Nonlinear Interaction of <i>N</i> Conservative Waves in Two Dimensions

Abstract

Kinematic Fourier (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 5 N parameters is considered.