Deterministic Chaos of Exponential Oscillons and Pulsons

An exact 3-D solution for deterministic chaos of J wave groups with M internal waves governed by the Navier-Stokes equations is presented. Using the Helmholtz decomposition, the Dirichlet problem for the Navier-Stokes equations is decomposed into the Archimedean, Stokes, and Navier problems. The exact solution is derived by the method of decomposition in invariant structures (DIS). A cascade differential algebra is developed for four families of invariant structures: deterministic scalar kinematic (DSK) structures, deterministic vector kinematic (DVK) structures, deterministic scalar dynamic (DSD) structures, and deterministic vector dynamic (DVD) structures. The Helmholtz decomposition of anticommutators, commutators, and directional derivatives is computed in terms of the dot and cross products of the DVK structures. Computation is performed with the help of the experimental and theoretical programming in Maple. Scalar and vector variables of the Stokes problem are decomposed into the DSK and DVK structures, respectively. Scalar and vector variables of the Navier problem are expanded into the DSD and DVD structures, correspondingly. Potentialization of the Navier field is possible since internal vortex forces, which are described by the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, external potential forces, which are expressed via the scalar potentials of the Helmholtz decomposition, superpose together to form the gradient of a dynamic pressure. Various constituents of the kinetic energy and the total pressure are visualized by the conservative, multi-wave propagation and interaction of three-dimensional, nonlinear, internal waves with a two-fold topology, which are called oscillons and pulsons.


Introduction
Conservative interaction of N three-dimensional internal waves controlled by the Navier-Stokes equations has been studied in [1], where the existence theorem is proved for a partial solution of the correspondent boundary-value problem via the Stationary Kinematic Euler-Fourier (SKEF) functions. However, an extreme sophistication of the partial solution derived with the help of experimental and theoretical programming in Maple doesn't permit development of a general solution for propagation and interaction of wave groups of internal waves.
To overcome this challenge, a structural approach to this problem has been developed in [2], where the differential algebra of SKEF structures is studied and   [1] and later developed in [2] as the SKEF structures. In a mod-

Scalar Kinematic Structures
a decay parameter 1 σ = (7) and, in a lower domain, It is a straightforward matter to show completeness of the DSK structures (1) with respect to differentiation in (x, y, z) of any order The first derivatives of the DSK structures [ ] , , , m m m m a b c d in (x, y) are covariant since they are proportional to costructures in the xand y-directions, respectively. The first derivatives of the DSK structures with respect to z are invariant because they are proportional to themselves.
The first spatial derivatives of the DSK structure , Connections between the theoretical DSK structures and the experimental DSK structures (1) and the corresponding values of the sign parameters i α and i β are 1, 1, , 1, , 1, , , The    represent a floating notation of the vertices of the differentiation diagram. The differentiation table (12) shows equivalence of the theoretical DSK structures with respect to the spatial differentiation of the first order since each of the theoretical DSK structures may be used to obtain the experimental derivatives (11 The differentiation diagram immediately explains invariance of the repeated second spatial derivatives. The second-order differentiation moves the DSK structure , where the gradients are calculated in accordance with (11).     Comparison of the definitions of the theoretical DVK structures (22) with the definition of the experimental DVK structures (20) again reveals the quadrality of theoretical formulas. The quadrality of the theoretical DVK structures is also confirmed by the tables of the divergences, the curls, the first spatial derivatives, the second spatial derivatives, the Laplacians and the first temporal derivatives. For the purpose of conciseness, further theoretical results will be shown mostly for the theoretical DVK structure , i m s that is sufficient for explanation of experimental results.
Calculation of the divergence of the theoretical DVK structure , shows that the theoretical and experimental DVK structures are divergence-free due to the Pythagorean identity (4).
Similarly, we find the curl of , i m s as follows: Thus, the DVK structures are curl-free, as well.
Computation of a differentiation table of the first spatial derivatives of the theoretical DVK structures yields In agreement with (25) and (12), the differentiation diagram of the DVK structures is identical to that of the DSK structures shown in Figure 1 because the differentiation tables of the DVK and DSK structures coincide up to the substitution , , , for all i and m. This property of the invariant DVK and DSK structures may be called a scalar-vector structural invariance. As we will find out later, the scalar-vector structural invariance also holds for the second spatial derivatives, the Laplacians, and the first temporal derivatives.  Harmonicity of the DVK structures may be readily obtained by summation of the repeated second spatial derivatives (26) and the parametric identity (4) as The DVK structures are closed regarding the spatial differentiation of any order due to (25). Completeness of the DVK structures with respect to temporal differentiation follows from Again, the first temporal derivative of the DVK structures ,
Expansion of (31) for all , , , i j m n and usage of (13) yield completeness of the experimental DSD structures with respect to spatial differentiation of any order.
The first derivative of the DSD structure , , i m j n s s with respect to z is invariant and with respect to (x, y) is covariant.
We then take second derivatives of (31) to obtain the repeated second derivatives of the theoretical DSD structures and the following mixed second derivatives of the theoretical DSD structures: Thus, Equations (32) demonstrate the invariance of the repeated second derivatives of the DSD structures in z and a partial invariance of the repeated second derivatives in (x, y). Equations (33) show the covariance of the mixed second derivatives in (x, y) and a partial covariance of the mixed second derivatives in (x, z) and (y, z).
Summation of the repeated second derivatives (32) establishes anharmonicity of the DSD structures since where f and g are the gradients of f and g, respectively, , , , , , .
Substitution of the theoretical DSK structures and simplification by (12) and Equation (12) where the scalar product of the DVK structures          ,   2  2  2  2  2  , , ,

Vector Dynamic Structures
after substituting (12) (  )   2  2  2  2  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  2  2   2  2  2  2  ,  ,  ,  ,  ,  ,  ,  ,   2  2  2  ,  ,  , , , , We substitute (12)   Third, the component definition of the cross product of two gradients f and g (37) yields We then find the following definition of the curl of the cross product of f and g , , x z x y x z y x y y x y Usage of the harmonicity conditions for f and g , Substitution of the theoretical DSK structures Eventually, we add to (56) and subtract from (56) the relevant directional derivatives of (45) to get

The Stokes Field
Internal waves of a Newtonian fluid with a constant density ρ and a constant kinematic viscosity ν in a field of gravity , , are governed by the momentum conservation law and the mass conservation law [3] 0.
An admissible decomposition of the boundary function in DSK boundary structures will be considered in Section 6 later. The conditions at infinity are satisfied due to (7) and (9). Configuration of the upper and lower domains of internal waves, which are generated through the boundary conditions (66)-(67) by surface waves propagating in a generation domain, is shown in Figure 2.
From viewpoint of the Fundamental Theorem of Vector Analysis [4] the Dirichlet problem (66)-(67) for the Navier-Stokes Equations (64)-(65) may be treated as a construction of the Helmholtz decomposition for the Archimedean field A F , the Stokes field S F , and the Navier field N F : The Archimedean, Stokes, and Navier fields are decomposed using the correspondent scalar potentials , h k p p and d p as follows: , , .
where h p stands in fluid dynamics for the hydrostatic pressure, k p for the kinematic pressure, and d p for the dynamic pressure. Summation of (68)-(71) yields the Helmholtz representation of the momentum conservation law (64) , , where t p , which is termed in fluid dynamics a total pressure, is a scalar potential of the superposition of the Archimedean, Stokes, and Navier fields. The problem of finding the scalar potential h p of the Archimedean field A F has a general solution [3]   In agreement with the Cauchy integral, we construct the field of kinematic pressure k p via general terms in the following form: The directional derivative ( ) u that describes propagation of the ith wave group in view of (103) may be written using the rectangular and triangular summations through the DVK structures as follows: , , , A non-orthogonal distribution of the kinetic energy between the wave groups, which is obtained using the triangular summation with respect to wave groups, has the following form:   [7], the solitary waves generated by crossed electric and magnetic fields [8], and the pulsatory waves of the Korteweg-de Vries equation [9], American Journal of Computational Mathematics which are exact nonlinear solutions for propagation and conservative interaction of aperiodic one-dimensional (1-D) waves. In view of (118), the corresponding solutions for the dynamic pressure d p and its propagation constituent are given by exponential pulsons of depression since they are strictly negative for all (x, y, z, t).

Conclusions
The most interesting properties of the scalar and vector kinematic structures that are studied in Section 2 and Section 3, respectively, are the scalar and vector structural oscillations, the scalar-vector duality, the quadrality of the theoretical DSK and DVK structures, and the equiprobability of the experimental DSK and DVK structures. The differentiation diagram in Figure 1 is actually an image of a mathematical two-dimensional dice, where all vertices have exactly the same probability despite the various scales of differentiation in the xand y-directions.
The success of the scalar and vector dynamic structures treated in Section 4 and Section 5, correspondingly, is explained by their exact similarity to the Navier-Stokes Equations (64)-(65) as the momentum conservation law (64) represents a superposition of kinematic terms and dynamic terms with an algebraic nonlinearity. Further attractive features of the DSD and DVD structures are the absence of a formal restriction on the functional amplitudes, the structural amplitudes, and the quartet weights and the global convergence the DSD and DVD structures for all (x, y, z, t). For instance, the general solution of fluid dynamics in the Boussinesq-Rayleigh series converges only locally and the radius of convergence diminishes with the Reynolds number [11].
The large number of the experimental DSD and DVD structures, 16 and 32, respectively, is required for completeness of the expansions of the directional derivative of the global velocity and the gradient of the total pressure since otherwise the vortical forces produced by the vector potential (63) of the Helmholtz decomposition may not compensate each other. Therefore, the experimental and theoretical programming in Maple that facilitates computation and verification of the numerous large arrays of scalar and vector terms is essential for development of the general solution for J wave groups with M internal waves. Since the exact general solution is not affected by viscous dissipation it may serve as the 3-D model of conservative propagation and interaction of kinetic energy of internal waves in ocean and atmosphere via the exponential oscillons and pulsons visualized in Figure 3 and Figure 4.
Research in the functional amplitudes of the DSK structures started in [2] has revealed existence of three types of wave lattices: 1) slanted up and down, 2) stepped up and down with rectangular steps, and 3) stepped up and down with smooth steps. It is interesting to explore the effect of the wave parameters on the Eulerian and Lagrangian properties of the DSD structures, as well. By the well-known conjecture [12], chaos of physical systems is explained by superpos-