Exact half pulse synthesis via the inverse scattering transform

In a paper of Nielson et al. it is shown, using the linear approximation, that it might be possible to create a pair of RF-pulses, which, after summation of the unrephased signals achieve a specified transverse magnetization. Such pulses, designed using the linear approximation, show rather poor slice selectivity. Using the inverse scattering transform formalism we give an algorithm to exactly achieve a specified "summed" transverse magnetization profile. Indeed for a constant phase transverse profile, our algorithm produces infinitely many solutions.     

Minimum energy pulse synthesis via the inverse scattering transform

This paper considers a variety of problems in the design of selective RF-pulses. We apply a formula of Zakharov and Manakov to directly relate the energy of an RF-envelope to the magnetization profile and certain auxiliary parameters used in the inverse scattering transform (IST) approach to RF-pulse synthesis. This allows a determination of the minimum possible energy for a given magnetization profile. We give an algorithm to construct both the minimum energy RF-envelope as well as any other envelope that produces a given magnetization profile. This includes an algorithm for solving the Gel'fand-Levitan-Marchenko equations with bound states. The SLR method is analyzed in terms of traditional scattering data, and shown to be a special (singular) case of the IST approach. RF-envelopes are computed for a variety of examples.     

Einstein-Rosen metrics generated by the inverse scattering transform

The most general Einstein-Rosen solutions obtainable by the inverse scattering transform, using the Levi-Cività metric as seed are analyzed. They can be classified as a family of evolving metrics without a clear physical interpretation and a family representing gravitational waves absorbed and radiated by a massive cylinder. The physical interpretation is based on a perturbative analysis, which shows that the solition waves have a peculiar superluminal effect, on the evaluation of the optical scalars and on Thorne's C energy.Work partially supported by research project CAICYT no. 0534/81.     

On relativistic-invariant formulation of the inverse scattering transform method

A manifestly relativistic-invariant formulation of the method of inverse scattering transform for relativistic-invariant equations is proposed. The sine-Gordon model and the massive Thirring model are considered.     

On the inverse scattering transform in two spatial and one temporal dimensions

1 IlltroductionThe.goal of this paper is to develop a formalism for (a) solving inverse problem in theplane for potentials decaying at infinity (i.e. given appropriate scattering data reconstruct thepotential q(x, y)); (b) .solving the initial value problem (for appropriately decaying initial data)of certain non-linear evolution equations in two spatial and one temporal dimension (i.e. givenq(x, ys 0) find q(x, y, t)). Several results thus obtained are also formallied. This formalism hasbeen d…     

On the inverse scattering transform,the cylindrical Korteweg-De Vries equation and similarity solutions

It is shown that the inverse scattering transform for the cylindrical KdV equation can be obtained directly from that for the 2-D KdV equation. Using this transform for the similarity solution, we obtain two representations of a Painlevé transcendent.     

On the use of discrete wavelet transform for solving integral equations of acoustic scattering

In this paper, the discrete wavelet transform (DWT) is used to solve the dense system of equations which arises from integral equation of acoustic scattering. The DWT using appropriate wavelet family for acquiring larger sparsification of the system matrix is used to obtain a sparse approximation to the transformed matrix that is used in place of the original matrix in an iterative solver. Alternatively DWT is also used to design sparse preconditioners for an iterative method. Also, DWT-based preconditioners are constructed to accelerate iterative Krylov subspace methods. Convergence rates and number of operations are discussed for each case.

     

Nonlinear shoaling of shallow water waves: perspective in terms of the inverse scattering transform

Summary We study nonlinear interactions in measured surface wave trains obtained in the Northern Adriatic Sea about 16 kilometres from Venice, Italy.Nonlinear Fourier analysis is discussed in terms of the exact spectral solution to the Korteweg-deVries (KdV) equation as given by theinverse scattering transform (IST). For the periodic and/or quasi-periodic boundary conditions assumed herein, the approach may be viewed as a nonlinear, broad-banded generalization of the ordinary, linear Fourier transform. In particular, we study solition interactions, their properties and the nonlinear dynamics of the radiation (or oscillation) modes as found from the inverse scattering transform analysis. We also conduct a number of computer experiments in which measured wave trains are numerically propagated forward in time toward shallow water and backward in time into deep water in order to assess how the nonlinear wave dynamics are influenced by propagation over variable bathymetry. On this basis we develop a scenario for the evolution of nonlinear wave trains, initially far offshore in deep water, as they propagate into shallow water regions. The deep-water waves have a small Ursell number and are hence not very nonlinear; as they propagate toward shallow water, the Ursell number gradually increases in the numerical experiments by about an order of magnitude. A useful parameterization of nonlinearity in these studies is the ?spectral modulus,? a number between 0 and 1, which is associated with each IST spectral frequency. Small values of the modulus mean that a particular spectral component is linear (a sine wave); large values of the modulus (≈1) indicate that the component is nonlinear (a soliton). There is a systematic increase of the modulus as the waves propagate into shallow water where nonlinear effects predominate; we describe how the modulus varies as a function of spectral frequency during this shoaling process. The results suggest that the effect of increasing nonlinearity ?saturates? the IST spectrum (i.e. the modulus ≈1 for all frequencies) to that virtually all spectral components become solitons in sufficiently shallow water.     

Delay-dependent amplification of a probe pulse via stimulated Rayleigh scattering

Stimulated Rayleigh scattering of pump and probe light pulses of close carrier frequencies is considered. A nonzero time delay between the two pulses is shown to give rise to amplification of the delayed (probe) pulse accompanied by attenuation of the pump, both on resonance and off resonance. In either case, phase-matching effects are shown to provide a sufficiently large gain, which can exceed significantly direct one-photon absorption losses.     

Optical inverse radon transform

Mathematical formulation of an optical processesor that implements the inverse radon transform is described. This operation is of use in X-ray computed tomography, transaxial scanning and other image construction systems. The optical system outlined uses the convolution of two two-dimensional functions to achieve the inverse radon transform by a new implementation method.     

基于多尺度变换的动态光散射粒径反演范围的自适应调整

通过多尺度变换实现了反演范围的自适应调整,使其更接近真实范围。分别采用反演范围固定算法与自适应算法对200～600 nm单峰和200～900 nm双峰分布颗粒的模拟相关函数进行了反演,结果表明：自适应算法的结果更接近理论分布,抗干扰能力更强。相对于固定算法,单峰分布颗粒最多可缩小峰值误差4.73%,缩小峰宽误差185 nm。双峰分布颗粒在0～0.001噪声水平时,峰值误差分别小于11.33%,12.45%,峰宽误差分别小于35,160 nm,而固定算法在噪声水平大于0.000 1时,难以得到合理的反演结果。反演范围自适应调整方法能够有效优化粒径反演结果。     

Three-body inverse scattering problem

Two methods are suggested to reconstruct three-body potentials from three-body scattering data. This was achieved by using the reduction of the corresponding Schrödinger equation to a system of ordinary differential equations (not integro-differential equations as usual in the direct problem). Exactly solvable three-body models are presented. A new simple method for solving the multi-dimensional inverse problem in a finite-difference approximation is considered in the Appendix.     

Improved discrete fractional Fourier transform

The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT's) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.     

A discrete fractional angular transform

A new discrete fractional transform defined by two parameters (angle and fractional order) is presented. All eigenvectors of the transform are obtained by an angle using recursion method. This transform is named as discrete fractional angular transform (DFAT). The computational load of kernel matrix of the DFAT is minimum than all other transforms with fractional order. This characteristics has very important practical applications in signal and image processing. Numerical results and the mathematical properties of this transform are also given. As fractional Fourier transform, this transform can be applied in one and two dimensional signal processing.     

Uncertainty relation for the discrete Fourier transform

We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=e(i phi) VU. Its most important application is to constrain how much a quantum state can be localized simultaneously in two mutually unbiased bases related by a discrete fourier transform. It provides an uncertainty relation which smoothly interpolates between the well-known cases of the Pauli operators in two dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the harmonic oscillator equation.     

Perturbation theory in inverse scattering

Perturbation theory of the inverse problem is discussed for a certain class of S-matrices. The convergence of the perturbation series is proved for this class.     

Vacuum decay by inverse scattering

We discuss the decay of false vacua which originate from quantum mechanical effects. In particular we examine the $\text{1}\text{N}$ expansion of the O(N) λφ⁴ model. This model has an effective potential which is complex for large values of the field. In a previous paper we showed that this effective potential could not be used to calculate the decay rate directly. In the present work we show that the vacuum can decay via poles in the effective action which are evident when it is written in terms of scattering variables.