Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1997"
TL;DR: In this paper, a unified transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced, based on the fact that linear and integrably nonlinear equations have the distinguished property that they possess a Lax pair formulation.
Abstract: A new transform method for solving initial boundary value problems for linear and for integrable nonlinear PDEs in two independent variables is introduced. This unified method is based on the fact that linear and integrable nonlinear equations have the distinguished property that they possess a Lax pair formulation. The implementation of this method involves performing a simultaneous spectral analysis of both parts of the Lax pair and solving a Riemann–Hilbert problem. In addition to a unification in the method of solution, there also exists a unification in the representation of the solution. The sine–Gordon equation in light–cone coordinates, the nonlinear Schrodinger equation and their linearized versions are used as illustrative examples. It is also shown that appropriate deformations of the Lax pairs of linear equations can be used to construct Lax pairs for integrable nonlinear equations. As an example, a new Lax pair of the nonlinear Schrodinger equation is derived.
571 citations
TL;DR: In this paper, a method is presented for selecting the type of actuator best suited to a given task, in the early stages of engineering design, based on matching performance characteristics of the actuator, such as force and displacement, to the requirements of the given task.
Abstract: A method is presented for selecting the type of actuator best suited to a given task, in the early stages of engineering design. The selection is based on matching performance characteristics of the actuator, such as force and displacement, to the requirements of the given task. The performance characteristics are estimated from manufacturers9 data and from simple models of performance limitation such as heat generation and resonance. Characteristics are presented in a graphical form which allows for a direct and systematic comparison of widely different systems of actuation. The actuators considered include man-made actuators (such as hydraulic, solenoid and shape memory alloy) and naturally occurring actuators (such as the muscles of animals and plants).
514 citations
TL;DR: In this article, the Hertz equations for the contact of elastic spheres are modifed by surface energy, and the force needed to separate the spheres is equal to (3/2)πΔ R γ, which is independent of the elastic modulus and so appears to be universally applicable and therefore to conflict with Bradley9s answer.
Abstract: Bradley (1932) showed that if two rigid spheres of radii R 1 and R 2 are placed in contact, they will adhere with a force 2πΔ R γ, where R is the equivalent radius R 1 R 1 /( R 1 + R 2 ) and Δγ is the surface energy or ‘work of adhesion’ (equal to γ1+γ2-γ12). Subsequently Johnson et al. (1971) (JKR theory) showed by a Griffith energy argument (assuming that contact over a circle of radius a introduces a surface energy -π a 2 Δγ) how the Hertz equations for the contact of elastic spheres are modifed by surface energy, and showed that the force needed to separate the spheres is equal to (3/2)πΔ R γ, which is independent of the elastic modulus and so appears to be universally applicable and therefore to conflict with Bradley9s answer. The discrepancy was explained by Tabor (1977), who identified a parameter 3 Δγ 2 / 3 / E * 2 / 3 \e governing the transition from the Bradley pull-off force 2π R Δ|γ to the JKR value (3/2)π R Δγ. Subsequently Muller et al. (1980) performed a complete numerical solution in terms of surface forces rather than surface energy, (combining the Lennard–Jones law of force between surfaces with the elastic equations for a half-space), and confirmed that Tabor9s parameter does indeed govern the transition. The numerical solution is repeated more accurately and in greater detail, confirming the results, but showing also that the load–;approach curves become S-shaped for values of μ greater than one, leading to jumps into and out of contact. The JKR equations describe the behaviour well for values of μ of 3 or more, but for low values of μ the simple Bradley equation better describes the behaviour under negative loads.
416 citations
TL;DR: In this paper, a thermodynamic framework is presented for the plasticity modelling of geotechnical materials, which is capable of modelling rigorously both friction and non-associated flow, and the strong connection between these phenomena is demonstrated.
Abstract: A thermodynamic framework is presented for the plasticity modelling of geotechnical materials. The framework is capable of modelling rigorously both friction and non-associated flow, and the strong connection between these phenomena is demonstrated. The formulation concentrates on the development of constitutive models from hypotheses about the form of an energy potential and a dissipation function. The reformulation of previous work, in which the Helmholtz free energy was used, to a new approach starting from the Gibbs free energy is found to be valuable. The relationship between the new functions and classical plasticity concepts (yield surface, plastic potential, isotropic and kinematic hardening, friction, dilation) is demonstrated. Comments are made on elastic-plastic coupling. Implications of the new approach for critical state soil models are discussed.
410 citations
TL;DR: In this article, the authors extended the fracture mechanics model to include both static and sliding friction, in which the rate of release of elastic strain energy is equated to the work done against surface forces, both frictional and adhesive.
Abstract: Attempts to establish the relationship between adhesion and friction at the contact of solid surfaces has been frustrated by their inevitable roughness. The recent development of nanotribology , in which a single asperity contact can be modelled in the surface force apparatus (SFA) or the atomic force microscope (AFM), has made possible the simultaneous measurement of friction and adhesion in a sliding experiment. For the case of pure adhesion, continuum mechanics models exist which assist in the interpretation of the measurements. In this paper these models are extended to include both static and sliding friction. The approach is through the concept of fracture mechanics, in which the rate of release of elastic strain energy is equated to the work done against surface forces, both frictional and adhesive. The model appears to be consistent with currently available experimental data.
323 citations
TL;DR: In this article, a nonlinear theory of elastic boundary coating (or reinforcement) of an elastic solid is developed for plane strain deformations, which is applied to the analysis of the equilibrium of a finitely deformed halfplane consisting of compressible elastic material coated along its edge.
Abstract: In this paper, a nonlinear theory of elastic boundary coating (or reinforcement) of an elastic solid is developed for plane strain deformations. The coating consists of a material curve endowed with intrinsic elastic properties associated with extensibility and bending stiffness bonded to part, or all, of the bounding curve of the elastic body. The equations describing the equilibrium of the coated body when subject to finite deformation are derived using a variational method. The incremental equations describing a small departure from an equilibrium configuration are then derived and used to investigate the stability of a deformed configuration and the possibility of bifurcation.
The theory is applied to the analysis of the equilibrium of a finitely deformed half-plane consisting of compressible elastic material coated along its edge. The influence of the coating on the bifurcation behaviour of the half–plane is assessed against known results for an uncoated half–plane. Numerical results are used to illustrate the influence of certain material parameters on bifurcation.
290 citations
TL;DR: In this article, the authors prove the existence and uniqueness of a renormalised solution of the nonlinear problem, where the data f and u0 belong to L1(Ω × (0, T)) and L1 (Ω), and where the function a:(0,T) × Ω × ℝN → ℘N is monotone (but not necessarily strictly monotonous) and defines a bounded coercive continuous operator from the space into its dual space.
Abstract: In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problemwhere the data f and u0 belong to L1(Ω × (0, T)) and L1 (Ω), and where the function a:(0, T) × Ω × ℝN → ℝN is monotone (but not necessarily strictly monotone) and defines a bounded coercive continuous operator from the space into its dual space. The renormalised solution is an element of C0 ([ 0, T] L1 (Ω)) such that its truncates TK(u) belong to withthis solution satisfies the equation formally obtained by using in the equation the test function S(u)φ, where φ belongs to and where S belongs to C∞(ℝ) with
249 citations
TL;DR: In this paper, the existence of multiple positive solutions of quasilinear problems of second order was proved using the fibrering method and the main part of the differential operator is p-Laplacian.
Abstract: Using the fibrering method, we prove the existence of multiple positive solutions of quasilinear problems of second order. The main part of our differential operator is p-Laplacian and we consider solutions both in the bounded domain Ω⊂ℝN and in the whole of ℝN. We also prove nonexistence results.
241 citations
TL;DR: In this article, a tensor tensor theory for small elastic deformations of either a compressible or incompressible isotropic elastic body, superposed on a known finite deformation, without assuming special forms for the strain energy function is developed.
Abstract: Using tensor notations a general theory is developed for small elastic deformations, of either a compressible or incompressible isotropic elastic body, superposed on a known finite deformation, without assuming special forms for the strain-energy function. The theory is specialized to the case when the finite deformation is pure homogeneous. When two of the principal extension ratios are equal the changes in displacement and stress due to the small superposed deformation are expressed in terms of two potential functions in a manner which is analogous to that used in the infinitesimal deformation of hexagonally aeolotropic materials. The potential functions are used to solve the problem of the infinitesimally small indentation, by a spherical punch, of the plane surface of a semi-infinite body of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation symmetrical about the normal to the force-free plane surface.
209 citations
TL;DR: In this paper, the results of a recent investigation carried out by the authors to verify the experimental results of Semenov in 1991 and Kulik and co-workers in 1991, who successfully demonstrated the ability of compliant surfaces to reduce the skin-friction drag and surface-flow noise in a turbulent boundary layer.
Abstract: Over the past forty years intensive investigations into the use of compliant surfaces have been undertaken, both theoretically and experimentally, in order to obtain turbulent drag reduction in boundary–layer flows. Although positive results were found in some of the studies, none of these had been successfully validated by independent researchers. In this paper the results are reported of a recent investigation carried out by the authors to verify the experimental results of Semenov in 1991 and Kulik and co–workers in 1991, who successfully demonstrated the ability of compliant surfaces to reduce the skin–friction drag and surface–flow noise in a turbulent boundary layer. A strain–gauge force balance was used in the present study to directly measure the turbulent skin–friction drag of a slender body of revolution in a water tunnel. Changes in the structure of turbulent boundary layer over a compliant surface in comparison with that over a rigid surface were also examined. The results clearly demonstrate that the turbulent skin friction is reduced for one of the two compliant coatings tested, indicating a drag reduction of up to 7 per cent within the entire speed range of the tests. The intensities of skin–friction and wall–pressure fluctuations measured immediately downstream from the compliant coating show reductions in the intensities of up to 7 and 19 per cent, respectively. The results also indicate reductions in turbulence intensity by up to 5 per cent across almost the entire boundary layer. Furthermore, an upwards shift of the logarithmic velocity profile is also evident indicating that the thickness of the viscous sublayer is increased as a result of turbulent drag reduction due to the compliant coating. It is considered that the results of the present experimental investigation convincingly demonstrate for the first time since the earlier work in Russia (by Semenov and Kulik) that a compliant surface can indeed produce turbulent drag reduction in boundary–layer flows.
197 citations
TL;DR: In this article, the maximum norm of the gradient of the passive scalar of the Boussinesq equation was shown to control the breakdown of smooth solutions of the problem.
Abstract: In this paper, we prove local existence and uniqueness of smooth solutions of the Boussinesq equations. We also obtain a blow-up criterion for these smooth solutions. This shows that the maximum norm of the gradient of the passive scalar controls the breakdown of smooth solutions of the Boussinesq equations. As an application of this criterion, we prove global existence of smooth solutions in the case of zero external force.
TL;DR: In this paper, the limiting form of the free energy governing a ferromagnetic film of vanishing thickness was determined by a scaling calculation, which generalizes Stoner and Wohlfarth's results for flat elli...
Abstract: We determine by a scaling calculation the limiting form of the free energy governing a ferromagnetic film of vanishing thickness. Our theory generalizes Stoner and Wohlfarth's results for flat elli...
TL;DR: In this article, the most general possible relations between the stress and strain-velocity components, which can be obeyed by an incompressible, visco-inelastic fluid, are derived.
Abstract: The classical theory of the hydrodynamics of viscous fluids depends on the assumption of a particular law governing the relations between the components of stress in a fluid and those of the strain-velocity. This assumption limits its applicability to Newtonian fluids. Here, the most general possible relations between the stress and strain-velocity components, which can be obeyed by an incompressible, visco-inelastic fluid, are derived. These relations also apply to an incompressible, visco-elastic fluid in a steady state of laminar flow. It is shown how equations of motion and boundary conditions can be obtained if these relations are known. Two problems involving laminar flow are then discussed in some detail. These are: (i) the torsional motion of a cylindrical mass of fluid, produced by means of forces applied to its plane ends, and (ii) the laminar flow of a mass of fluid contained between two coaxial cylinders rotating with different angular velocities.
TL;DR: In this paper, the authors characterised the possible limit points of the ratio (λe −λ)/e as e→0 for a planar convex, classical polygon with sides of rational or infinite slopes and showed that there is often a continuum of such limit points.
Abstract: Let λe be a Dirichlet eigenvalue of the ‘periodically, rapidly oscillating’ elliptic operator –∇·(a(x/e)∇) and let ∇ be a corresponding (simple) eigenvalue of the homogenised operator –∇·(A∇). We characterise the possible limit points of the ratio (λe–λ)/e as e→0. Our characterisation is quite explicit when the underlying domain is a (planar) convex, classical polygon with sides of rational or infinite slopes. In particular, in this case it implies that there is often a continuum of such limit points.
TL;DR: In this paper, the effect of adding a surfactant (sodium dodecyl sulphate) to droplets boiling on a hot stainless steel surface was studied and the results compared with those for droplets of pure water.
Abstract: The effect of adding a surfactant (sodium dodecyl sulphate) to droplets boiling on a hot stainless steel surface was studied. Experiments were done using solutions containing 100 ppm and 1000 ppm by weight of surfactant and the results compared with those for droplets of pure water. Surface temperature was varied from 60 to 340 degrees C. Droplet impact and evaporation was photographed using both video and 35 mm cameras. Addition of the surfactant significantly reduced lifetimes of droplets in a state of evaporation or nucleate boiling. For surface temperatures below those required to initiate nucleate boiling, the principal effect of the surfactant was to reduce liquid–solid contact angle, increasing the surface area wetted by the drop. At higher surface temperatures, the surfactant promoted vapour bubble nucleation and foaming in the liquid, greatly enhancing heat transfer. Increasing surfactant concentration reduced the Leidenfrost temperature, above which droplets were levitated above the surface on a thin film of their own vapour. The surfactant had no effect on evaporation time of droplets in film boiling.
TL;DR: In this paper, a discrete analogue of the Moutard transformation is constructed by means of discrete analogues of the kink solutions of the continuous system, and it is shown that, in a particular form, this system is an integrable discretization of a (2+1)-dimensional sine-Gordon system.
Abstract: Superposition principles, both linear and nonlinear, associated with the Moutard transformation are found. On suitable reinterpretation, these constitute an integrable discrete nonlinear system and its associated linear system. Further, it is shown that, in a particular form, this system is an integrable discretization of a (2+1)–dimensional sine–Gordon system. Solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the Moutard transformation. Included in these solutions are discrete analogues of the kink solutions of the continuous system.
TL;DR: In this article, the problem of determining the effective thermal conductivity of a composite material consisting of periodic arrays of spheres with interfacial resistance was considered, and the authors applied Rayle...
Abstract: We consider the problem of analytically determining the effective thermal conductivity of a composite material consisting of periodic arrays of spheres with interfacial resistance. We applied Rayle...
TL;DR: In this article, the effects of spanwise flexibility on the propulsive efficiency of oscillating foils with span-wise flexibility were studied using a time-domain panel method, and it was shown that passive spanwise flexibility reduces the propulsion efficiency of these planforms, but that propulsive performance can be increased, over the value for an equivalent rigid foil, by careful control of the phase of the spanwise flexible relative to other motion parameters.
Abstract: The propulsive performance of oscillating foils with spanwise flexibility was studied using a time-domain panel method. The work was done to assess the effects of spanwise flexibility on the propulsive efficiency of these propulsors, especially those employed by relatively fast swimming marine animals. The method is valid for three-dimensional attached flows around the actual planforms found on these animals and was used in the study reported here to assess the performance of the flukes of an immature fin whale ( Balaenoptera physalus ). It is shown that passive spanwise flexibility reduces propulsive efficiency, but that propulsive efficiency of these planforms can be increased, over the value for an equivalent rigid foil, by careful control of the phase of the spanwise flexibility relative to other motion parameters.
TL;DR: In this article, the traditional combustion problems of calculating flame speeds for a premixed gaseous fuel and for a pre-mixed solid fuel are revisited using a simpler (than previously) non-dimensional temperature.
Abstract: The traditional combustion problems of calculating flame speeds for a premixed gaseous fuel and for a premixed solid fuel are revisited using a simpler (than previously) non–dimensional temperature. It turns out to be possible to carry out asymptotic calculations for flame speed and the agreement with corresponding numerical calculations is remarkably good. In each case the uniqueness of the speed is considered using phase plane methods, with a little effort to determine the nature of the ‘cold’ critical point. Consideration of the stability of the travelling combustion wave fronts suggests a period doubling route to chaos for the premixed solid fuel (as the exothermicity is decreased) and corresponds with previous work using different non–dimensional temperature and parameters.
TL;DR: In this paper, the transition from stable periodic non-impacting motion to impacting motion is analyzed for a mechanical oscillator and it is shown that a grazing impact leads to an almost one-dimensional stretching in state space.
Abstract: The transition from stable periodic non-impacting motion to impacting motion is analysed for a mechanical oscillator. By using local methods, it is shown that a grazing impact leads to an almost one-dimensional stretching in state space. A condition can then be formulated, such that a grazing trajectory will be stable if the condition is fulfilled. If this is the case, the bifurcation will be continuous and the motion after the bifurcation can be understood by a one-dimensional mapping. This mapping is known to exhibit chaotic solutions as well as arbitrary long stable cycles, depending on parameters.
TL;DR: In this article, a general method for the rigorous solution of a problem associated with a circular inclusion embedded within an infinite matrix in antiplane shear is developed, where the bonding at the inclusion-matrix interface is assumed to be imperfect.
Abstract: A general method is developed for the rigorous solution of a problem associated with a circular inclusion embedded within an infinite matrix in antiplane shear. The bonding at the inclusion–matrix interface is assumed to be imperfect. Most significant is the fact that the imperfection in the interface is assumed to be circumferentially inhomogeneous. Using analytic continuation, the basic boundary–value problem for two analytic functions is reduced to a first‐order differential equation for a single analytic function and the closed‐form solution is obtained. The method is illustrated using several specific examples. The results from these examples are compared to the corresponding results when the imperfection in the interface is homogeneous. These comparisons illustrate how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the stresses induced within the inclusion.
TL;DR: In this paper, the quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins al...
Abstract: The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins al...
TL;DR: In this paper, the authors used a rate-sensitive polycrystal plasticity model together with the Marciniak-Kuczynski approach for the computation of forming limit diagrams (FLDs).
Abstract: This paper is concerned with the computation of forming limit diagrams (FLDs) using a rate-sensitive polycrystal plasticity model together with the Marciniak–Kuczynski approach. Sheet necking is initiated from an initial imperfection in terms of a narrow band. The deformations inside and outside the band are assumed to be homogeneous and conditions of compatibility and equilibrium are enforced across the band interfaces. Thus, the polycrystal model needs only to be applied to two polycrystalline aggregates, one inside and one outside the band. Each grain is modelled as an FCC crystal with 12 distinct slip systems. The response of an aggregate comprised of many grains is based on an elastic–viscoplastic Taylor‐type polycrystal model developed by Asaro and Needleman (1985). The effects of initial imperfection intensity and orientation, initial distribution of grain orientations, crystal elasticity, strain rate sensitivity, single slip hardening and latent hardening on the FLD are discussed in detail. The predicted FLD is compared with experimental data for an aluminium alloy sheet.
TL;DR: In this paper, the problem of factorization of large numbers on a quantum computer which is realized within a linear ion trap was investigated, and upper bounds on the size of the numbers that can be factorized on such a computer were derived.
Abstract: We investigate the problem of factorization of large numbers on a quantum computer which we imagine to be realized within a linear ion trap. We derive upper bounds on the size of the numbers that can be factorized on such a quantum computer. These upper bounds are independent of the power of the applied laser. We investigate two possible ways to implement qubits, in metastable optical transitions and in Zeeman sublevels of a stable ground state, and show that in both cases the numbers that can be factorized are not large enough to be of practical interest. We also investigate the effect of quantum error correction on our estimates and show that in realistic systems the impact of quantum error correction is much smaller than expected. Again no number of practical interest can be factorized.
TL;DR: In this article, a multisymplectic structure where a distinct differential two-form is assigned to each space direction and time was proposed to characterize Hamiltonian PDEs on unbounded domains.
Abstract: Action, symplecticity, signature and complex instability are fundamental concepts in Hamiltonian dynamics which can be characterized in terms of the symplectic structure. In this paper, Hamiltonian PDEs on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two-form is assigned to each space direction and time. This leads to a new geometric formulation of the conservation of wave action for linear and nonlinear Hamiltonian PDEs, and, via Stokes's theorem, a conservation law for symplecticity. Each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode. The first invariant in this family is classical Krein signature (or energy sign, when the energy is time independent) and the other (spatial) signatures are energy flux signs, leading to a classification of instabilities that includes information about directional spatial spreading of an instability. The theory is applied to several examples: the Boussinesq equation, the water-wave equations linearized about an arbitrary Stokes's wave, rotating shallow water flow and flow past a compliant surface. Some implications for non-conservative systems are also discussed.
TL;DR: In this paper, a transformation of the free boundary problem together with an asymptotic analysis (performed about the solution when the transaction cost is zero) leads to solutions which are shown to be good approximations for cases which can be solved by numerical methods.
Abstract: It is known that the optimal trading strategy for a certain portfolio problem featuring fixed transaction costs is obtained from the solution of a free boundary problem. The latter can only be solved with numerical methods, and computations become formidable when the number of available securities is larger than three or four. This paper shows how a transformation of the free boundary problem together with an asymptotic analysis (performed about the solution when the transaction cost is zero) leads to solutions which are shown to be good approximations for cases which can be solved by numerical methods. These approximately optimal trading strategies are easy to compute, even when there are many risky securities, as is illustrated for the case of the 30 Dow Jones Industrials.
TL;DR: In this article, a theoretical model for predicting the partition ratio at the grain level is presented, which is correlated with measured results to validate the model and establish the effective thermal conductivity of cubic boron nitride (CBN) and aluminium oxide abrasives.
Abstract: In grinding, the heat generated is removed from the area of contact by conduction into the workpiece, conduction into the grinding wheel grains, convection with the material removed and convection to the fluid. The partition ratios in surface grinding were estimated from the measured temperatures using a thermocouple technique. A theoretical model for predicting the partition ratio at the grain level is presented. The theoretical model was correlated with measured results to validate the model and establish the effective thermal conductivity of cubic boron nitride (CBN) and aluminium oxide abrasives. The effective thermal conductivity of CBN was found to be considerably lower than the reported theoretical value albeit much higher than the thermal conductivity of aluminium oxide. The findings provide the basis for improved prediction of workpiece temperatures in grinding.
TL;DR: The coordination sequence S(n) of a lattice or net gives the number of nodes that are n bonds away from a given node as discussed by the authors, where S(1) is the familiar coordination number.
Abstract: The coordination sequence S(n) of a lattice or net gives the number of nodes that are n bonds away from a given node. S(1) is the familiar coordination number. Extending the work of O'Keeffe and ot...
TL;DR: In this paper, the authors generalize these results to multidimensional Laplace-type integrands with quadratic critical points, integrated over infinite complex domains, and show that dimensionality only trivially affects the form of the exact multiddimensional remainder.
Abstract: The method of steepest descents for single dimensional Laplace-type integrals involving an asymptotic parameter k was extended by Berry & Howls in 1991 to provide exact remainder terms for truncated asymptotic expansions in terms of contributions from certain non-local saddlepoints. This led to an improved asymptotic expansion (hyperasymptotics) which gave exponentially accurate numerical and analytic results, based on the topography of the saddle distribution in the single complex plane of the integrand. In this paper we generalize these results to similar well-behaved multidimensional integrands with quadratic critical points, integrated over infinite complex domains. As previously pointed out the extra complex dimensions give rise to interesting problems and phenomena. First, the conventionally defined surfaces of steepest descent are no longer unique. Second, the Stokes's phenomenon (whereby contributions from subdominant saddles enter the asymptotic representation) is of codimension one. Third, we can collapse the representation of the integral onto a single complex plane with branch cuts at the images of critical points. The new results here demonstrate that dimensionality only trivially affects the form of the exact multidimensional remainder. Thus the growth of the late terms in the expansion can be identified, and a hyperasymptotic scheme implemented. We show by a purely algebraic method how to determine which critical points contribute to the remainder and hence resolve the global connection problem, Riemann sheet structure and homology associated with the multidimensional topography of the integrand.
TL;DR: In this paper, the authors considered the model equations for gravity waves in horizontally stratified fluids and gave rigorous mathematical treatment to the well-posedness of the initial value problem, the question of existence of solitary wave solutions, and theoretical results about the stability of these solitary waves.
Abstract: Model equations for gravity waves in horizontally stratified fluids are considered. The theories to be addressed focus on stratifications featuring either a single pycnocline or neighbouring pairs of pycnoclines. Particular models investigated include the general version of the intermediate long-wave equation derived by Kubota, Ko and Dobbs to simulate waves in a model system consisting of two homogeneous layers separated by a narrow region of variable density, and the related system of equations derived by Liu, Ko and Pereira for the transfer of energy between waves running along neighbouring pycnoclines. Issues given rigorous mathematical treatment herein include the well-posedness of the initial value problem for these models, the question of existence of solitary wave solutions, and theoretical results about the stability of these solitary waves.