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(KF) structures, exponential kinematic Fourier (KEF) structures, dynamic
exponential (DEF) Fourier structures, and KEF-DEF structures with constant and
space-dependent structural coefficients are developed in the current paper to
treat kinematic and dynamic problems for nonlinear interaction of N conservative waves in the
two-dimensional theory of the Newtonian flows with harmonic velocity. The
computational method of solving partial differential equations (PDEs) by
decomposition in invariant structures, which continues the analytical methods
of undetermined coefficients and separation of variables, is extended by using
an experimental and theoretical computation in Maple?. For internal waves
vanishing at infinity, the Dirichlet problem is formulated for kinematic and
dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and
Bernoulli equations in the upper and lower domains. Exact solutions for upper and
lower cumulative flows are discovered by the experimental computing, proved by
the theoretical computing, and verified by the system of Navier-Stokes PDEs.
The KEF and KEF-DEF structures of the cumulative flows are visualized by
instantaneous surface plots with isocurves. Modeling of a deterministic wave
chaos by aperiodic flows in the KEF, DEF, and KEF-DEF structures with 5N parameters is considered.
A new exact solution for nonlinear interaction of two pulsatory waves of
the Korteweg-de Vries (KdV) equation is computed by decomposition in an
invariant zigzag hyperbolic tangent (ZHT) structure. A computational algorithm
is developed by experimental programming with lists of equations and
expressions. The structural solution is proved by theoretical programming with
symbolic general terms. Convergence, tolerance, and summation of the ZHT
structural approximation are discussed. When a reference level vanishes, the
two-wave solution is reduced to the two-soliton solution of the KdV equation.
Kinematic exponential Fourier (KEF) structures, dynamic exponential (DEF)
Fourier structures, and KEF-DEF structures with time-dependent structural
coefficients are developed to examine kinematic and dynamic problems for a
deterministic chaos of N stochastic
waves in the two-dimensional theory of the Newtonian flows with harmonic velocity.
The Dirichlet problems are formulated for kinematic and dynamics systems of the
vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in
the upper and lower domains for stochastic waves vanishing at infinity.
Development of the novel method of solving partial differential equations
through decomposition in invariant structures is resumed by using experimental
and theoretical computation in Maple?. This computational method generalizes
the analytical methods of separation of variables and undetermined
coefficients. Exact solutions for the deterministic chaos of upper and lower
cumulative flows are revealed by experimental computing, proved by theoretical
computing, and justified by the system of Navier-Stokes PDEs. Various scenarios
of a developed wave chaos are modeled by 3N parameters and 2N boundary
functions, which exhibit stochastic behavior.