A mathematical analysis of the time-dependent, baroclinic, Boussinesq, Eliassen balanced model of tropical cyclone dynamics is presented. An efficient numerical method is developed to solve the equations in momentum c...
A mathematical analysis of the time-dependent, baroclinic, Boussinesq, Eliassen balanced model of tropical cyclone dynamics is presented. An efficient numerical method is developed to solve the equations in momentum coordinates with cylindrical symmetry. Computer simulations illustrate the effects of an experimental parameterization for the heating due to condensation. The model reproduces realistic characteristics of a tropical cyclone and provides a useful tool for analyzing the dynamic relationship between the energy influx due to latent heating and the vortex structure.
We analyze the problem of stationary self-propagating fronts in potential flow. The issues of local existence and uniqueness for solutions of the ODE describing stationary fronts, multiplicity of solutions and lineari...
We analyze the problem of stationary self-propagating fronts in potential flow. The issues of local existence and uniqueness for solutions of the ODE describing stationary fronts, multiplicity of solutions and linearized stability of a stationary front as a solution of the (hyperbolic) evolution equation are addressed. The results are illustrated in the case of the dual-source system, which is a simple model of a combustion system in which local extinction may arise. Model extensions for combustion applications are presented.
Although normal ordering (NO) is often used as a quantization procedure for classical problems that are based on complex mode amplitudes, an alternate choice, called ''symmetric ordering'' (SO), is cha...
Although normal ordering (NO) is often used as a quantization procedure for classical problems that are based on complex mode amplitudes, an alternate choice, called ''symmetric ordering'' (SO), is championed here. In normal form the SO operator is simply related to zero order Laguerre polynomials that are implied by Weyl's quantization rule. Thus SO is as convenient as NO, and it is more accurate when the rotating wave approximation is used for a mass-spring oscillator with a slightly nonlinear spring. The two quantization methods are compared for a Hartree analysis of the discrete self-trapping equation with an arbitrary power of the nonlinearity.
A new mechanism for the creation of structures in two-dimensional turbulence is investigated. The forced Navier-Stokes equations are solved numerically in a periodic square in the limit of zero viscosity. The force is...
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A new mechanism for the creation of structures in two-dimensional turbulence is investigated. The forced Navier-Stokes equations are solved numerically in a periodic square in the limit of zero viscosity. The force is a white-in-time random noise acting in a narrow band of high wavenumbers. The inverse-cascade process and the presence of the boundary lead ultimately to a pile-up of energy in the lowest wavenumber (Bose condensation). In the asymptotic limit where the enstrophy cascade range is negligible, Bose condensation is solely responsible for the generation of coherent vortices and intermittency in the system. We present the evolution of the velocity and vorticity fields through the later stages of the condensate state, and explore the possible implications for atmospheric turbulence constrained by the periodic domain about the earth.
An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is prove...
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An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is proved for the McKendrick equation that governs the dynamics of the age distribution function. This proof shows how a decoupled ordinary differential equation for the total population size can be derived. This result makes a study of the population's asymptotic dynamics (indeed, often its global asymptotic dynamics) mathematically tractable. Several applications to models for intra-specific competition and predation are given.
We study the motion of surfaces in an intrinsic formulation in which the surface is described by its metric and curvature tensors. The evolution equations for the six quantities contained in these tensors are reduced ...
We study the motion of surfaces in an intrinsic formulation in which the surface is described by its metric and curvature tensors. The evolution equations for the six quantities contained in these tensors are reduced in number in two cases: (i) for arbitrary surfaces, we use principal coordinates to obtain two equations for the two principal curvatures, highlighting the similarity with the equations of motion of a plane curve;and (ii) for surfaces with spatially constant negative curvature, we use parameterization by Tchebyshev nets to reduce to a single evolution equation. We also obtain necessary and sufficient conditions for a surface to maintain spatially constant negative curvature as it moves. One choice for the surface's normal motion leads to the modified Korteweg-de Vries equation, the appearance of which is explained by connections to the AKNS hierarchy and the motion of space curves.
The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat liner operator L, is developed in sufficient detail for later use...
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The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat liner operator L, is developed in sufficient detail for later use in studies of perturbations of the NLS equation. ''Counting lemmas'' for the non-selfadjoint operator L, are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite disc D in the complex eigenvalue plane. The radius of the disc D is controlled by the H-1 norm of the potential q. For this integrable NLS Hamiltonian system, unstable tori are identified, and Backlund transformations are then used to construct global representations of their stable and unstable manifolds - ''whiskered tori'' for the NLS pde. The Floquet discriminant DELTA(lambda;q) used to introduce a natural sequence of NLS constants of motion, [F(j)(q) = DELTA(lambda = lambda(j)c(q);q), where lambda(j)c denotes the j(th) critical point of the Floquet discriminant DELTA(lambda)]. A Taylor series expansion of the constants F(j)(q), with explicit representations of the first and second variations, is then used to study neighborhoods of the whiskered tori. In particular, critical tori with hyperbolic structure are identified through the first and second variations of F(j)(q), which themselves are expressed in terms of quadratic products of eigenfunctions of L. The second variation permits identification, within the disc D, of important bifurcations m the spectral configurations of the operator L. The constant F(j)(q), as the height of the Floquet discriminant over the critical point lambda(j)c, admits a natural interpretation as a Morse function for NLS isospectral level sets. This Morse interpretation is studied in some detail. It is valid globally for the infinite tail, {F(j)(q)}\j\>N, which is associated with critical points outside the disc D. Within this disc, the interpretation is only valid locally, with the s
Hamiltonian integration schemes for the Nonlinear Schroedinger Equation are examined. The efficiency with respect to accuracy and integration time of an integrable scheme, a standard conservative scheme, and a symplec...
Hamiltonian integration schemes for the Nonlinear Schroedinger Equation are examined. The efficiency with respect to accuracy and integration time of an integrable scheme, a standard conservative scheme, and a symplectic method is compared.
The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poiss...
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The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poisson structure separately, then form the the resulting ODE's. The stability, accuracy, and dispersion of different explicit splitting methods are analyzed, and we give the circumstances under which die best results can be obtained;in particular, when the Hamiltonian can be split into linear and nonlinear terms. Many different treatments and examples are compared.
This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long-time intervals. Standard error estimates comp...
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