Nonlinear Gauge Transformations and Exact Solutions of the Doebner-Goldin Equation

Invariants of nonlinear gauge transformations of a family of nonlinear Schr\"odinger equations proposed by Doebner and Goldin are used to characterize the behaviour of exact solutions of these equations.Comment: 12 pages, LaTeX, to appear in "Nonlinear, Deformed and Irreversible Quantum Systems", Proceedings of an International Symposium on Mathematical Physics, World Scientific, Singapore 199

Unsteady undular bores in fully nonlinear shallow-water theory

We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for ``depth ratios'' across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9 figure

Gauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schr\"odinger Equations

Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically-motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schr\"odinger equations. The general equation in this family, including terms considered by Kostin, by Bialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent coefficients, can be obtained from the linear Schr\"odinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time-evolutions in quantum mechanics.Comment: RevTeX, 20 pages, no figures. also available on http://www.pt.tu-clausthal.de/preprints/asi-tpa/021-96.htm

Relative Periodic Solutions of the Complex Ginzburg-Landau Equation

A method of finding relative periodic orbits for differential equations with continuous symmetries is described and its utility demonstrated by computing relative periodic solutions for the one-dimensional complex Ginzburg-Landau equation (CGLE) with periodic boundary conditions. A relative periodic solution is a solution that is periodic in time, up to a transformation by an element of the equation's symmetry group. With the method used, relative periodic solutions are represented by a space-time Fourier series modified to include the symmetry group element and are sought as solutions to a system of nonlinear algebraic equations for the Fourier coefficients, group element, and time period. The 77 relative periodic solutions found for the CGLE exhibit a wide variety of temporal dynamics, with the sum of their positive Lyapunov exponents varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8. Preliminary work indicates that weighted averages over the collection of relative periodic solutions accurately approximate the value of several functionals on typical trajectories.Comment: 32 pages, 12 figure