Now showing 1 - 2 of 2
  • Publication
    Quadratic invariants for discrete clusters of weakly interacting waves
    We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix A with entries 1, −1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ≡ N−M∗ N−M, where M∗ is the number of linearly independent rows in A. Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M∗ < M. We illustrate our findings by presenting examples from the Charney–Hasegawa–Mima wave model, and by showing a classification of small (up to three-triad) clusters.
    Scopus© Citations 14  272
  • Publication
    Derivation of the Biot-Savart equation from the Nonlinear Schrödinger equation
    (American Physical Society, 2015-11-25) ;
    We present a systematic derivation of the Biot-Savart equation from the nonlinear Schrödinger equation, in the limit when the curvature radius of vortex lines and the intervortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines, [equation not represented here], with cutoff length [equation not represented here], where ρ0 is the background condensate density far from the vortex lines and κ is the quantum of circulation.
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