# Showing papers in "Studies in Applied Mathematics in 1994"

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68 citations

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56 citations

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35 citations

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TL;DR: In this article, the internal layer motion associated with the Ginzburg-Landau equation was analyzed for various boundary conditions on x = ± 1, and the effect of various types of boundary conditions and nonlinearities was highlighted.

Abstract: The internal layer motion associated with the Ginzburg-Landau equation
is analyzed, in the limit , for various boundary conditions on x = ±1. The nonlinearity Q(u) results either from a double-well potential or a periodic potential, each having wells of equal depth. Using a systematic asymptotic method, some previous work in deriving equations of motion for the internal layers corresponding to metastable patterns is extended. The effect of the various types of boundary conditions and nonlinearities will be highlighted. A dynamical rescaling method is used to numerically integrate these equations of motion. Using formal asymptotic methods, certain canonical problems describing layer collapse events are formulated and solved numerically. A hybrid asymptotic-numerical method, which incorporates these layer collapse events, is used to give a complete quantitative description of the coarsening process associated with the Ginzburg-Landau equation. For the Neumann problem with a double-well potential, the qualitative description of the coarsening process given in Carr and Pego [4] will be confirmed quantitatively. In other cases, such as for a periodic potential with Dirichlet boundary conditions, it will be shown that, through layer collapse events, a metastable pattern can tend to a stable equilibrium solution with an internal layer structure.

33 citations

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TL;DR: In this article, the critical Rayleigh number and wave number at marginal stabilities are calculated for both free and rigid boundaries, and it is noted that there exist ranges for which the stability criteria is intermediate to the low porosity Darcy approximation and to high porosity single viscous fluid.

Abstract: This paper addresses the problem of the onset of Rayleigh-Benard convection in a porous layer using Brinkman's equation and anisotropic permeability. The critical Rayleigh number and wave number at marginal stabilities are calculated for both free and rigid boundaries. In both cases, it is noted that there exist ranges for which the stability criteria is intermediate to the low porosity Darcy approximation and to high porosity single viscous fluid. The permeability anisotropy is found to select the mode of instability.

30 citations

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29 citations

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TL;DR: In this article, the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids was studied, and it was shown that for a weak interaction, the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0).

Abstract: In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ1=|cm/cn−cosδ|, Δ2=|cn/cm−cosδ|,cm,n, are the linear, long wave speeds for waves with mode numbers m, n, δ is the angle between the respective propagation directions, and α measures the wave amplitude. In this case, each wave is governed by its own Kortweg-de Vries (KdV) equation for a shallow fluid, or intermediate long-wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ1,2 are 0(α), and this case is governed by two coupled Kadomtsev-Petviashvili (KP) equations for a shallow fluid, or two coupled two-dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ1 is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ1 leads to a local distortion of the wave speed of the mode n. When the interacting waves belong to the same mode (i.e., m = n) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity as the interacting waves.

27 citations

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TL;DR: In this paper, the amplitude and phase of the captured oscillations after crossing the separatrix are matched to a transition region consisting of a large sequence of nearly solitary pulses along the boundary of the saddle point, both before and after capture.

Abstract: Dissipative perturbations of strongly nonlinear oscillators that correspond to slowly varying double-well potentials are considered. The method of averaging, which describes the solution as nearly periodic, fails as the trajectory approaches the unperturbed separatrix, a homoclinic orbit of the saddle point, significantly before it is captured in either well. Nevertheless, perturbed initial conditions corresponding to the boundary of the basin of attraction for each well, which are the perturbed stable manifolds of the saddle point, are accurately determined using only the method of averaging modified by Melnikov energy ideas near the separatrix. To determine the amplitude and phase of the captured oscillations after crossing the separatrix, a transition region is constructed consisting of a large sequence of nearly solitary pulses along the separatrix. The amplitude and phases of the slowly varying nonlinear oscillations away from the separatrix, both before and after capture, are matched to this transition region. In this way, analytic connection formulas across the separatrix are obtained and are shown to depend on the perturbed initial conditions.

21 citations

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15 citations

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12 citations

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TL;DR: In this paper, a wave packet effect is proposed in which wave packet effects are dominant in the critical layer and it is argued that in many applications, this is the appropriate choice.

Abstract: In the linear inviscid theory of shear flow stability, the eigenvalue problem for a neutral or weakly amplified mode revolves around possible discontinuities in the eigenfunction as the singular critical point is crossed. Extensions of the linear normal mode approach to include nonlinearity and/or wave packets lead to amplitude evolution equations where, again, critical point singularities are an issue because the coefficients of the amplitude equations generally involve singular integrals. In the past, viscosity, nonlinearity, or time dependence has been introduced in a critical layer centered upon the singular point to resolve these integrals. The form of the amplitude evolution equation is greatly influenced by which choice is made. In this paper, a new approach is proposed in which wave packet effects are dominant in the critical layer and it is argued that in many applications this is the appropriate choice. The theory is applied to two-dimensional wave propagation in homogeneous shear flows and also to stratified shear flows. Other generalizations are indicated.

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TL;DR: In this article, it was shown that Sheffer polynomials can be represented as moments of convolution semigroups of probability measures, and that shift-invariant operators and umbral operators can also be expressed as moments.

Abstract: In this paper we investigate which Sheffer polynomials can be represented as moments of convolution semigroups of probability measures. We also obtain general integral representations for shift-invariant operators and for umbral operators. As a corollary, we obtain new proofs for representation theorems for Sheffer polynomials due to Sheffer and Thorne.

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TL;DR: In this paper, the authors characterize the Sheffer sequences by a single convolution identity where F(y) is a shift-invariant operator, and show that these solutions can then be interpreted as cocommutative coalgebras.

Abstract: We characterize the Sheffer sequences by a single convolution identity
where F(y) is a shift-invariant operator. We then study a generalization of the notion of Sheffer sequences by removing the requirement that F(y) be shift-invariant. All these solutions can then be interpreted as cocommutative coalgebras. We also show the connection with generalized translation operators as introduced by Delsarte. Finally, we apply the same convolution to symmetric functions where we find that the “Sheffer” sequences differ from ordinary full divided power sequences by only a constant factor.

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TL;DR: In this paper, a simple model equation for western boundary outflow in the Stommel model of the large scale ocean circulation is obtained by evaluating the potential vorticity equation at the western boundary.

Abstract: A simple model equation for western boundary outflow in the Stommel model of the large scale ocean circulation is obtained by evaluating the potential vorticity equation at the western boundary. A series solution to this model equation demonstrates similar behavior to the boundary layer solution of the potential vorticity equation, in particular that “resonances” are present at a discrete series of parameter values which necessitate the addition of logarithms to the series; these resonances occur because the model equation has a logarithmic branch point at these values.

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TL;DR: The geometric, algebraic, and combinatorial explanations of Dobinski's formula are presented by mixed volumes of compact convex sets, Mobius inversion, difference operator, and species as mentioned in this paper.

Abstract: The geometric, algebraic, and combinatorial explanations of Dobinski's formula are presented by mixed volumes of compact convex sets, Mobius inversion, difference operator, and species. The employed method may be useful in proving some other combinatorial identities.

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TL;DR: In this paper, the problem of counting surjections from an n-set to a k-set is generalized to enumerating solutions of a1 ∨ ⋯ ∨ an = y, with each ai an atom of the k-interval [x, y] in a binomial lattice L. When L is modular, the number of such solutions is representable as a q-difference.

Abstract: The elementary problem of counting surjections from an n-set to a k-set is generalized to that of enumerating solutions of a1 ∨ ⋯ ∨ an = y, with each ai an atom of the k-interval [x, y] in a binomial lattice L. When L is modular, the number of such solutions is representable as a q-difference and satisfies a simple recurrence.

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TL;DR: In this paper, the effects of strong bores and small surface disturbances on weakly nonlinear shallow water waves over a variable bottom were investigated. But the authors focused on the effect of a strong bore with quiescent water over an isolated bottom disturbance.

Abstract: This paper gives an extension of previous work [2] on weakly nonlinear shallow water waves over a variable bottom to include the effects of strong bores and small surface disturbances. We first consider the interaction of a strong bore with quiescent water over an isolated bottom disturbance to highlight some of the modifications that are introduced in our results for both noncritical and transcritical Froude numbers. We also exhibit the secular effect on the bore trajectory of a bottom disturbance that has a nonzero average. In a second example, we consider the interaction of a strong bore with a small amplitude periodic surface disturbance upstream. We show that downstream of the bore, the wave length of this disturbance increases, whereas its amplitude increases (decreases) depending on whether the bore speed is larger (smaller) than a critical value. We also use this example to illustrate the derivation of the solution and bore trajectory to second order accuracy. All our asymptotic results, obtained in the form of multiple scale expansions, are compared with numerical solutions for a number of illustrative cases.

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TL;DR: In this article, the authors proposed a numerical solution for travelling combustion waves in a porous medium based on a shooting method used in an existence proof. But their numerical result suggests that there is a limit for the inlet gas velocity below which no travelling wave solution can be constructed.

Abstract: Numerical solutions for travelling combustion waves in a porous medium are sought. The algorithm of computation is based on a shooting method used in an existence proof. The numerical result suggests that there is a limit for the inlet gas velocity below which no travelling wave solution can be constructed.

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TL;DR: In this article, a comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given and the height of the free surface for each family of solutions is found explicitly.

Abstract: A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.

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TL;DR: In this article, the authors employed the previously formulated constitutive relations for an elastic heat conductor to examine the nonlinear influence of the temperature on the propagation of second sound and showed that large values of second sounds travel more slowly than do smaller ones.

Abstract: In this paper we employ our previously formulated constitutive relations for an elastic heat conductor to examine the nonlinear influence of the temperature on the propagation of second sound. Utilizing the method of geometric optics we ascertain that large values of second sound travel more slowly than do smaller ones. As well, numerical results for a signaling problem are displayed graphically in order to highlight the qualitative properties of the temperature and strain waves. The choice of signaling data was motivated by experimental studies.

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TL;DR: In this article, the Schrodinger equation in the complex k-plane with jumps on Im(k2) = 0 is considered and the solution of this problem can be reduced to solving two Riemann-Hilbert (RH) problems.

Abstract: We consider the nonlinear Schrodinger equation in the variable q(x,t) with the forcing iu(t)δ(x)+iu1(t)δ′(x). We assume that q(x,0),u(t),u1(t) are given and that these functions as well as their first two derivatives belong to L1∩L2(ℝ+). We show that the solution of this problem can be reduced to solving two Riemann—Hilbert (RH) problems in the complex k-plane with jumps on Im(k2) = 0. Each RH problem is equivalent to a linear integral equation that has a unique global solution. These linear integral equations are uniquely defined in terms of certain functions (scattering data) b(k), c(k), d(k), and f(k). The functions b(k) and d(k) can be effectively computed in terms of q(x,0). However, although the analytic properties of c(k) and f(k) are completely determined, the relationship between c(k), f(k), q(x,0), u(t), and u1(t) is highly nonlinear. In spite of this difficulty, we can give an effective description of the asymptotic behavior of q(x,t) for large t. In particular, we show that as t ∞, solitons are generated moving away from the origin. It is important to emphasize that the analysis of this problem, in addition to techniques of exact integrability, requires the essential use of general PDE techniques.

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TL;DR: In this article, the authors introduced Geissinger multiplication on the vector space generated by indicator functions of closed convex sets, and obtained a polynomial identity for the mixed volume.

Abstract: This paper introduces Geissinger multiplication on the vector space generated by indicator functions of closed convex sets. Minkowski's mixed volume for compact convex sets is naturally represented in terms of the volume of the Geissinger multiplication of their indicator functions. Some properties of mixed volumes and new results are obtained by this representation, including a polynomial identity.

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TL;DR: In this article, a higher order extension to Moore's equation governing the evolution of a thin layer of uniform vorticity in two dimensions is obtained, which is valid for consideration of motion whereby the layer thickness is uniformly small compared with the local radius of curvature of the center line.

Abstract: A higher order extension to Moore's equation governing the evolution of a thin layer of uniform vorticity in two dimensions is obtained. The equation, in fact, governs the motion of the center line of the layer and is valid for consideration of motion whereby the layer thickness is uniformly small compared with the local radius of curvature of the center line. It extends Birkoff's equation for a vortex sheet. The equation is used to examine the growth of disturbances on a straight, steady layer of uniform vorticity. The growth rate for long waves is in good agreement with the exact result of Rayleigh, as required. Further, the growth of waves with length in a certain range is shown to be suppressed by making this approximate allowance for finite thickness. However, it is found that very short waves, which are quite outside the range of validity of the equation but which are likely to be excited in a numerical integration of the equation, are spuriously amplified as in the case of Moore's equation. Thus, numerical integration of the equation will require use of smoothing techniques to suppress this spurious growth of short wave disturbances.

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TL;DR: Explicit solutions for the stream function satisfying the Navier Stokes equations representing the steady two-dimensional motion of a viscous incompressible liquid were found in this paper. But these solutions are restricted to certain regions of the x, y plane.

Abstract: Explicit solutions are found for the stream function satisfying the Navier Stokes equations representing the steady two-dimensional motion of a viscous incompressible liquid. The solutions contain two arbitrary analytic functions and in general are confined to certain regions of the x, y plane.