Nonlinear Superposition Law Research Articles

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov–Milne–Pinney equation and associated first integral. The special relativistic extension of the Ray–Reid system and invariant is obtained. General properties of the relativistic Ermakov–Milne–Pinney are analyzed. The conservative case of the relativistic Ermakov–Milne–Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered to be well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov–Milne–Pinney equation has additional nonlinearities, due to the relativistic effects.

Nonlinear gravitational waves and their polarization

Vacuum gravitational fields invariant for a non-Abelian Lie algebra generated by two Killing fields whose commutator is light-like are analyzed. It is shown that they represent non-linear gravitational waves obeying to two non-linear superposition laws. The energy and the polarization of this family of waves are explicitly evaluated.

An integrable model in general relativity

It is shown that gravitational fields invariant for a non Abelian 2-dimensional Lie algebra of Killing fields are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order partial differential equation (the Laplace equation or the Darboux equation) on the plane. Those determined via Laplace or Darboux equations are exact nonlinear gravitational waves obeying to two nonlinear superposition laws.

Equivalence among the propagators of three-dimensional time-dependent quadratic systems and free particles, by solving the Schrödinger equation

In this article we evaluate Feynman’s propagator exactly for a threedimensional time-dependent quadratic Lagrangian, by solving the Schrödinger equation. Through a rotation and a non-linear superposition law of coordinates, we show that such a propagator can be obtained from thefree- particle propagator in a new space-time coordinate system.

Second constant of motion for generalized Ermakov systems

We derive a second invariant for generalized Ermakov-Ray-Reid systems. This invariant and the one previously known form a non-linear system of equations that can be solved for the velocities. The results are general solutions to one of the equations. This exhibits openly the nonlinear superposition law.

Quantum-mechanical Green’s function and nonlinear superposition law

We derive the quantum-mechanical Green’s function for the problem of a time-dependent variable mass particle subject to a time-dependent forced harmonic-oscillator potential by taking direct recourse of the corresponding Schrödinger equation. Through the usage of the nonlinear superposition law of Ray and Reid, we show that such a Green’s function can be obtained from that for the problem of a particle with unit (constant) mass subject to either a forced harmonic potential with constant frequency or only to a time-dependent linear field.

Nonlinear superposition law and Feynman propagator

We demonstrate the usefulness of the nonlinear superposition law of Ray and Reid in connection with the Feynman propagator. We derive the propagator for the problem of a time-dependent variable mass particle subject to a harmonic potential with a time-dependent frequency by taking direct recourse to the corresponding Schrödinger equation.

Superposition of solutions to Bäcklund transformations for the SU(n) principal σ-model

We show that the Bäcklund transformations for the SU(n) principal σ-model may be linearized using a geometrical interpretation of these equations involving the minimal orbit of SU(n, n) in the Grassmann manifold Gn (C2n). Linearization puts the equations in Zakharov–Mikhailov–Shabat (ZMS) form. Using this form of the equations, we prove inductively a nonlinear superposition law and a permutability theorem for iterated Bäcklund transformations analogous to known results in the theory of the sine–Gordon and KdV equations. From the superposition law we get an explicit form for multisoliton solutions to the σ-model.

Homogeneous quadratic invariants for one-dimensional classical homogeneous quadratic Hamiltonians: Some properties of the auxiliary equation

Using a previously derived characterization of the homogeneous quadratic invariants for an arbitrary one-dimensional classical homogeneous quadratic Hamiltonian we obtain various ‘‘nonlinear superposition laws.’’ We establish a close connection between one of these (the quadratic superposition law) and the original characterization.

Invariants for forced time-dependent oscillators and generalizations

Several methods for deriving invariants for time-dependent systems have been developed. Four of these methods are as follows: (1) dynamical algebra, (2) Noether's theorem, (3) transformation group, and (4) Ermakov's method. These four methods are related by their use in deriving invariants for the forced, time-dependent oscillator. For this system the methods give the same results. We also discuss forced, nonlinear oscillators possessing analogous invariants. The nonlinear superposition laws for the forced oscillator are derived and discussed.

A generalization of the nonlinear superposition idea for Ermakov systems

The “nonlinear superposition law” for systems of the Ermakov-type is generalized to systems of n + 1 second-order differential equations having n invariants, quadratic in the velocities.

Nonlinear superposition, higher-order nonlinear equations, and classical linear invariants

We find a class of nonlinear ordinary differential equations, in ρ, of any order m?2 whose solutions are given by the nonlinear superposition law ρ(t) = x(t)r(τ), dτ = μ(t)dt, where dmr/dτm = F(r, r ′,...) and x(t) satisfies a special self-adjoint linear equation of order m. The coefficients of the self-adjoint equation in x, which are identical to those of the nonlinear equations in ρ, can be deduced from the well known invariants of the classical theory of linear differential equations.

Classification scheme for two-dimensional Ermakov-type systems and generalizations

We describe a classification scheme for certain pairs of coupled, nonlinear second-order ordinary differential equations, having a first integral which can be obtained by an Ermakov-type elimination procedure. The classification produces a hierarchy of systems having different levels of generality. Different cases within the same level are distinguished by the integrability of a certain one-form, which is related to a nonlinear superposition law between the two equations. The study of these one-forms and the classification are made systematic by using the general reduction theorem of Pfaffian forms. The paper contains illustrations of the way previous examples fit into the classification scheme, and a discussion of its generalization to higher dimensions.

Ermakov systems, velocity dependent potentials, and nonlinear superposition

We derive new Ermakov systems with velocity-dependent potentials. The extended Ermakov system presented contains all known one-dimensional cases and many new systems. These Ermakov systems lead to a nonlinear superposition law for the solutions.

Noether’s theorem and Ermakov systems for nonlinear equations of motion

We present a simple but important generalization of the use of Noether’s theorem to generate invariants for time-dependent nonlinear equations of motion. Our main result is that Noether’s theorem produces invariants even when the equation of motion contains arbitrary functions. The results obtained by using Noether’s theorem are the most general results for which one obtains explicit nonlinear superposition laws for the class of equations of motion considered. We also correct a recent paper on this same subject.

Nonlinear superposition law for generalized Ermakov systems

We discuss a generalized Ermakov system for which a general nonlinear superposition law exists. We give an explicit example of the use of the new superposition law.

Ermakov systems, nonlinear superposition, and solutions of nonlinear equations of motion

We report several important additions to our original discussion of Ermakov systems. First, we show how to derive the Ermakov system from more general equations of motion. Second, we show that there is a general nonlinear superposition law for Ermakov systems. Also, we give explicit examples of the nonlinear superposition law. Finally, we point out that any ordinary differential equation can be included in many Ermakov systems.