Theoretical Quantization of Exact Wave Turbulence in Exponential Oscillons and Pulsons

Abstract

In a preceding paper, the theoretical and experimental, deterministic and random, scalar and vector, kinematic structures, the theoretical and experimental, deterministic-deterministic, deterministic-random, random-deterministic, random-random, scalar and vector, dynamic structures have been developed to compute the exact solution for wave turbulence of exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. The rectangular, diagonal, and triangular summations of matrices of the turbulent kinetic energy and general terms of numerous sums have been used in the current paper to develop theoretical quantization of the kinetic energy of exact wave turbulence. Nested structures of a cumulative energy pulson, a deterministic energy pulson, a deterministic internal energy oscillon, a deterministic-random internal energy oscillon, a random internal energy oscillon, a random energy pulson, a deterministic diagonal energy oscillon, a deterministic external energy oscillon, a deterministic-random external energy oscillon, a random external energy oscillon, and a random diagonal energy oscillon have been established. In turn, the energy pulsons and oscillons include deterministic group pulsons, deterministic internal group oscillons, deterministic-random internal group oscillons, random internal group oscillons, random group pulsons, deterministic diagonal group oscillons, deterministic external group oscillons, deterministic-random external group oscillons, random external group oscillons, and random diagonal group oscillons. Sequentially, the group pulsons and oscillons contain deterministic wave pulsons, deterministic internal wave oscillons, deterministic-random internal wave oscillons, random internal wave oscillons, random wave pulsons, deterministic diagonal wave oscillons, deterministic external wave oscillons, deterministic-random external wave oscillons, random external wave oscillons, random diagonal wave oscillons. Consecutively, the wave pulsons and oscillons are composed of deterministic elementary pulsons, deterministic internal elementary oscillons, deterministic-random internal elementary oscillons, random internal elementary oscillons, random elementary pulsons, deterministic diagonal elementary oscillons, deterministic external elementary oscillons, deterministic-random external elementary oscillons, random-deterministic external elementary oscillons, random external elementary oscillons, and random diagonal elementary oscillons. Symbolic computations of exact expansions have been performed using experimental and theoretical programming in Maple.