Equation (1) withα=β. This yields solutions expressible as the sum of two solutions of the Korteweg-de Vries equation.     

Time-Resolved Fluorescence and Transient Spectroscopy in Determining Photochemical and Photophysical Channels in Reacting Systems in Solutions and Microheterogeneous Media

Abstract— Characterization of short-lived intermediates in homogeneous and microheterogeneous systems has been carried out using time-resolved spectroscopic techniques. The data obtained from these techniques have been analyzed in a relatively unconventional manner to elucidate complex transient behavior for two reactive systems. The highly nonexponential fluorescence decay for a series of fraws-stilbene-derivatized amphiphiles that readily form bilayer systems in aqueous media has been analyzed using a distribution of lifetimes analysis (DLA). The utility of DLA for quantitative studies was first determined by simulation of artificial decay data. Despite some limitations in DLA, qualitative conclusions as to the nature of the fluorescing species may be drawn when supplementary information such as steady-state spectroscopic data are also considered. The results indicate that the observed fluorescence originates from different types of excited-state species that consist of two or more trans-stilbene units; one of the emissions is attributed to the excited state of a ground-state aggregate while the other is assigned to an excimer that may arise from a 'defect'in the bilayer. The nonexponential nature of the decays is attributed to distributions of environments experienced by the fluorescing species. Electron transfer (ET) reactions between several excited pinacols and carbon tetrachloride in solution have been found to yield products with quantum yields that are higher than unity in the presence of oxygen, suggesting a chain mechanism for product formation. In these systems both the donor and the acceptor undergo bond fragmentation following the initial ET step. The individual steps involved in the proposed mechanism for these systems have been investigated in part using different steady-state and time-resolved laser spectroscopic techniques. However, it was also necessary to utilize pulse radiolysis in order to confirm the involvement of certain radical intermediates that were not observable by the usual flash photolysis techniques.     

Alkynylcyclohexadienyl tricarbonyliron complexes: A new directing effect for organoiron-mediated synthesis

The first example of an alkyne-substituted tricarbonyl(η⁵-cyclohexadienyl)iron(1+) complex has been prepared and the ω directing effect of the phenylethynyl substituent has been demonstrated in nucleophile addition reactions. Addition of NC⁻ also occurs at the a position to form an unusual η¹, η³-structure.     

Fast peroxyoxalate chemiluminescence for minimized analytical separation systems

The maximum intensity, Imax, and time required to reach the maximum emission, taumax, for 1-aminopyrene monitored in 1,1'-oxalyldi-4-methylimidazole (OD4MI) chemiluminescence (CL) reactions are approximately 61 times higher and 16 times faster than their respective values for bis(2,4,6-trichlorophenyl)oxalate (TCPO) CL reactions in the presence of imidazole (ImH).     

Mercury determination by CV-AAS in wastewater and sewage sludge from a stabilization pond system

The mercury concentrations in wastewater and sewage sludge of a stabilization pond system have been evaluated. The system is built by three parallel facultative ponds followed by two systems of three maturation ponds in series. The samples of wastewater and sludge were digested using nitric acid and placed into a Parr-type bomb for 4 h at 110° C. Mercury was measured by Cold Vapour Atomic Absorption Spectrometry (CV-AAS) at 253.7 nm with sodium tetrahydroborate as reductant. The methodologies were checked with an USEPA quality control sample, a standard reference material from NIST and with another method of mineralization (cold mineralization) showing good results. Concentrations of mercury in wastewater between 1.47 ± 0.75 gl^–1 have been found at the entrance of the system and 0.74 ± 0.0 gl^–1 at the exit, while in sludge the results were between 0.29 ± 0.12 gkg^–1 in the facultative pond and 0.04 ± 0.02 gkg^–1 in the second maturation pond (exit).     

Synthesis and binding affinities of novel SRIF-mimicking beta-D-glucosides satisfying the requirement for a pi-cloud at C1

[reaction: see text] The synthesis of four bioactive analogues of the somatostatin (SRIF-14) mimetic, beta-d-glucoside (+)-2, in which the C1 indole side chain is replaced with indole surrogates, has been achieved. These congeners, possessing the naphthyl, benzothiophene, benzyl, and benzofuran substituents, were predicted to satisfy the electrostatic requirements of the tryptophan binding pocket of SRIF. Unlike the previously described C4 picolyl and pyrazinyl congeners, these ligands bind the hSST4 receptor.     

A comparative analysis of the total syntheses of the amphidinolide T natural products

In this article we compare and contrast the strategies and tactics used in the syntheses of the amphidinolide T family of natural products that have been reported by Fürstner, Ghosh and ourselves. Similar approaches to the trisubstituted THF ring present in the targets are utilized in all of the syntheses, but each strategy showcases a different means of macrocyclization.     

cis-Enhanced cyclopropanation catalysts: reaction chemistry of three isomers of Rh2[N(C6H5)COCH3]4

The catalytic activities of three structural isomers of Rh₂[N(C₆H₅)COCH₃]₄ in cyclopropanation reactions were surveyed. These studies showed cis cyclopropanation selectivity with bulky alkenes for 2,2-cis- and 2,2-trans-Rh₂[N(C₆H₅)COCH₃]₄.     

Rapid serial analysis of multiple oligonucleotide samples on a microchip using optically-gated injection

Optically-gated injection of fluorescently-labeled DNA has been accomplished for the first time. Rapid, serial analysis of oligonucleotide ladders has been shown on a microchip using this injection technique. Separations of five- and six-component samples have been completed in 60 s or less with a capability to carry out serial injections of these samples every 15 s. The technique has been shown to have better than five base resolution for small oligonucleotides and excellent reproducibility in migration times (< or = 0.75% RSD). Currently, the limit of detection for the system is 0.23 microM. Additionally, multiple unique samples of DNA have been consecutively analyzed in a single separation lane using optical gating. Six consecutive injections of three different samples have been achieved with no sample carryover and a total analysis time of approximately 10 min. These results show the potential of optical gating as an alternative injection technique for high-throughput DNA applications, such as genotyping and monitoring dynamic processes.     

Second-order nonlinear optical properties of the Ag(111)/electrolyte interface in the presence of self-assembled monolayers containing conjugated pi systems. I. Alpha-functionalized terthiophene films on Ag(111)

The nonlinear optical properties of self-assembled monolayers obtained from bonding two different alpha-functionalized terthiophenes (alpha-T3) to (111) silver electrode surfaces have been investigated using second harmonic generation (SHG). The two (alpha-T3) compounds used were functionalized with alkane chains of different lengths (C8 and C4), and each was terminated with a thiol anchoring group. A nitrile group was attached to the terminal thiophene ring of the (alpha-T3) compound with the C4 chain. The orientation of the polarization of the incident beam was changed systematically and gradually between "p" and "s" orientation and the SH signal (isotropic and anisotropic contributions) analyzed in both directions ("P" and "S"). The symmetry of the system was reduced by the presence of the adlayers from C3v to C3. The dependence on the applied potential and the incident wavelength has also been studied. The relative magnitudes and phases of the various second-order tensor elements have been estimated and compared with the values for a bare surface. A resonance process in the (alpha-T3) pi moiety has been investigated, and from this, the effective "band-gap" energies of the organic semiconductor SAMs (i.e., the energy difference between the pi-pi bands) have been estimated.