Spherical aberration and its correction using Lie algebraic methods

We discuss three methods to correct spherical aberration for a point to point imaging system. First, results obtained using Fermat's principle and the ray tracing method are described briefly. Next, we obtain solutions using Lie algebraic techniques. Even though one cannot always obtain analytical results using this method, it is often more powerful than the first method. The result obtained with this approach is compared and found to agree with the exact result of the first method.

Application of dynamical Lie algebraic method to atom-diatomic molecule scattering

The dynamical Lie algebraic approach developed by Alhassid and Levine combined with intermediate picture is applied to the study of translational-vibrational energy transfer in the collinear collision between an atom and an anharmonic oscillator. We find that the presence of the anharmonic terms indeed has an effect on the vibrational probabilities of the oscillator. The computed probabilities are in good agreement with those obtained using exact quantum method. It is shown that the approach of dynamical Lie algebra combining with intermediate picture is reasonable in the treating of atom-anharmonic oscillator scattering.

Affine Lie algebraic origin of constrained KP hierarchies

An affine sl(n + 1) algebraic construction of the basic constrained KP hierarchy is presented. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and it is shown that these approaches are equivalent. The model is recognized to be the generalized non-linear Schrödinger (GNLS) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-Bäcklund transformations and interpolate between GNLS and multi-boson KP-Toda hierarchies. Our construction uncovers the origin of the Toda lattice structure behind the latter hierarchy. © 1995 American Institute of Physics.

Álgebras de Lie e aplicações à sistemas alternantes

Pós-graduação em Matemática - IBILCE

复杂条件地层滤波预测的理论与关键技术研究

The theory researches of prediction about stratigraphic filtering in complex condition are carried out, and three key techniques are put forward in this dissertation. Theoretical aspects: The prediction equations for both slant incidence in horizontally layered medium and that in laterally variant velocity medium are expressed appropriately. Solving the equations, the linear prediction operator of overlaid layers, then corresponding reflection/transmission operators, can be obtained. The properties of linear prediction operator are elucidated followed by putting forward the event model for generalized Goupillaud layers. Key technique 1: Spectral factorization is introduced to solve the prediction equations in complex condition and numerical results are illustrated. Key technique 2: So-called large-step wavefield extrapolation of one-way wave under laterally variant velocity circumstance is studied. Based on Lie algebraic integral and structure preserving algorithm, large-step wavefield depth extrapolation scheme is set forth. In this method, the complex phase of wavefield extrapolation operator’s symbol is expressed as a linear combination of wavenumbers with the coefficients of this linear combination in the form of the integral of interval velocity and its derivatives over depth. The exponential transform of the complex phase is implemented through phase shifting, BCH splitting and orthogonal polynomial expansion. The results of numerical test show that large-step scheme takes on a great number of advantages as low accumulating error, cheapness, well adaptability to laterally variant velocity, small dispersive, etc. Key technique 3: Utilizing large-step wavefield extrapolation scheme and based on the idea of local harmonic decomposition, the technique generating angle gathers for 2D case is generalized to 3D case so as to solve the problems generating and storing 3D prestack angle gathers. Shot domain parallel scheme is adopted by which main duty for servant-nodes is to compute trigonometric expansion coefficients, while that for host-node is to reclaim them with which object-oriented angle gathers yield. In theoretical research, many efforts have been made in probing into the traits of uncertainties within macro-dynamic procedures.

T-duality in affine NA Toda models

The construction of non-Abelian affine Toda models is discussed in terms of its underlying Lie algebraic structure. It is shown that a subclass of such non-conformal two-dimensional integrable models naturally leads to the construction of a pair of actions, which share the same spectra and are related by canonical transformations.

T-duality of axial and vector dyonic integrable models

A general construction of affine nonabelian (NA)-Toda models in terms of the axial and vector gauged two loop WZNW model is discussed. They represent integrable perturbations of the conformal sigma -models (with tachyons included) describing (charged) black hole type string backgrounds. We study the off-critical T-duality between certain families of axial and vector type integrable models for the case of affine NA-Toda theories with one global U(1) symmetry. In particular we find the Lie algebraic condition defining a subclass of T-selfdual torsionless NA-Toda models and their zero curvature representation. (C) 2001 Academic Press.

Nonautonomous mixed mKdV-sinh-Gordon hierarchy

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Negative even grade mKdV hierarchy and its soliton solutions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Estrutura algébrica de hierarquias integráveis e problemas de valor de contorno

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)