bluuebunny/crossref_metadata_2025 huggingface dataset:
Resonance in Periodically Perturbed Hopf Bifurcation
Authors: Kath William L.
Published: September 2015

We examine in detail the effect of a periodic perturbation upon a system undergoing a Hopf bifurcation. Particular attention is paid to the case of resonance between the natural Hopf bifurcation frequency and the perturbation frequency. It is shown that for a certain parameter range this Hopf bifurcation is modified by the perturbation to exhibit frequency (phase) pulling and locking, and the nature of this frequency locking is described. In addition, the effect of adding diffusion to the system is discussed, and it is shown that diffusive waves are possible.
Spinup from Rest of Immiscible Fluids
Authors: Baker G. R. , Israeli M.
Published: September 2015

We investigate the spinup from rest of two immiscible fluids of different densities and kinematic viscosities in a vertically mounted cylinder. Our attention is restricted to small internal rotational Froude numbers, in which case the interface remains essentially horizontal. By requiring small enough Ekman numbers, Wedemeyer's (1964) approximation may be used to obtain partial differential equations describing the inward convection and diffusion of the azimuthal velocity in both layers. Solutions are found illustrating the effects of varying the density, viscosity, and height of each layer. Qualitative agreement with experiment is reported.
Some Results on Systems of Finite Sets That Satisfy a Certain Intersection Condition
Authors: Ein Lawrence M. H. , Richman David Ross , Kleitman Daniel J. , Shearer James , Sturtevant Dean
Published: September 2015

Let S be a finite set, and fix K>2. Let F be a family of subsets of S with the property that whenever A1,...,Ak are sets in F, not necessarily distinct, and A1 ⋂ ⋯ ⋂ Ak = ∅, then A1 ⋃ ⋯ ⋃ Ak = S. We prove here that the maximum size of such a family is 2|S|−1 + 1. If we require that the sets A1,...,Ak be distinct, then the maximum size of F is again 2|S|−1 + 1, provided that |S| ≥ log2(K−2)+3.
On Some Optimal Constructions of Chess Tournament Combinations
Authors: Iwahori Nobuko
Published: September 2015

An optimal solution for the following “chess tournament” problem is given. Let n, r be positive integers such that r<n. Put N=2n, R=2r+1. Let XN,R be the set of all ordered pairs (T, A) of matrices of degree N such that T=(tij) is symmetric, A=(aij) is skew‐symmetric, tij ∈,{0, 1, 2,…, R), aij ∈{0,1,–1}. Moreover, suppose tii=aii=0 (1⩽i⩽N). tij = tik>0 implies j=k, tij=0 is equivalent to aij=0, and |ai1|+|ai2|+…+|aiN|=R (1⩽i⩽N). Let p(T, A) be the number of i such that 1⩽i⩽N and ai1 + ai2 + … + aiN >0. The main result of this note is to show that max p(T, A) for (T, A)∈XN, R is equal to [n(2r+1)/(r+1)], and a pair (T0, A0) satisfying p(T0, A0)=[n(2r+1)/(r+1)] is also given.
Poset Measure and Saturated Partitions
Authors: Griggs Jerrold R.
Published: September 2015

The theory of saturated partitions is applied to solve the problem of maximizing the measure of a finite poset when each point has measure in [0, 1] and each chain has measure at most λ>0.
The Existence of One Dimensional Steady Detonation Waves in a Simple Model Problem
Authors: Holmes Philip , Stewart D. S.
Published: September 2015

We study a system of three nonlinear ordinary differential equations arising from one dimensional steady combustion problems. The four singular points in the phase space correspond to equilibrium solutions before and after burning has taken place, and the desired detonation wave is a trajectory connecting two of these points (the cold and hot boundary points). In this paper we show that the problem of finding steady detonations is equivalent to proving the existence and uniqueness of a transversal heteroclinic orbit. The presence of a large parameter, the activation energy, permits us to show analytically that such an orbit exists for an open set of parameter values. In doing so, we are able to give a geometric interpretation of the cold boundary difficulty. We conclude with some numerical results for lower activation energies.
Homotopy Spheres in Formal Language
Authors: Farmer Frank D.
Published: September 2015

The set of nonempty proper subwords of a word is either contractible or a homotopy n‐sphere. There is a simple algorithm which computes n. The existence of spherical words is investigated, and the words which yield spheres are determined. A language which is closed under subwords has a finite number of components, and each component has a finitely generated fundamental group. For each n greater than 1, there is a language on two letters which has the homotopy type of an infinite cluster of n‐sphere. There is a language on two letters which has nontrivial homology in each dimension greater than 1. If a language is closed under subwords and has bounded period, then it has the homotopy type of a finite polyhedron.
On the Crenation of Coated Droplets
Authors: Landman K. A. , Greenspan H. P.
Published: September 2015

The dynamic response of a small, very viscous, coated droplet is examined as additional fluid is deposited into a compressible surface layer. At early times, the surface material behaves like an incompressible fluid and a number of crenations form. These protuberances decrease in size and in number as the compressibility of the coating takes effect, until finally, the droplet again becomes spherical. This sequence of events and its dependence on the compressibility and rheological properties of the fluids are studied. Similar ruffling effects are observed in cell biology, and some of the possible implications of the fluid dynamical model are discussed.
On Crapo's Beta Invariant for Matroids
Authors: Oxley James G.
Published: September 2015

In this paper we characterize the matroids for which Crapo's beta invariant does not exceed 4. In addition, we relate the beta invariant of a matroid to its chromatic number and to its connectivity.
Marginal Separation
Authors: Stewartson Keith , Smith F. T. , Kaups Kalle
Published: September 2015

An analytical and numerical study is presented on the response of high Reynolds number flow when, starting from a fully attached state, it is forced to gradually approach separation as a certain critical parameter is increased. The end of attached flow near a rounded leading edge is the particular problem considered, although the application of the theory is much wider. The critical parameter is the angle of incidence. The Goldstein singularity first appears only weakly, and then the attached‐flow concept can even admit a small local separation bubble. As the singularity increases in strength, however, a solution of this type suddenly ceases to exist and a dramatic transition to another, separated form of motion seems to be implied. Nonuniqueness of the attached‐flow solution is also found.
Singularly Perturbed Semilinear Robin Problems
Authors: Howes F A.
Published: September 2015

Several phenomena of current interest in catalytic reaction theory can be phrased in terms of solutions of singularly perturbed, second‐order, semilinear differential equations satisfying boundary conditions of Robin type. This paper offers a mathematical theory for such problems in both the scalar and the vector case, and treats some examples involving the pseudo‐steady‐state hypothesis and a parallel reaction inside of a porous catalyst pellet.
Limit Behavior and the Existence of Combustion Shock Layers
Authors: Gaalswyk Arie
Published: September 2015

This paper contains a comprehensive study of the global and limit behavior of integral curves in the three‐dimensional phase space for the system of nonlinear ordinary differential equations governing the structure of combustion shock layers. A significant general result is a list of admissible routes for the integral curves representing the four combustion types. Perhaps most noteworthy is a theorem on the nonexistence of strong deflagrations. A special aspect of the study points to the significance of the stoichiometric parameters for a reaction, and the need to distinguish between first‐order and higher‐order reactions. The detailed information contained in this paper concerning the various possible modes of combustion for the different reactions should have potential application in the search for efficient energy conversion by combustion processes.
On the Onset of Rayleigh‐Bénard Convection in a Fluid Layer of Slowly Increasing Depth
Authors: Walton I. C.
Published: September 2015

A two‐scaling approach is used to investigate the onset of convection in a fluid layer whose depth is a slowly increasing function of horizontal distance. It is shown that whatever the value of the imposed temperature difference between the boundaries (provided, of course, that the lower one is hotter) there are regions which are stable and regions which are unstable to small perturbations. As the depth increases the amplitude of steady solutions increases from exponentially small values to take on the familiar square‐root behavior of weakly nonlinear solutions. The solution in this narrow transition region is described in terms of the second Painlevé transcendent. In the exceptional case when the perturbation takes the form of longitudinal rolls, this equation needs some modification in that the second derivative is replaced by the fourth. The flow in a horizontal layer when the temperature difference between the boundaries increases slowly may be treated in exactly the same way. The necessary modifications to theory and results are given in an Appendix.
A Diffusion Equation Illustrating Spectral Theory for Boundary Layer Stability
Authors: Herron Isom H.
Published: September 2015

The stability of a special solution of a nonlinear diffusion equation is examined. It is shown that the spectral resolution of the linearized stability equation is best accomplished by requiring sufficiently strong decay of the eigenfunctions at infinity. The techniques employed illustrate what is involved in determining the spectral resolution for the equations of hydrodynamic stability when the boundary layer has a transverse component at infinity.
Hysteresis, Period Doubling, and Intermittency at High Prandtl Number in the Lorenz Equations
Authors: Fowler A. C. , McGuinness M. J.
Published: September 2015

We analyse a recently derived difference equation for the Lorenz equations, and thereby predict previously observed phenomena of period doubling and intermittent transitions. We also predict a hysteretic effect in such transitions, and give quantitative approximations to the bifurcation curves in (r, σ) parameter space. These are in agreement with the results of direct numerical simulation.
Dynamics of a Couple‐Stress Fluid Membrane
Authors: Waxman Allen M.
Published: September 2015

This work concerns the dynamics of a two‐dimensional (surface) fluid membrane which resists bending in a manner reminiscent of elastic shells. Such bending stiffness can arise from the molecular structure of the membrane, the molecules being treated as “directors” oriented normal to the surface. In this paper we unify the hydrodynamics of a two‐dimensional viscoelastic fluid with the mechanics of an elastic shell in the spirit of a Cosserat continuum. The kinematics of the evolving surface geometry is developed, along with the dynamics of this surface phase, in the time‐dependent, non‐Euclidean metric space of the surface itself. Considerations of linear‐ and angular‐momentum balance lead to an asymmetric surface stress tensor as well as a moment (or couple‐stress) tensor for which a variety of constitutive relations are considered. The antisymmetric component of stress, along with a transverse shear force, couples the membrane bending motion to the surface flow vorticity. The relevance of this theory to the dynamics of coated droplets in general, and to biological cell membranes in particular, is briefly discussed.
Compressible Ekman Layers on Curved Boundaries
Authors: Greenspan H. P.
Published: September 2015

The effects of the viscous Ekman layers on a rotating compressible gas in a container of arbitrary shape are described by a set of equivalent boundary conditions on the inviscid flow.
On the Effects of Nonparallelism, Three‐Dimensionality, and Mode Interaction in Nonlinear Boundary‐Layer Stability
Authors: Hall P. , Smith F. T.
Published: September 2015

A self‐consistent theoretical investigation is described for the nonlinear stability, and spatial development, of disturbances in a plane boundary layer subject to a number of three‐dimensional modes, their nonlinear interactions, and the effects of nonparallelism of the basic flow. For the largest weakly nonlinear disturbances considered, nonparallel‐flow effects appear to be negligible at first sight, and primary, secondary, and/or tertiary bifurcations, usually supercritical but not always so, can occur when two fundamental modes are present. As a result the flow downstream then always has three ultimate possibilities: a unique stable disturbed state, two or more possible stable states, or no stable state possible. It is here that the nonparallel‐flow effects exert their crucial influence. For nonparallelism comes into play significantly during the initial growth or decay of a disturbance, and that initial spatial development, from given initial conditions upstream, controls what happens subsequently as the disturbance increases. Thus in the first possibility above, the stable state is achieved through a smooth bifurcation, due to nonparallelism; in the second possibility the nonparallelism decides which stable state is attained (smoothly) from the initial conditions; and in the third possibility the nonparallel flow effects force the disturbance to terminate in a singular fashion. This singularity then leads to a fully nonlinear effect, locally on the boundary‐layer flow. More complicated interactions can arise if more than two three‐dimensional modes are present. The novel effect of the nonparallelism has a connection with related Navier‐Stokes calculations even at near‐critical Reynolds numbers.
Analysis of the Lorenz Equations for Large r
Authors: Fowler A. C.
Published: September 2015

In the limit of large r, the Lorenz equations become “almost” conservative. In this limit, one can use the method of averaging (or some equivalent) to obtain a set of two autonomous differential equations for two slowly varying amplitude functions B and D. A stable fixed point of these equations represents the stable periodic solution which is observed at large r. There is an invariant line B = D on which the method breaks down and the averaged equations are no longer valid. In this paper we show how to extend the validity of the analysis by Poincaré mapping B and D across this line. This extended analysis provides (in principl ) a complete recipe for constructing approximate solutions, and shows how a strange invariant set can occur in connection with an essentially analytically constructed two‐dimensional mapping.
Weak and Strong Interactions between Internal Solitary Waves
Authors: Gear J. A. , Grimshaw R.
Published: September 2015

The interaction of weakly nonlinear long internal gravity waves is studied. Weak interactions occur when the wave phase speeds are unequal; this case includes that of a head‐on collision. It is shown that each wave satisfies a Korteweg‐de Vries equation, and the main effect of the interaction is described by a phase shift. Strong interactions occur when the wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a pair of coupled Korteweg‐de Vries equations, for which some preliminary numerical results are presented.
Resonantly Interacting Weakly Nonlinear Hyperbolic Waves. I. A Single Space Variable
Authors: Majda Andrew , Rosales Rodolfo
Published: September 2015

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves in a single space variable. This theory includes as a special case the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small‐amplitude periodic initial data are prescribed. In the important physical example of the 3 × 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. (In the general case we give many new conditions guaranteeing nonresonance for a given hyperbolic system with prescribed initial data, as well as other new structural conditions which imply that resonance occurs.) A method for treating resonantly interacting waves in several space variables, together with applications, will be developed by the authors elsewhere.
A Multidimensional Inverse‐Scattering Method
Authors: Nachman Adrian I. , Ablowitz Mark J.
Published: September 2015

A formal solution of the inverse scattering problem for the n‐dimensional; time‐dependent and time‐independent Schrödinger equations is given. Equations are found for reconstructing the potential from scattering data purely by quadratures. The solution also helps elucidate the problem of characterizing admissible scattering data.
Hamiltonian Hierarchies on Semisimple Lie Algebras
Authors: Sattinger D. H.
Published: September 2015

A recursion formula is described which generates infinite hierarchies of completely integrable Hamiltonian systems of nonlinear partial differential equations. These equations govern the evolution of a function u of x, t which takes its values in a semisimple Lie algebra. A Hamiltonian for the hierarchy is given in terms of a meromorphic connection matrix.
Nonlinear Waves in a Rod: Results for Incompressible Elastic Materials
Authors: Wright T. W.
Published: September 2015

A nonlinear intrinsic theory is used to describe the motions of a straight round elastic rod including the influence of radial shear and inertia. Consideration of steady wave motions reduces the two coupled partial differential equations to ordinary differential equations for which two integrals of the motion may be found. For incompressible elastic materials with the restriction of small strain gradients, but arbitrary finite strains, a large variety of exact solutions may be found by quadrature. These include large amplitude periodic waves (which may contain shocks), solitary waves, and in some cases waves that are transitional from one stress level to another. Such solutions may be found for uniform stress strain curves that are concave up or down or that contain inflections, and even for nonmontonic curves, which have been used to represent phase transitions.
Unimodality and Lie Superalgebras
Authors: Stanley Richard P.
Published: September 2015

It is well‐known how the representation theory of the Lie algebra sl(2, ℂ) can be used to prove that certain sequences of integers are unimodal and that certain posets have the Sperner property. Here an analogous theory is developed for the Lie superalgebra osp(1,2). We obtain new classes of unimodal sequences (described in terms of cycle index polynomials) and a new class of posets (the “super analogue” of the lattice L(m,n) of Young diagrams contained in an m × n rectangle) which have the Sperner property.
Evolution Equations for Weakly Nonlinear, Long Internal Waves in a Rotating Fluid
Authors: Grimshaw R.
Published: September 2015

Evolution equations are derived for weakly nonlinear, long internal waves in a frame of reference rotating about the vertical axis. When the internal Rossby radius is at most comparable with the wavelength, the evolution equation is a Korteweg‐de Vries equation for shallow fluids, and its counterpart for deep fluids. To leading order the transverse structure is that for a linear internal Kelvin wave. For weaker rotation when effects due to the internal Rossby radius are comparable with the effects due to nonlinearity and dispersion, the evolution equation is a rotation‐modified two dimensional Korteweg‐de Vries equation for shallow fluids, or its counterpart for deep fluids. Some comparisons are made with the recent experiments of Maxworthy (1983).
Pseudospherical Surfaces and Evolution Equations
Authors: Chern S. S. , Tenenblat K.
Published: September 2015

We consider evolution equations, mainly of type ut = F(u, ux,..., ∂ku/∂xk), which describe pseudo‐spherical surfaces. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. Moreover, we investigate how the geometrical properties of surfaces provide analytic information for such equations.
Linear Stability of Miscible Displacement Processes in Porous Media in the Absence of Dispersion
Authors: Hickernell F. J. , Yortsos Y. C.
Published: September 2015

The linear stability of miscible displacement processes in porous media is examined in the absence of diffusion and dispersion. Bounds for the rate of growth of the disturbance are derived. The asymptotic behavior of the rate of growth as a function of the wavenumber of the disturbance and the mobility profile characteristics is obtained for both small and large wavenumbers. A closed‐form solution is also presented for a particular mobility profile. It is shown that such displacement processes are linearly unstable in the case when the mobility profile contains any segments of decreasing mobility, and marginally stable in the opposite case. The effect of gravity on linear stability, in the case of compressible flows, is also briefly discussed.
Dendrites in the Small Undercooling Limit
Authors: Pelce P. , Pomeau Y.
Published: September 2015

Dendrites of a solid growing steadily in an undercooled melt have a shape given by the solution of certain nonlinear integrodifferential equations. If one does not take account of the Gibbs‐Thomson condition, these equations have an exact solution due to Ivantsov and to Cahn and Horway. For elementary dimensional reasons, the tip velocity in these solutions is left undetermined. This degeneracy is removed by the effect of interface curvature on the melting temperature (Gibbs‐Thomson effect). We show that, in the physically relevant limit of small undercooling, the shape of the dendrite can be deduced from parameterless similarity equations, where the Gibbs‐Thomson effect is included. Every physical quantity is known in this limit, up to a nonlinear eigenvalue that has to be found numerically.
Analytic and Numerical Solutions of a Nonlinear Boundary‐Layer Problem
Authors: Ierley Glenn R. , Ruehr Otto G.
Published: September 2015

Solutions of a viscoinertial western‐boundary‐layer problem arising in physical oceanography are discussed for both no‐slip and stress‐free boundary conditions. A fairly complete characterization of the behavior of the solutions in a number of regimes is presented using the techniques of regular perturbation expansion, matched asymptotic expansion, and rescaling of both regular and singular solutions. A comparison with the actual solutions found by extensive numerical computations is given. Parametrization of solutions in terms of the unknown first (or second) derivative at the origin, for which a useful analytic approximation is developed, provides a convenient characterization of the entire family.
The Recursion Operator of the Kadomtsev‐Petviashvili Equation and the Squared Eigenfunctions of the Schrödinger Operator
Authors: Fokas A. S. , Santini P. M.
Published: September 2015

The recursion operator of the Kadomtsev‐Petviashvili equation is algorithmically derived. This recursion operator is the two‐spatial‐dimensional analogue of the Lenard operator of the Korteweg‐deVries equation. It is also the “squared” eigenfunction operator of the time‐dependent Schrödinger operator. The existence of the recursion operator suggests that the Kadomtsev‐Petviashvili equation is a hi‐Hamiltonian system.
Exact Representations for Acoustical Waves When the Sound Speed Varies in Space and Time
Authors: Seymour Brian , Varley Eric
Published: September 2015

This paper describes the propagation of acoustical disturbances in materials where the sound speed c(x,t) varies in space and time. It is shown that there are forms of c(x,t) for which the general solution to the one dimensional wave equation can be found in closed form. The c(x,t) are characterized by the condition that the expansion schemes used in the theory of geometrical acoustics should terminate after a finite number of terms to yield exact solutions to the wave equation.
Integrable Nonlinear Equations for Water Waves in Straits of Varying Depth and Width
Authors: David D. , Levi D. , Winternitz P.
Published: September 2015

A considerable amount of information is currently available on the creation and propagation of large solitary waves in marine straits. In order to be able to analyze such data we develop a theoretical model, extending previous one‐dimensional models to the case of straits with varying width and depth, and nonvanishing vorticity. Starting from the Euler equations for a three‐dimensional homogeneous incompressible inviscid fluid, we derive, in the quasi‐one‐dimensional long‐wave and shallow‐water approximation, a generalized KadomtsevPetviashvili (GKP) equation, together with its appropriate boundary conditions. In general, the coefficients of this equation depend on the form of the bottom and on the vorticity; the sides of the straits figure only in the boundary conditions. Under certain restrictions on the vorticity and the geometry of the straits we reduce the GKP equation to one of several completely integrable partial differential equations, in order to study the evolution of solitons which originate in the straits.
Solutions of the Ginzburg‐Landau Equation of Interest in Shear Flow Transition
Authors: Landamn Michael J.
Published: September 2015

The Ginzburg‐Landau equation may be used to describe the weakly nonlinear 2‐dimensional evolution of a disturbance in plane Poiseuille flow at Reynolds number near critical. We consider a class of quasisteady solutions of this equation whose spatial variation may be periodic, quasiperiodic, or solitarywave‐ like. Of particular interest are solutions describing a transition from the laminar solution to finite amplitude states. The existence of these solutions suggests the existence of a similar class of solutions in the Navier‐Stokes equations, describing pulses and fronts of instability in the flow.
Triad Resonance for Weakly Coupled, Slowly Varying Oscillators
Authors: Grimshaw R.
Published: September 2015

A dynamical system with weak nonlinearity is considered for the situation when three modes come into resonance. Interaction equations are derived, and various analytical approximations and numerical results obtained. If is found that either the interaction is explosive, when all modes generally grow during the interaction, with the possibility of an algebraic singularity developing, or the interaction is contained, and the modes exchange action during the coupling. The explosive interaction requires both positive‐ and negative‐energy modes to participate, while the contained interaction can accommodate modes with the same‐sign or opposite‐sign energies.
Linear Problems Associated with Evolution Equations of the Type utt = F(u, ux, uxx, ut)
Authors: Jorge L. P. , Tenenblat K.
Published: September 2015

Evolution equations which describe pseudospherical surfaces are considered. A systematic procedure to determine a linear problem for which a given equation is the integrability condition is obtained. The theory is applied to families of equations. In particular, the Liouville equation is obtained as the integrability condition of a linear problem.
Canonical and Noncanonical Recursion Operators in Multidimensions
Authors: Boiti M. , Léon J. J‐P. , Pempinelli F.
Published: September 2015

The hierarchies of evolution equations associated with the spectral operators ∂x∂y − R∂y − Q and ∂x∂y − Q in the plane are considered. In both cases a recursion operator Ф12, which is nonlocal and generates the hierarchy, is obtained. It is shown that only in the first case does the recursion operator satisfy the canonical geometrical scheme in 2 + 1 dimensions proposed by Fokas and Santini. The general procedure proposed allows one to derive, at the same time, the evolution equations associated with a given linear spectral problem and their Backlund transformations (if they exist), with no need to verify by long and tedious computations the algebraic properties of Ф12. Two equations in the first hierarchy can be considered as two different integrable generalizations in the plane of the dispersive long wave equation. All equations in this hierarchy are shown to be both a dimensional reduction of bi‐Hamiltonian n × n matrix evolution equations in multidimensions and a generalization in the plane of bi‐Hamiltonian n × n matrix evolution equations on the line.
A Method for Obtaining Exact Solutions to Partial Differential Equations with Variable Coefficients
Authors: Varley Eric , Seymour Brian
Published: September 2015

This paper describes techniques that can be used to transform PDEs with variable coefficients into equations with constant coefficients. The techniques are illustrated by calculating shear flows over quite general surfaces, by solving the signaling problem for diffusive processes in inhomogeneous materials, and by solving the signaling problem for acoustical waves when the sound speed varies with distance. The techniques may also be used to solve equations governing processes in inhomogeneous, anisotropic materials.
On Marcel Riesz's Ultrahyperbolic Kernel
Authors: Trione Susana Elena
Published: September 2015

Let t = (t1, …, tn) be a point of ℝn. We shall write

. We put by definition Rα(u) = u(α−n)/2/Kn(α); here α is a complex parameter, n the dimension of the space, and Kn(α) is a constant. First we evaluate □Rα(u) = Rα(u), where □ the ultrahyperbolic operator. Then we obtain the following results: R−2k(u) = □kδ; R0(u) = δ; and □kR2k(u) = δ, k = 0, 1, …. The first result is the n‐dimensional ultrahyperbolic correlative of the well‐known one‐dimensional formula
. Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.
A Variational Principle for the Ackerberg‐O'Malley Resonance Problem
Authors: Srinivasan R.
Published: September 2015

A new variational principle is proposed for determining the asymptotic expansion of the solution of the Ackerberg‐O'Malley resonance problem [Stud. Appl. Math. 49:277–295 (1970)] to any order in ε. The method used yields new higher‐order results not permitted by the technique of Grasman and Matkowsky [SIAM J. Appl. Math. 32:588–597 (1977)]. Explicit results using the method are reported to O(ε) and confirmed with asymptotic expansions of the exact solution; the O(1) results agree with those reported in the literature. In the case where the coefficient functions are analytic, an exact solution is presented. It is not difficult to perform the higher‐order calculations using the proposed variational approach, in contrast to the current methods in use.
Solitons in Shallow Seas of Variable Depth and in Marine Straits
Authors: David D. , Levi D. , Winternitz P.
Published: September 2015

A previously derived equation and boundary condition [the generalized Kadomtsev‐Petviashvili system] is used to describe the propagation of stable solitary waves in open seas and marine straits. The GKP system is transformed, under specific geophysical conditions, into a simpler system that allows exact soliton type solutions. The curved wave crests corresponding to these solutions are plotted for several choices of the depth function and side boundaries.
A Mean Flow First Harmonic Theory for Hydrodynamic Instabilities
Authors: Benney D. J. , Chow K.
Published: September 2015

A general theory is presented for nonlinear instabilities arising in steady hydrodynamic motions. For quasiparallel flows at high values of the Reynolds number it is found that for relatively small disturbance levels the usual ideas concerning the generation of higher harmonics and the subsequent modification of the fundamental may be overwhelmed by three dimensional interactions between the evolving mean flow and the first harmonic wave. The differences from and similarities to existing asymptotic and numerical studies are discussed. The theory developed applies to a variety of flow configurations. Numerical results are given for Poiseuille flow and the Blasius boundary layer. In addition the theory developed here is applied to simulate the instabilities produced in a boundary layer due to the presence of free stream disturbances.
On the Derivation of the Modified Kadomtsev‐Petviashvili Equation
Authors: Grimshaw R. , Melville W K.
Published: September 2015

It is well known that the Kadomtsev‐Petviashvili (KP) equation is the two‐dimensional analogue of the Korteweg—de Vries (KdV) equation. We reconsider the derivation of the KP equation, modified to include the effects of rotation, in order to determine the nature of the initial conditions. The motivation for this is that if the solutions of the modified KP equation are assumed to be locally confined, then they satisfy a certain constraint, which appears to restrict considerably the class of allowed initial conditions. The outcome of the analysis presented here is that in general it is not permissible to assume that solutions of the modified KP equation are locally confined, and hence the constraint cannot be applied. The reason for this is the radiation of Poincaré waves, which appear behind the main part of the solution described by the modified KP equation.
The Shunted Homopolar Dynamo—An Analytic Approach to a Poincaré Map
Authors: Lu Ya Yan
Published: September 2015

We study the set of ordinary differential equations governing the homopolar disk dynamo. It is found that this set, which is a modification of the Lorenz system, has strange attractors of Lorenz type when R (which corresponds to the Rayleigh number of the Lorenz system) tends to infinity. A central aspect of this study is that the Poincare map for this limit can be obtained through Melnikov's perturbation method, in contrast to the usual dependence on numerical computation.
A Bijective Proof of the Fundamental Theorem on Symmetric Functions
Authors: Senato Domenico
Published: September 2015

In this note we prove with a bijective argument that every monomial symmetric function is a polynomial with integer coefficients in the elementary symmetric functions.
Some Self‐Similar Two‐Phase Flows of Non‐Newtonian Fluids Through a Porous Medium
Authors: Pascal H.
Published: September 2015

Pascal This paper addresses the question of the rheological effects of non‐Newtonian fluids in a flow system, in which a two‐phase flow zone is coupled to a single‐phase flow zone by a moving fluid interface. This flow system is involved in a technique for oil displacement in a porous medium, where a non‐Newtonian displacing fluid (a polymer solution) is used to displace a non‐Newtonian heavy oil. The self‐similar solutions of the equations governing the dynamics of the moving interface, separating the displacing and displaced fluids, are obtained for the one‐dimensional and plane radial flows. The effects associated with the presence of a two‐phase flow zone, behind the moving interface, on the interface movement are analyzed. The existence of a pressure front ahead of the moving interface, moving with a finite velocity, is also shown. The relevance of this result to the propagation of pressure disturbances in a non‐Newtonian fluid flowing through a porous medium is discussed with regard to interpretation of the transient pressure response in an injection well for polymer‐solution floods.
The Inverse Problem for Ward's System
Authors: Villarroel Javier
Published: September 2015

We study the direct and inverse problems for a 2 + 1‐dimensional model recently formulated by Richard Ward. We recover the instanton solutions, and we also obtain new ones.
Operational Representations for the Classical Hermite Polynomials
Authors: Chatterjea S. K. , Srivastava H. M.
Published: September 2015

The authors present a systematic analysis of the various known (or seemingly new) operational representations for the classical Hermite polynomials. The interrelationships between many of these results, and the potential for their applications in other related areas of interest, are also indicated.
On Stirling Functions of the Second Kind
Authors: Butzer P. L. , Hauss M.
Published: September 2015

We continue the study of Stirling functions of the second kind, S(α,k) where α > 0 is real. In particular, we consider integral and fractional derivative representations of the S(α,k), as well as connections with generalized Bernoulli and Stirling polynomials.
Effects of Radiative Damping on Resonantly Generated Internal Gravity Waves
Authors: Mitsudera H. , Grimshaw R.
Published: September 2015

Internal gravity waves can propagate indefinitely upwards when a fluid is continuously stratified and unbounded vertically. This causes radiative damping of waves which are resonantly generated by the interaction between a mean flow and a topographic feature at the lower boundary. An evolution equation of Benjamin‐Davis‐Acrivos type, including temporal evolution, detuning, nonlinearity, dispersion, radiative damping, and forcing, is derived to describe the wave behavior. Using a pseudospectral method, we obtain and discuss various numerical results, especially for the resonant cases, covering forcing whose polarity is the same as, or opposite to, that of a free solitary wave, and for weak or strong radiative damping. In practice, weak‐radiative cases are likely to be realized when an inversion layer exists at a certain low level over a mountain range.
Branching Near Plane Waves in Perturbed Dispersive Systems
Authors: Newton Paul K.
Published: September 2015

A Poincaré‐Lindstedt type technique for partial differential equations is used to study branching phenomena in perturbed dispersive systems arising in hydrodynamic stability theory. Multi‐periodic waves with two frequencies which branch from a family of neutrally stable nonlinear periodic plane waves are constructed, the second frequency as a power series expansion in ε. The branching is compared with that of the unperturbed equations described in an earlier paper for the purpose of understanding how higher order perturbation terms effect the properties of the lower order amplitude equations. We find that in general the perturbation terms alter the leading order frequency shifts, thus changing the bifurcation from pitchfork to transverse type. The method is used to study the perturbed nonlinear Schrödinger equation and the perturbed MKdV equation.
Fractional Calculus and the Sum of Powers of Natural Numbers
Authors: Srivastava H. M. , Joshi J. M. C. , Bisht C. S.
Published: September 2015

By employing a certain operator of fractional calculus, some interesting results were proven recently for the sums of powers of natural numbers. This paper presents a systematic analysis of these (as well as of numerous related) classes of summation or expansion formulas which have already been considered in the literature in various contexts other than that of fractional calculus. Some historical details are also provided.
On the Kadomtsev‐Petviashvili Equation and Associated Constraints
Authors: Ablowitz Mark J. , Villarroel Javier
Published: September 2015

The initial‐boundary‐value problem for the Kadomtsev‐Petviashvili equation in infinite space is considered. When formulated as an evolution equation, found that a symmetric integral is the appropriate choice in the nonlocal term; namely,
. If one simply chooses
, then an infinite number of constraints on the initial data in physical space are required, the first being
. The conserved quantities are calculated, and it is shown that they must be suitably regularized from those that have been used when the constraints are imposed.
A Generalization of the Pfaff‐Saalschütz Theorem
Authors: Constantineau Ivan
Published: September 2015

We generalize a well‐known theorem of Pfaff and Saalschütz and give algebraic and combinatorial proofs.
Observations on the Role of Vorticity in the Stability Theory of Wall Bounded Flows
Authors: Herron Isom H.
Published: September 2015

The linearized equations for the evolution of disturbances to four wall bounded flows are treated. The flows are plane Couette flow and plane Poiseuille flow, Hagen‐Poiseuille pipe flow, and the asymptotic suction profile. By looking at the vorticity it is proved simply that plane Couette flow and Hagen‐Poiseuille flow are linearly stable. Further study is made of the structure of the disturbance equation by the introduction of a special vorticity adjoint.
On the Foundation of Combinatorial Theory. X. A Categorical Setting for Symmetric Functions
Authors: Bonetti Flavio , Rota Gian‐Carlo , Senato Domenico , Venezia Antonietta M.
Published: September 2015

A categorical setting is developed for the theory of symmetric functions.
The Painlevé‐Kowalevski and Poly‐Painlevé Tests for Integrability
Authors: Kruskal Martin D. , Clarkson Peter A.
Published: September 2015

The characteristic feature of the so‐called Painlevé test for integrability of an ordinary (or partial) analytic differential equation, as usually carried out, is to determine whether all its solutions are single‐valued by local analysis near individual singular points of solutions. This test, interpreted flexibly, has been quite successful in spite of various evident flaws. We review the Painleve test in detail and then propose a more robust and generally more appropriate definition of integrability: a multivalued function is accepted as an integral if its possible values (at any given point in phase space) are not dense. This definition is illustrated and justified by examples, and a widely applicable method (the poly‐Painlevé method) of testing for it is presented, based on asymptotic analysis covering several singularities simultaneously.
Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem)
Authors: Venturini G.
Published: September 2015

Given p ∈ N = {1, 2, 3 ...}, as a generalization of the 3x + 1 problem [7], we study the behavior of the sequences s(m) = {mn}n ≥ 0, m ∈ Z (the set of the integers), defined by the iterative formula
urn:x-wiley:00222526:media:sapm1992863185:sapm1992863185-math-0001
and ar = tr/p (tr ∈ N), and br are chosen in such a way that mn ∈ Z for every n. Our aim is to establish when these sequences are divergent, or convergent into a cycle, considering also a fixed point as a cycle. Moreover, the structure of possible cycles is investigated.
The Painlevé Connection Problem: An Asymptotic Approach. I
Authors: Joshi Nalini , Kruskal Martin D.
Published: September 2015

The connection problem for the first and second Painlevé equations is the problem of relating the asymptotic behavior of a solution on a path of approach to infinity (in the complex plane of the independent variable) to those along another such path. A direct natural asymptotic method of solving this problem is described in detail in this paper. In particular, a uniformly valid description of the general (two‐complex‐parameter) asymptotic behaviors—given to leading order by elliptic functions—is derived by a generalization of the multiple‐scales method.
On the Surface Waves in a Shallow Channel with an Uneven Bottom
Authors: Benilov Eugene S.
Published: September 2015

This paper examines surface gravity waves in a shallow channel with periodic or random bottom irregularities. Three types of bottom topography are distinguished, allowing a simplification of the basic shallow‐water‐wave equations. For two of them, asymptotic equations of the Korteveg‐de Vries type are derived (the third type has been considered earlier by other authors).
Vector Eigenfunction Expansions for Plane Channel Flows
Authors: Hennington Dan S. , Schmid Peter J.
Published: September 2015

The full nonlinear initial‐boundary value problem for the evolution of disturbances in plane Poiseuille flow is considered. The problem is formulated in vector form using the normal velocity and normal vorticity as components. The solution is presented as an expansion in linear eigenmodes. These modes consist of both Orr‐Sommerfeld modes and modes of the normal vorticity (Squire) equation. The case of degenerating eigenmodes is also considered and it is shown that the Benney‐Gustavsson normal velocity‐normal vorticity resonance is a special case of a degeneracy between the vector eigenmodes. The solution to the nonlinear problem is presented as an expansion in the linear eigenmodes as well as in modes of the self‐adjoint part of the linear equation. The full nonlinear solution is further reduced to small systems of coupled amplitude equations using the center manifold theorem.
Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials
Authors: Ward Michael J.
Published: September 2015

Solutions exhibiting an internal layer structure are constructed for a class of nonlinear singularly perturbed boundary value problems with translation invariant potentials. For these problems, a routine application of the method of matched asymptotic expansions fails to determine the locations of the internal layer positions. To overcome this difficulty, we present an analytical method that is motivated by the work of Kath, Knessl and Matkowsky [4]. To construct a solution having n internal layers, we first linearize the boundary value problem about the composite expansion provided by the method of matched asymptotic expansions. The eigenvalue problem associated with the homogeneous form of this linearization is shown to have n exponentially small eigenvalues. The condition that the solution to the linearized problem has no component in the subspace spanned by the eigenfunctions corresponding to these exponentially small eigenvalues determines the internal layer positions. These “near” solvability conditions yield algebraic equations for the internal layer positions, which are analyzed for various classes of nonlinearities.
On the Generalizations of a Theorem of Beppo Levi
Authors: Trion S. E.
Published: September 2015

In this paper we give two generalizations of a theorem of Beppo Levi ([1], p. 347, Formula (12)). This theorem affirms that, under certain conditions, the following assertion is true:
urn:x-wiley:00222526:media:sapm1992873195:sapm1992873195-math-0001
where
urn:x-wiley:00222526:media:sapm1992873195:sapm1992873195-math-0002
φ(x) is a function that verifies φ(0) > 0; f(x) is defined and bounded in the interval (a, b) and continuous in the point 0 with f(0) ≠ 0; f(x) and φ(x) are integrable functions in the interval [a, b]; c >, 0 and υ > 1. This problem was studied by Laplace [2], Darboux [3], Stieltjes [4], Lebesgue [5], Romanovsky [6], and Fowler [7]. The first generalization (Section 1, Theorem 1.2, Formula (1.35)) says that, under certain conditions, the following formula is valid: urn:x-wiley:00222526:media:sapm1992873195:sapm1992873195-math-0003 where φn(x) is a sequence of functions and Bn(a) designates the n‐dimentional ball of radius a and center in the origin. The extension follows by Romanovsky's method. The absolute maximum of φ(x) in the extremes of the interval of definition is treated in the second generalization of the Theorem of Beppo Levi (Section 2, Theorem 2.2, Formulas (2.1), (2.2)). We note that Beppo Levi proves this assertion in the interior of the interval.