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...
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
Recent experimental results of high gain C VI x‐ray lasing in rapidly recombining plasmas are described. 7 μm diameter carbon fiber targets of up to 5 mm length were irradiated at intensities between 3×1015∼1&...
Recent experimental results of high gain C VI x‐ray lasing in rapidly recombining plasmas are described. 7 μm diameter carbon fiber targets of up to 5 mm length were irradiated at intensities between 3×1015∼1×1016 W/cm2 by a 2 ps, 20 TW chirped pulse amplification (CPA) beam from the VULCAN Nd‐glass laser (λ=1.053 μm) at RAL. The gain length product on the 18.2 nm Balmer α transition of C VI ions was measured to be 6.5±1. The ratio of intensities of resonance lines of H‐like and He‐like ions in the rapidly recombining plasmas was used as a useful diagnostic of initial conditions for high gain operation of the C VI recombination x‐ray lasing.
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.
There exist many comets with near-parabolic orbits in the Solar System. Among various theories proposed to explain their origin, the Oort cloud hypothesis seems to be the most reasonable (Oort, 1950). The theory assum...
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There exist many comets with near-parabolic orbits in the Solar System. Among various theories proposed to explain their origin, the Oort cloud hypothesis seems to be the most reasonable (Oort, 1950). The theory assumes that there is a cometary cloud at a distance 10(3) - 10(5) AU from the Sun and that perturbing forces from planets or stars make orbits of some of these comets become of near-parabolic type. Concerning the evolution of these orbits under planetary perturbations, we can raise the question: Will they say in the Solar System forever or will they escape from it? This is an attractive dynamical problem. If we go ahead by directly solving the dynamical differential equations, we may encounter the difficulty of long-time computation. For the orbits of these comets are near-parabolic and their periods are too long to study on their long-term evolution. With mapping approaches the difficulty will be overcome. In another aspect, the study of this model has special meaning for chaotic dynamics. We know that in the neighbourhood of any separatrix i.e. the trajectory with zero frequency of the unperturbed motion of an Hamiltonian system, some chaotic motions have to be expected. Actually, the simplest example of separatrix is the parabolic trajectory of the two body problem which separates the bounded and unbounded motion. From this point of view, the dynamical study on near-parabolic motion is very important. Petrosky's elegant but more abstract deduction gives a Kepler mapping which describes the dynamics of the cometary motion (Petrosky, 1988). In this paper we derive a similar mapping directly and discuss its dynamical characters.
We report results showing that spatially periodic Bernstein-Greene-Kruskal (BGK) waves, which are exact nonlinear traveling wave solutions of the Vlasov-Maxwell equations for collisionless plasmas, satisfy a nonlinear...
We report results showing that spatially periodic Bernstein-Greene-Kruskal (BGK) waves, which are exact nonlinear traveling wave solutions of the Vlasov-Maxwell equations for collisionless plasmas, satisfy a nonlinear principle of superposition in the small-amplitude limit. For an electric potential consisting of N traveling waves, cphi(x,t)= Ji=1Ncphi(i)(x-νit), where νi is the velocity of the ith wave and each wave amplitude cphi(i) is of order ε which is small, we first derive a set of quantities scrĒ(i)(x,u,t) which are invariants through first order in ε for charged particle motion in this N-wave field. We then use these functions scrĒ(i)(x,u,t) to construct smooth distribution functions for a multispecies plasma which satisfy the Vlasov equation through first order in ε uniformly over the entire x-u phase plane for all time. By integrating these distribution functions to obtain the charge and current densities, we also demonstrate that the Poisson and Ampère equations are satisfied to within errors that are O(ε3/2). Thus the constructed distribution functions and corresponding field describe a self-consistent superimposed N-wave solution that is accurate through first order in ε. The entire analysis explicates the notion of small-amplitude multiple-wave BGK states which, as recent numerical calculations suggest, is crucial in the proper description of the time-asymptotic state of a plasma in which a large-amplitude electrostatic wave undergoes nonlinear Landau damping.
The structural properties of the electric double layer surrounding an infinitely long isolated cylindrical polyelectrolyte are studied in the modified Poisson-Boltzmann theory. The mean electrostatic potential and the...
The structural properties of the electric double layer surrounding an infinitely long isolated cylindrical polyelectrolyte are studied in the modified Poisson-Boltzmann theory. The mean electrostatic potential and the singlet density distributions are obtained for a 2:2, 1:2 and 2:1 restricted primitive model electrolyte and compared with those from the classical Poisson-Boltzmann theory and available results from the hypernetted chain/mean spherical approximation theory.
We consider a family of three-dimensional, volume preserving maps depending on a small parameter epsilon. As epsilon --> 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for sma...
We consider a family of three-dimensional, volume preserving maps depending on a small parameter epsilon. As epsilon --> 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for small epsilon the heteroclinic connection breaks up and that the splitting between its components scales with epsilon like epsilon(gamma) exp(-beta/epsilon). We estimate beta using the singularities of the epsilon --> 0+ heteroclinic orbit in the complex plane. We then estimate gamma using linearization about orbits in the complex plane. While these estimates are not proven, they are well supported by our numerical calculations. The work described here is a special case of the theory derived by Amick et al. which applies to q-dimensional volume preserving mappings.
A new algorithm based on spectral element discretization and non-oscillatory ideas is developed for the solution of hyperbolic partial differential equations. A conservative formulation is proposed based on cell avera...
A new algorithm based on spectral element discretization and non-oscillatory ideas is developed for the solution of hyperbolic partial differential equations. A conservative formulation is proposed based on cell averaging and reconstruction procedures, that employs a staggered grid of Gauss-Chebyshev and Gauss-Lobatto Chebyshev discretizations. The non-oscillatory reconstruction procedure is based on ideas similar to those proposed by Cai et al. ( Math. Comput. 52, 389 (1989)) but employs a modified technique which is more robust and simpler in terms of determining the location and strength of a discontinuity. It is demonstrated through model problems of linear advection, inviscid Burgers equation, and one-dimensional Euler system that the proposed algorithm leads to stable, non-oscillatory accurate results. Exponential accuracy away from the discontinuity is realized for the inviscid Burgers equation example.
1 Introduction We want to investigate the following boundary value problem:where m>1, n>1, p≥1 and m>p. As for a, the following properties will be assumed: (A.1) a(r)∈C^1 ([0, ∞)) and a(r)&...
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1 Introduction We want to investigate the following boundary value problem:where m>1, n>1, p≥1 and m>p. As for a, the following properties will be assumed: (A.1) a(r)∈C^1 ([0, ∞)) and a(r)>0 for any r∈(0, ∞); (A.2) there exists α>0 such that (r-α)a(r)≥0 for any r∈[0, ∞). The initial value problem corresponding to (1.1) is the following Cauchy problem:
Results of a numerical study of the dynamics of a collection of disks colliding inelastically in a periodic two-dimensional enclosure are presented. The properties of this system, which is perhaps the simplest model f...
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