On Darboux transformations for the derivative nonlinear Schr\"odinger equation

We consider Darboux transformations for the derivative nonlinear Schr\"odinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the derivative nonlinear Schr\"odinger equation are given as explicit examples.Comment: 14 page

Analyses of mean and turbulent motion in the tropics with the use of unequally spaced data

Wind velocities from 25 km to 60 km over Ascension Island, Fort Sherman and Kwajalein for the period January 1970 to December 1971 are analyzed in order to achieve a better understanding of the mean flow, the eddy kinetic energy and the Eulerian time spectra of the eddy kinetic energy. Since the data are unequally spaced in time, techniques of one-dimensional covariance theory were utilized and an unequally spaced time series analysis was accomplished. The theoretical equations for two-dimensional analysis or wavenumber frequency analysis of unequally spaced data were developed. Analysis of the turbulent winds and the average seasonal variance and eddy kinetic energy of the turbulent winds indicated that maximum total variance and energy is associated with the east-west velocity component. This is particularly true for long period seasonal waves which dominate the total energy spectrum. Additionally, there is an energy shift for the east-west component into the longer period waves with altitude increasing from 30 km to 50 km

Yang-Baxter Maps from the Discrete BKP Equation

We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation

On solutions to the non-Abelian Hirota-Miwa equation and its continuum limits

In this paper, we construct grammian-like quasideterminant solutions of a non-Abelian Hirota-Miwa equation. Through continuum limits of this non-Abelian Hirota-Miwa equation and its quasideterminant solutions, we construct a cascade of noncommutative differential-difference equations ending with the noncommutative KP equation. For each of these systems the quasideterminant solutions are constructed as well.Comment: 9 pages, 1 figur

Darboux dressing and undressing for the ultradiscrete KdV equation

We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We then show how to reconstruct the potential in the scattering problem at any time, using an ultradiscrete analogue of a Darboux transformation. This is achieved by obtaining data uniquely characterising the soliton content and the `background' from the initial potential by Darboux transformation.Comment: 41 pages, 5 figures // Full, unabridged version, including two appendice

Darboux and binary Darboux transformations for discrete integrable systems 1. Discrete potential KdV equation

The Hirota-Miwa equation can be written in `nonlinear' form in two ways: the discrete KP equation and, by using a compatible continuous variable, the discrete potential KP equation. For both systems, we consider the Darboux and binary Darboux transformations, expressed in terms of the continuous variable, and obtain exact solutions in Wronskian and Grammian form. We discuss reductions of both systems to the discrete KdV and discrete potential KdV equations, respectively, and exploit this connection to find the Darboux and binary Darboux transformations and exact solutions of these equations