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At least 937 records · Page 52

A Prediction of Uranium(VI) Diffusion in Montmorillonite at Various Chemical Solution Conditions

Most nuclear waste disposal options include compacted bentonite, predominately consisting of montmorillonite clay, as part of a barrier system to minimize contaminant mobility. Uranium (U) is the primary element in spent nuclear fuel, and a potential contaminant of water resources. Uranium sorption onto clay and its slow diffusive transport away from waste canisters are expected to limit U(VI) mobility, and possibly also control nuclear fuel degradation rates based on mass action considerations. A prediction of U(VI) diffusion in these systems is complicated by two main factors: (1) the complex mineralogical structure of montmorillonite clay, leading to two types of clay porosities and surface sites, and (2) the dependence of U(VI) sorption on U(VI) solution speciation and the specific chemical system conditions. For instance, depending on pH and ionic strength, U(VI) may sorb onto montmorillonite due to weak ion exchange reactions or the formation of stable surface complexes. Furthermore, in contrast to conventional porous media, U(VI) diffusion in montmorillonite is not only limited by contaminant sorption, but also by a (potential) lack of access to the full clay porosity for some U(VI) solution species. For instance, at the high degrees of clay compaction expected in future barrier systems, the partial or full exclusion of U(VI) from negatively-charged clay interlayer spaces may substantially decrease U(VI) fluxes. In this modeling study, we specifically investigated how the charge of U(VI) solution species affects U(VI) sorption and diffusion behavior. For this purpose, we first simplified complex chemical solution speciation diagrams to charge speciation diagrams. Then, we coupled an existing U(VI) surface complexation model (Tournassat et al., Geochim. Cosmochim. Acta, 2018) with an analytical solution of the diffusion equation to simulate U(VI) mobility in lab-scale through-diffusion experiments for various chemical conditions. Our results suggest that there are two important parameters that govern U(VI) diffusion in these systems. First, U(VI) surface complexation on clay edge surfaces leads to contaminant retardation. On the field scale, this will determine how early a plume may cross site boundaries and pose risks to drinking water resources. Second, the degree of access of U(VI) solution species to diffuse layers in clay interlayer spaces largely controls U(VI) diffusive fluxes. In the field, this parameter will govern the level of U(VI) concentrations that can be expected in contaminated waters.

Pistorino, Jonathan C.↗

Hybrid Analytics Solution for Power Plant Operations

Expert Microsystems, Inc., MapEx Software Inc. and XMPLR Energy LLC have teamed to submit a proposal to the US Department of Energy’s Office of Fossil Energy to improve the efficiency, reliability and flexibility of existing coal-based power plants. The project, titled Hybrid Analytics Solution to Improve Coal Power Plant Operations (the “Hybrid Analytics” project), will develop, demonstrate and commercialize a novel approach that integrates two proven real-time monitoring techniques. The Hybrid Analytics approach integrates into a single, integrated solution: a) an established, advanced data-driven analytics solution that includes artificial intelligence, advanced pattern recognition and machine-learning techniques and b) a well-proven, first principle thermal heat balance model solution. The future role of coal plants will depend on the ability to cycle and follow loads to meet marginal power demands, accommodate renewables and support the grid. Essential to success in this new role is a plant’s ability to maintain high reliability, efficiency and flexibility. However, these changing operational demands put significant stresses on power plant components that directly impact the ability to deliver these requirements. In the past, plants have independently monitored performance using physics-based solutions (heat rate, boiler losses, etc.) and reliability using data-driven analytics (advanced pattern recognition). The Hybrid Analytics solution integrates both approaches to enhance fault detection, provide automated diagnostics and estimates of remaining time to act, and provide critical, enhanced and accurate real-time information to coal plant operators. As a result, coal plants can improve efficiency, reliability, and flexibility and can cost-effectively maintain their key role in the power delivery system.

01 COAL, LIGNITE, AND PEAT↗

Adjusting the pH of a pretreatment solution using carbon dioxide useful for integrating saccharification and fermentation

The present invention provides for a method of fermenting or saccharifying a biomass comprising: (a) (i) contacting a biomass comprising a polysaccharide, and an ionic liquid (IL) to form a first solution, or (ii) providing the first solution comprising the biomass and the IL, (b) contacting the first solution and carbon dioxide such that the first solution results in a lower pH, (c) introducing (i) an enzyme capable of enzymatically to breakdown at least one bond in the polysaccharide or a breakdown product of the polysaccharide, and/or (ii) a microorganism that capable of producing the enzyme and/or fermenting the polysaccharide or a breakdown product of the polysaccharide, such that the polysaccharide is at least partially broken down and the first solution is transformed into a second solution.

Sun, Jian↗

Lipophilic diglycolamide compounds for extraction of rare earth metals from aqueous solutions

A method for extracting rare earth metals (e.g., lanthanides and/or actinides) from aqueous solution, the method comprising: (i) acidifying an aqueous solution containing said rare earth metals with sulfuric acid to result in an acidified aqueous solution containing 1-12 M concentration of sulfuric acid; and (ii) contacting the acidified aqueous solution with an aqueous-insoluble hydrophobic solution comprising a rare earth extractant molecule dissolved in an aqueous-insoluble hydrophobic solvent to result in extraction of one or more of the rare earth metals into the aqueous hydrophobic solution, wherein the rare earth extractant molecule has the following structure: wherein R1, R2, R3, and R4 are independently selected from hydrocarbon groups containing 1-20 carbon atoms, provided that the total carbon atoms in R1, R2, R3, and R4 is at least 12; and R5 and R6 are independently selected from hydrogen atom and hydrocarbon groups containing 1-3 carbon atoms.

Brigham, Derek↗

Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge–Kutta fourth–order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler–Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, ƒ(u), through a novel process. The roots of ƒ(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.

Laos, Hector↗

Corrosion Testing of Carbon Steel Exposed to Sludge Heel Chemical Cleaning Solutions - 16109

As efforts continue to treat and dispose of millions of gallons of legacy radioactive materials from the production of nuclear weapons, non-compliant waste storage tanks will gradually be emptied of the bulk waste volume leaving heel materials requiring removal prior to tank closure. The waste heel slurries are distributed on the floor of large, million gallon tanks, which frequently contain numerous obstructions that limit the effectiveness of mechanical removal methods. As a result, chemical cleaning methods are needed for the effective removal of the heels as well as chemical scales that are present on the tank walls and other interior surfaces. Two tank cleaning technologies have already been implemented at the Savannah River Site (SRS): Low Temperature Aluminum Dissolution (LTAD) and Bulk Oxalic Acid Cleaning (BOAC). Recent chemical cleaning efforts on SRS Tank 12 were very successful with regard to bulk sludge heel (especially for Al, Fe, and U phases) and beta/gamma radionuclide removal. Although chemical cleaning using these technologies has been shown to be effective, no disposition path has been identified for oxalate added during BOAC, and insoluble oxalate salts are accumulating within the SRS tank farm and waste processing facilities (evaporators, etc.). Extensive sludge washing is also required to remove moderately soluble sodium oxalate salts prior to sludge vitrification in the SRS Defense Waste Processing Facility (DWPF). As a result, oxalate additions to the tank farm need to be minimized by the use of supplementary acids to assist sludge removal in OA or the use of other cleaning reagents or processing strategies. Previous SRNL testing revealed the importance of pH control for BOAC and recommended the use of a supplementary acid (dilute HNO{sub 3}) with OA to minimize oxalate additions. Primary drivers in SRS Tank Closure Performance Assessments, which evaluate the fate and impact of tank sludge residuals on a geological timescale, are the removal of alpha emitting radionuclides present at low concentrations such as Pu, Am, and Np, which are not highly soluble in currently utilized chemical cleaning reagents. Scoping studies conducted at SRNL revealed promising methods to dissolve the actinides within the HLW tank heels. Oxidation of the actinides with permanganate in either strong caustic (10 M NaOH) or dilute acidic (0.2 M HNO{sub 3}) solutions was shown to result in dissolution of oxy/hydroxide phases of these metals in the absence of major sludge phases. The waste tanks at SRS and Hanford are constructed of carbon steel. Carbon steel corrodes rapidly in most acidic and/or oxidizing environments. Previous electrochemical corrosion testing indicated that the corrosion rates in dilute HNO{sub 3} with OA may result in manageable corrosion rates, over short exposure periods. Additional corrosion testing was performed to provide data for the nitric acid/OA blend as well as the acid/permanganate and caustic permanganate based chemical cleaning solutions. Corrosion rates from the passive coupon tests for the nitric acid/oxalic acid blend were significantly greater than those observed previously with the electrochemical testing. This result may be due to simultaneous electrochemical and chemical dissolution of the steel. The corrosion rate data from the coupon tests in the nitric acid/oxalic acid blend were evaluated to determine the degree of potential structural damage. The corrosion rates, although relatively high, would not be expected to cause damage that would reduce the capacity of the tank primary wall if the process is completed within a month. The corrosivity of sodium permanganate in four proposed cleaning solutions, 1 nitric acid solution and 3 sodium hydroxide solutions, was studied by electrochemical methods at room temperature. The corrosion rates were significantly less aggressive than in the nitric acid/oxalic acid blend corrosion tests. Passive coupon tests needs to be performed in these solutions to ensure that chemical dissolution of the metal is not occurring simultaneously. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Integrated solution techniques for security constrained unit commitment problem

Apparatus and methods are disclosed for solving Mixed Integer Programming (MIP) problems, such as Security Constrained Unit Commitment (SCUC) problems used by power grid authorities to perform day-ahead market clearing. In certain examples, a plurality of threads of a software tool implementing a concurrent optimizer can be executed concurrently and sequentially to generate new solutions to a SCUC problem for an upcoming planning horizon. Data can be shared among the concurrently executing threads, such as intermediate/incumbent solutions and hints regarding the fixing of variables and constraints to reduce the size of the SCUC problem. In some examples, the threads are seeded with historical solutions from prior planning horizons. The software tool can select a best solution from the solutions generated by the threads, and determine dispatch instructions for a device coupled to the power grid for the upcoming planning horizon based at least in part on the selected solution.

Pan, Feng↗

Method of adjusting the pH of a pretreatment solution using carbon dioxide useful for integrating saccharification and fermentation of a biomass

The present invention provides for a method of fermenting or saccharifying a biomass comprising: (a) (i) contacting a biomass comprising a polysaccharide, and an ionic liquid (IL) to form a first solution, or (ii) providing the first solution comprising the biomass and the IL, (b) contacting the first solution and carbon dioxide such that the first solution results in a lower pH, (c) introducing (i) an enzyme capable of enzymatically to breakdown at least one bond in the polysaccharide or a breakdown product of the polysaccharide, and/or (ii) a microorganism that capable of producing the enzyme and/or fermenting the polysaccharide or a breakdown product of the polysaccharide, such that the polysaccharide is at least partially broken down and the first solution is transformed into a second solution.

Sun, Jian↗

Efficient traveltime solutions of the acoustic TI eikonal equation

Numerical solutions of the eikonal (Hamilton–Jacobi) equation for transversely isotropic (TI) media are essential for imaging and traveltime tomography applications. Such solutions, however, suffer from the inherent higher-order nonlinearity of the TI eikonal equation, which requires solving a quartic polynomial for every grid point. Analytical solutions of the quartic polynomial yield numerically unstable formulations. Thus, it requires a numerical root finding algorithm, adding significantly to the computational load. Using perturbation theory we approximate, in a first order discretized form, the TI eikonal equation with a series of simpler equations for the coefficients of a polynomial expansion of the eikonal solution, in terms of the anellipticity anisotropy parameter. Such perturbation, applied to the discretized form of the eikonal equation, does not impose any restrictions on the complexity of the perturbed parameter field. Therefore, it provides accurate traveltime solutions even for models with complex distribution of velocity and anisotropic anellipticity parameter, such as that for the complicated Marmousi model. The formulation allows for large cost reduction compared to using the direct TI eikonal solver. Furthermore, comparative tests with previously developed approximations illustrate remarkable gain in accuracy in the proposed algorithm, without any addition to the computational cost.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Roles of Bi, M and VO{sub 4} tetrahedron in photocatalytic properties of novel Bi{sub 0.5}M{sub 0.5}VO{sub 4} (M=La, Eu, Sm and Y) solid solutions for overall water splitting

Novel Bi{sub 0.5}M{sub 0.5}VO{sub 4} (BMV; M=La, Eu, Sm and Y) solid solutions were prepared and studied in this paper. All the samples were proved to produce H{sub 2} and O{sub 2} simultaneously from pure water under the irradiation of UV light. M-O bond lengths were proved to increase with M cations by refining cell parameters and atomic positions. Besides, band gaps, energy gaps and photocatalytic activities of BMV also changed with M cations. Both of M-O and V-O bond lengths were suggested to account for this phenomenon. Inactive A{sub 0.5}Y{sub 0.5}VO{sub 4} (A=La, Ce) for water splitting proved incorporation of Bi rather than distortion of VO{sub 4} tetrahedron was a critical factor for improving efficiency of overall water splitting by facilitating the generation of electron and hole with lighter effective masses. Replacement of Bi by M cations not only gave indirect effect on band structure but also raised position of conduction band minimum to meet requirement of H{sub 2} production. - Graphical abstract: Novel Bi{sub 0.5}M{sub 0.5}VO{sub 4} (M=La, Eu, Sm and Y) solid solutions showed the high and stable photocatalytic activities for overall water splitting with their crystal radii of M elements. Highlights: Black-Right-Pointing-Pointer BMV solid solutions were novel highly efficient V-based photocatalysts for overall water splitting. Black-Right-Pointing-Pointer Photocatalytic activity of BMV solid solution related to the effective ionic radii of M cations. Black-Right-Pointing-Pointer Incorporation of Bi is one of key factors for the highly efficient activity of BMV solid solution. Black-Right-Pointing-Pointer Incorporation of Y is dispensable for H{sub 2} production.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Ion–Molecule Complex Dissociation and Formation Dynamics in LiCl Aqueous Solutions from 2D IR Spectroscopy

Ion–molecule complex dynamics as well as water dynamics in concentrated lithium chloride (LiCl) solutions are examined using ultrafast two-dimensional infrared (2D IR) spectroscopy with the CN stretching mode of methyl thiocyanate (MeSCN) as the vibrational probe. In pure water, MeSCN has a narrow symmetric absorption line shape. 2D IR spectral diffusion measurements of the CN stretch give the identical time dependence of water dynamics, as previously observed using the OD stretch of HOD in H 2 O. In concentrated LiCl solutions, the IR absorption spectrum of MeSCN displays two distinct peaks, one corresponding to water H-bonded to the N lone pair of MeSCN (W) and the other corresponding to Li + associated with the N (L). These two species are in equilibrium, and switching of the CN bonding partner from Li + to H 2 O and vice versa was observed and explicated with 2D IR chemical exchange spectroscopy. The MeSCN·Li + complex dissociation time constant, τ LW , and the MeSCN·H 2 O dissociation time constant, τ WL , were determined. The observed τ LW chemical exchange dissociation time constant changes from 60 to 40 ps as the LiCl concentration decreases from ~10.7 to ~7.7 M, mainly due to the increase of the water concentration as the LiCl concentration is reduced. The observed time constants are independent of the model for the chemical reaction. With the assumption of a simple chemical equation, MeSCN·Li + + H 2 O ⇌ MeSCN·H 2 O + Li + , the equilibrium equation rate constants were obtained from the observed chemical exchange time constants. It was determined that the equilibrium rate constants barely change even though the viscosity changes by a factor of 2 and the ionic strength changes by a factor of 1.4. Extrapolation to dilute LiCl solution estimates the τ LW to be ~30 ps. The orientational relaxation (anisotropy decay) of both the W and L complexes was measured using polarization selective 2D IR experiments. The lithium-bonded species undergoes orientational relaxation ~3 times slower than the water-bonded species in each LiCl solution studied. In conclusion, the difference demonstrates the distinct interactions with the medium experienced by the neutral and charged species in the concentrated salt solutions.

14 SOLAR ENERGY↗

Exact self-similar solutions for the magnetized Noh Z pinch problem

A self-similar solution is derived for a radially imploding cylindrical plasma with an embedded, azimuthal magnetic field. The plasma stagnates through a strong, outward propagating shock wave of constant velocity. This analysis is an extension of the classic Noh gasdynamics problem to its ideal magnetohydrodynamics (MHD) counterpart. The present exact solution is especially suitable as a test for MHD codes designed to simulate linear Z pinches. To demonstrate the application of the new solution to code verification, simulation results from the cylindrical R-Z version of Mach2 and the 3D Cartesian code Athena are compared against the analytic solution. Alternative routines from the default ones in Athena lead to significant improvement of the results, thereby demonstrating the utility of the self-similar solution for verification.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Low-frequency four-wave mixing spectroscopy of biomolecules in aqueous solutions

Four-wave mixing (FWM) spectroscopy is used to detect the rotational resonances of H{sub 2}O and H{sub 2}O{sub 2} molecules in DNA and denatured DNA aqueous solutions in the range {+-}10 cm{sup -1} with a spectral resolution of 3 GHz. It is found that the resonance contribution of the rotational transitions of these molecules increases significantly in solutions rather than in distilled water. This fact is interpreted as a manifestation of specific properties of a hydration layer at DNA-water and denatured DNA-water interfaces. Analysis of the FWM spectra shows that the concentration of H{sub 2}O{sub 2} molecules in the hydration layer of the DNA solution increases by a factor of 3 after denaturation. The FWM spectra of aqueous solutions of {alpha}-chymotrypsin protein are obtained in the range {+-}7cm{sup -1} at the protein concentrations between 0 and 20 mg cm{sup -3}. It is found that the hypersound velocity in the protein aqueous solution, measured by the shift of Brillouin components in the scattering spectrum, obeys a cubic dependence on the protein concentration and reaches a value of about 3000 m s{sup -1} at 20 mg cm{sup -3}. (application of lasers and laser-optical methods in life sciences)

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Solution for (1+1)-dimensional surface solitons in thermal nonlinear media

Analytical solutions for (1+1)-dimensional surface fundamental solitons in thermal nonlinear media are obtained. The stationary position and the critical power of surface solitons are obtained using these analytical solutions. The analytical solutions are verified by numerical simulations. The solutions for surface breathers and their breathing period, along with solutions for surface dipole and tripole solitons, are also given.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Late time solutions for inhomogeneous {Lambda}CDM cosmology, their characterization, and observation

Assuming homogeneous isotropic {Lambda}CDM cosmology allows {Lambda}, spatial curvature, and dark matter density to be inferred from large scale structure observations such as supernovae. The purpose of this paper is to extend this to allow observations to measure or constrain inhomogeneity and anisotropy. We obtain the general inhomogeneous anisotropic {Lambda}CDM solution which is locally asymptotic to an expanding de Sitter solution as a late time expansion using Starobinsky's method (analogous to the 'holographic renormalization' technique in AdS/CFT) together with a resummation of the expansion. The dark matter is modeled as perfect dust fluid. The terms in the expansion systematically describe inhomogeneous and anisotropic deformations of an expanding Friedmann-Lemaitre-Robertson-Walker solution and are given as a spatial derivative expansion in terms of data characterizing the solution--a 3-metric and a perturbation of that 3-metric. Leading terms describe inhomogeneity and anisotropy on the scale set by the cosmological constant, approximately the horizon scale today. Higher terms in the expansion describe shorter scale variations. We compute the luminosity-distance-redshift relation and argue that comparison with current and future observations would allow a partial reconstruction of the characterizing data. We also comment on smoothing these solutions noting that geometric flows (such as Ricci flow) applied to the characterizing data provide a canonical smoothing method.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

AdS black hole solutions in dilatonic Einstein-Gauss-Bonnet gravity

We find that anti-de Sitter (AdS) spacetime with a nontrivial linear dilaton field is an exact solution in the effective action of the string theory, which is described by gravity with the Gauss-Bonnet curvature terms coupled to a dilaton field in the string frame without a cosmological constant. The AdS radius is determined by the spacetime dimensions and the coupling constants of curvature corrections. We also construct the asymptotically AdS black hole solutions with a linear dilaton field numerically. We find these AdS black holes for hyperbolic topology and in dimensions higher than four. We discuss the thermodynamical properties of those solutions. Extending the model to the case with the even-order higher Lovelock curvature terms, we also find the exact AdS spacetime with a nontrivial dilaton. We further find a cosmological solution with a bounce of three-dimensional space and a solitonic solution with a nontrivial dilaton field, which is regular everywhere and approaches an asymptotically AdS spacetime.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Americium/Lanthanide Separations in Alkaline Solutions for Advanced Nuclear Fuel Cycles

Project goals: Can used nuclear fuel be partitioned by dissolution in alkaline aqueous solution to give a solution of uranium, neptunium, plutonium, americium and curium and a filterable solid containing nearly all of the lanthanide fission products and certain other fission products? What is the chemistry of Am/Cm/Ln in oxidative carbonate solutions? Can higher oxidation states of Am be stabilized and exploited? Conclusions: Am(VI) is kinetically stable in 0.5-2.0 M carbonate solutions for hours. Aliquat 336 in toluene has been successfully shown to extract U(VI) and Pu(VI) from carbonate solutions. (Stepanov et al 2011). Higher carbonate concentration gives lower D, SF{sub U/Eu} for = 4 in 1 M K{sub 2}CO{sub 3}. Experiments with Am(VI) were unsuccessful due to reduction by the organics. Multiple sources of reducing organics...more optimization. Reduction experiments of Am(VI) in dodecane/octanol/Aliquat 336 show that after 5 minutes of contact, only 30-40% of the Am(VI) has been reduced. Long enough to perform an extraction. Shorter contact times, lower T, and lower Aliquat 336 concentration still did not result in any significant extraction of Am. Anion exchange experiments using a strong base anion exchanger show uptake of U(VI) with minimal uptake of Nd(III). Experiments with Am(VI) indicate Am sorption with a Kd of 9 (10 minute contact) but sorption mechanism is not yet understood. SF{sub U/Nd} for = 7 and SF{sub U/Eu} for = 19 after 24 hours in 1 M K{sub 2}CO{sub 3}.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Expansion dynamics of a two-component quasi-one-dimensional Bose–Einstein condensate: Phase diagram, self-similar solutions, and dispersive shock waves

We investigate the expansion dynamics of a Bose–Einstein condensate that consists of two components and is initially confined in a quasi-one-dimensional trap. We classify the possible initial states of the two-component condensate by taking into account the nonuniformity of the distributions of its components and construct the corresponding phase diagram in the plane of nonlinear interaction constants. The differential equations that describe the condensate evolution are derived by assuming that the condensate density and velocity depend on the spatial coordinate quadratically and linearly, respectively, which reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas–Fermi approximation. We have obtained self-similar solutions of these differential equations for several important special cases and write out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms may be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves is considered. We compare the numerical solutions of the Gross–Pitaevskii equations with their approximate analytical solutions and numerically study the situations where the analytical method being used admits no exact solutions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗