Wave Turbulence of Exponential Oscillons and Pulsons

Abstract

This paper represents a novel approach to wave turbulence, which may be called exact wave turbulence, since it is based on an exact solution for nonlinear (both resonant and nonresonant) interactions of turbulent waves, which are governed by the nonstationary Navier-Stokes equations in three dimensions. Exact wave turbulence is aimed to complement the well-known theories of wave turbulence: statistical wave turbulence and resonance wave turbulence. Presented results are focused on derivation and justification of the exact solution, which may be later used to explore the Eulerian, Lagrangian, and statistical properties of wave turbulence. The computed exact solution for wave turbulence generalizes the exact solutions for deterministic chaos of exponential oscillons and pulsons and for stochastic chaos of random exponential oscillons and pulsons, which have been developed with the help of the method of decomposition in invariant structures. For the aim of completeness, experimental and theoretical, deterministic, random, and time-complementary, scalar and vector, kinematic structures are briefly discussed. We also define and study experimental and theoretical, deterministic-deterministic, deterministic-random, random-deterministic, and random-random, scalar dynamic structures together with experimental and theoretical, deterministic-deterministic, deterministic-random, random-deterministic, and random-random, vector dynamic structures of the m th and n th families. The Helmholtz decomposition is used to expand the Dirichlet problems for the turbulent Navier-Stokes equations into the Archimedean, the turbulent Stokes, and the turbulent Navier problems. The kinematic structures are used to find solutions to the deterministic, random, and turbulent Stokes problems, which include the Dirichlet boundary conditions and conditions at infinities. The dynamic structures are employed to compute necessary and sufficient conditions of existence of the exact solution for wave turbulence of exponential oscillons and pulsons with the help of experimental and theoretical programming in Maple. The cumulative pressure field of the turbulent Navier-Stokes problem is derived in the scalar, kinematic, and dynamic structures, as well. Concluding remarks deal with the most interesting properties of the invariant structures and the exact solution and briefly review open problems.