The Nonlinear Amplitude Equation in Harmonic Phenomena 1 The Nonlinear Amplitude Equation in Harmonic Phenomena

Oscillatory motion is exhibited in physical systems in a vast range of scales allencompassing from molecular vibrations to celestial mechanics. Propagating wave phenomena are also encountered in many systems embracing man-made linear and non-linear devices used as information carriers, as well as a myriad of acoustical vibrations and electromagnetic phenomena observed in nature. The dynamic equilibrium between two forms of energy is shared by all these systems. Amplitude and phase categories provide a suitable conceptual and mathematical framework to describe these physical processes. Invariants arising from this formalism play a crucial role in the obtention of solutions and conserved quantities. The amplitude and phase representation of coordinate or disturbance variables in second order differential equations leads to an Ermakov pair of equations. Under appropriate circumstances these equations may be decoupled into a second order non linear differential equation for the amplitude and a third order nonlinear equation for the phase. Various results scattered in the literature will be presented here in a unified manner as well as several new developments in three broad areas, namely classical dynamical systems, quantum mechanics and electromagnetic propagation. Classical dynamical systems: The one dimensional time dependent harmonic oscillator is studied in detail. The Ermakov-Lewis invariant or a closely related orthogonal functions invariant is economically derived. The nonlinear superposition principle is revised. General solutions, given a particular solution, to the non linear amplitude and phase equations are presented. The analysis of these results lead to novel analytical solutions of the time dependent oscillator equation with a sub-period monotonic parameter. This procedure, as we shall demonstrate, seems promising for the development of analytical solutions with more general parameters. The indeterminacy of the amplitude and phase functions has been recently addressed and is further discussed in this paper. The relationship between exact invariants and previously known adiabatic invariants is evaluated. Quantum domain: The orthogonal functions invariant is generalized to the quantum realm. The time dependent quantized quadratic Hamiltonian gives rise to two 2 Manuel Fernández Guasti linear invariants that may be casted into a quadratic form. The time dependent Schrodinger equation with one spatial dimension is solved with this formalism. It is shown that in the time dependent case, the Ermakov-Lewis invariant plays the role that the Hamiltonian does in the time independent case. The invariant operator is expressed in terms of amplitude and phase operators. Squeezed and coherent states are simply dealt with unitary transformations that stem from the invariant form. The evolution of minimum uncertainty states are studied under an abrupt frequency or mass change. Wave propagation: The time dependent oscillator problem is shown to be formally equivalent to the electromagnetic propagation of monochromatic waves in one dimension. The nonlinear amplitude differential equation allows for an arbitrary refractive index variation. A hyperbolic tangent refractive index profile will be treated in detail. The orthogonal functions invariant translates into a conserved quantity for the 1+3 dimensional scalar wave equation. The assessed quantity and its associated flow can then be evaluated using different criteria. The important topic of phase retrieval from intensity measurements and the irradiance propagation equation is discussed. This formalism leads to radiometric quantities that differ for ultra-short pulses from the prevailing optical radiometric theory.