Spatial-Temporal Structure of Mixed-Age Forest Boundary: The Simplest Mathematical Model

The modeling of forest ecosystems is one of IIASA's continuous research activities in the Environment Program. There are two main approaches to this modeling: a) simulation and b) qualitative (analytical). This paper belongs to the latter. Analytical models allow the prediction of the behavior of key variables of ecosystems and can be used to organize and analyze data produced by simulation models or obtained by observations. This paper is devoted to the study of a simple mathematical model of spatially distributed non-even-age forests. The main tools used in the paper are new methods of qualitative theory of non-linear differential equations. This work is a continuation of the cooperation in forest modeling at IIASA started in 1986-1989 by W. Clark, H. Shugart, R. Fleming and the authors of this paper

Angular momenta creation in relativistic electron-positron plasma

Creation of angular momentum in a relativistic electron-positron plasma is explored. It is shown that a chain of angular momentum carrying vortices is a robust asymptotic state sustained by the generalized nonlinear Schrodinger equation characteristic to the system. The results may suggest a possible electromagnetic origin of angular momenta when it is applied to the MeV epoch of the early Universe.Comment: 20 pages, 6 figure

A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schr\"odinger equation (NLS) $i\partial_t u + \Delta u + |u|^2u=0$ scatter, i.e. approach the solution to a linear Schr\"odinger equation as $t\to \pm \infty$. The criteria is expressed in terms of the scale-invariant quantities $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}$ and $M[u]E[u]$, where $u_0$ denotes the initial data, and $M[u]$ and $E[u]$ denote the (conserved in time) mass and energy of the corresponding solution $u(t)$. The focusing NLS possesses a soliton solution $e^{it}Q(x)$, where $Q$ is the ground-state solution to a nonlinear elliptic equation, and we prove that if $M[u]E[u]<M[Q]E[Q]$ and $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} < \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution $e^{it}Q(x)$, for which equality in these conditions is obtained, is global but does not scatter. We further show that if $M[u]E[u] \|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle \cite{KM06a} in their study of the energy-critical NLS

Finite time singularities in a class of hydrodynamic models

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form ${\cal L}\sim\int k^\alpha|{\bf v_k}|^2d^3{\bf k}$ in 3D Fourier representation, where $\alpha$ is a constant, $0<\alpha< 1$. Unlike the case $\alpha=0$ (the usual Eulerian hydrodynamics), a finite value of $\alpha$ results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel vortex filaments and an analog of the Crow instability is found at small wave-numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like $(t^*-t)^{1/(2-\alpha)}$, where $t^*$ is the singularity time.Comment: LaTeX, 17 pages, 3 eps figures. This version is close to the journal pape

On the stability of standing matter waves in a trap

We discuss excited Bose-condensed states and find the criterion of dynamical stability of a kink-wise state, i.e., a standing matter wave with one nodal plane perpendicular to the axis of a cylindrical trap. The dynamical stability requires a strong radial confinement corresponding to the radial frequency larger than the mean-field interparticle interaction. We address the question of thermodynamic instability related to the presence of excitations with negative energy.Comment: 4 pages, 3 figure

Instabilities in the two-dimensional cubic nonlinear Schrodinger equation

The two-dimensional cubic nonlinear Schrodinger equation (NLS) can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. In this paper, we establish that every one-dimensional traveling wave solution of NLS with trivial phase is unstable with respect to some infinitesimal perturbation with two-dimensional structure. If the coefficients of the linear dispersion terms have the same sign then the only unstable perturbations have transverse wavelength longer than a well-defined cut-off. If the coefficients of the linear dispersion terms have opposite signs, then there is no such cut-off and as the wavelength decreases, the maximum growth rate approaches a well-defined limit.Comment: 4 pages, 4 figure

Transverse instability and its long-term development for solitary waves of the (2+1)-Boussinesq equation

The stability properties of line solitary wave solutions of the (2+1)-dimensional Boussinesq equation with respect to transverse perturbations and their consequences are considered. A geometric condition arising from a multi-symplectic formulation of this equation gives an explicit relation between the parameters for transverse instability when the transverse wavenumber is small. The Evans function is then computed explicitly, giving the eigenvalues for transverse instability for all transverse wavenumbers. To determine the nonlinear and long time implications of transverse instability, numerical simulations are performed using pseudospectral discretization. The numerics confirm the analytic results, and in all cases studied, transverse instability leads to collapse.Comment: 16 pages, 8 figures; submitted to Phys. Rev.

Generation and evolution of vortex-antivortex pairs in Bose-Einstein condensates

We propose a method for generating and controlling a spatially separated vortex--antivortex pair in a Bose-Einstein condensate trapped in a toroidal potential. Our simulations of the time dependent Gross-Pitaevskii equation show that in toroidal condensates vortex dynamics are different from the dynamics in the homogeneous case. Our numerical results agree well with analytical calculations using the image method. Our proposal offers an effective example of coherent generation and control of vortex dynamics in atomic condensates.Comment: 4 pages, 2 figure

On topological charge carried by nexuses and center vortices

In this paper we further explore the question of topological charge in the center vortex-nexus picture of gauge theories. Generally, this charge is locally fractionalized in units of 1/N for gauge group SU(N), but globally quantized in integral units. We show explicitly that in d=4 global topological charge is a linkage number of the closed two-surface of a center vortex with a nexus world line, and relate this linkage to the Hopf fibration, with homotopy $\Pi_3(S^3)\simeq Z$; this homotopy insures integrality of the global topological charge. We show that a standard nexus form used earlier, when linked to a center vortex, gives rise naturally to a homotopy $\Pi_2(S^2)\simeq Z$, a homotopy usually associated with 't Hooft-Polyakov monopoles and similar objects which exist by virtue of the presence of an adjoint scalar field which gives rise to spontaneous symmetry breaking. We show that certain integrals related to monopole or topological charge in gauge theories with adjoint scalars also appear in the center vortex-nexus picture, but with a different physical interpretation. We find a new type of nexus which can carry topological charge by linking to vortices or carry d=3 Chern-Simons number without center vortices present; the Chern-Simons number is connected with twisting and writhing of field lines, as the author had suggested earlier. In general, no topological charge in d=4 arises from these specific static configurations, since the charge is the difference of two (equal) Chern-Simons number, but it can arise through dynamic reconnection processes. We complete earlier vortex-nexus work to show explicitly how to express globally-integral topological charge as composed of essentially independent units of charge 1/N.Comment: Revtex4; 3 .eps figures; 18 page

Nonlinear coupled Alfv\'{e}n and gravitational waves

In this paper we consider nonlinear interaction between gravitational and electromagnetic waves in a strongly magnetized plasma. More specifically, we investigate the propagation of gravitational waves with the direction of propagation perpendicular to a background magnetic field, and the coupling to compressional Alfv\'{e}n waves. The gravitational waves are considered in the high frequency limit and the plasma is modelled by a multifluid description. We make a self-consistent, weakly nonlinear analysis of the Einstein-Maxwell system and derive a wave equation for the coupled gravitational and electromagnetic wave modes. A WKB-approximation is then applied and as a result we obtain the nonlinear Schr\"{o}dinger equation for the slowly varying wave amplitudes. The analysis is extended to 3D wave pulses, and we discuss the applications to radiation generated from pulsar binary mergers. It turns out that the electromagnetic radiation from a binary merger should experience a focusing effect, that in principle could be detected.Comment: 20 pages, revtex4, accepted in PR