Showing papers on "Inverse scattering transform published in 2006"

Inverse Scattering Transform for the Camassa-Holm equation

Abstract: An inverse scattering method is developed for the Camassa–Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard inverse scattering transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.

Introduction to non-Kerr Law Optical Solitons

Abstract: INTRODUCTION History Optical Waveguides THE NONLINEAR SCHRODINGER EQUATION Introduction Traveling Waves Integrals of Motion Parameter Evolution Quasi-Stationary Solution KERR LAW NONLINEARITY Introduction Traveling Wave Solution Inverse Scattering Transform Integrals of Motion Variational Principle Quasi-Stationary Solution Lie Transform POWER LAW NONLINEARITY Introduction Traveling Wave Solution Integrals of Motion Quasi-Stationary Solution PARABOLIC LAW NONLINEARITY Introduction Traveling Wave Solution Integrals of Motion Quasi-Stationary Solution DUAL-POWER LAW NONLINEARITY Introduction Traveling Wave Solution Integrals of Motion Quasi-Stationary Solution SATURABLE LAW NONLINEARITY Introduction The NLSE Bistable Solitons Arbitrary Pulse Propagation SOLITON-SOLITON INTERACTION Introduction Mathematical Formulation Quasi-Particle Theory STOCHASTIC PERTURBATION Introduction Kerr Law Power Law Parabolic Law Dual-Power Law OPTICAL COUPLERS Introduction Twin-Core Couplers Multiple-Core Couplers Magneto-Optic Waveguides OPTICAL BULLETS Introduction 1 + 3 Dimensions EPILOGUE HINTS AND SOLUTIONS BIBLIOGRAPHY INDEX

Qualitative methods in inverse scattering theory : an introduction

Abstract: Functional Analysis and Sobolev Spaces.- Ill-Posed Problems.- Scattering by an Imperfect Conductor.- The Inverse Scattering Problem for an Imperfect Conductor.- Scattering by an Orthotropic Medium.- The Inverse Scattering Problem for an Orthotropic Medium.- The Factorization Method.- Mixed Boundary Value Problems.- A Glimpse at Maxwell?s Equations.

Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions

Abstract: The inverse scattering transform for the vector defocusing nonlinear Schrodinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system.

The Cauchy Problem on the Plane for the Dispersionless Kadomtsev - Petviashvili Equation

Abstract: We construct the formal solution to the Cauchy problem for the dispersionless Kadomtsev-Petviashvili equation as an application of the inverse scattering transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev-Petviashvili equation, associated with the inverse scattering transform of the time-dependent Schrodinger operator for a quantum particle in a time-dependent potential.

Stability estimates for the inverse boundary value problem by partial Cauchy data

Abstract: In this paper we study the inverse boundary value problem for the Schrodinger equation with a potential and the conductivity equation using partial Cauchy data. We derive stability estimates for these inverse problems.

Explicit Solutions to the Korteweg-De Vries Equation on the Half Line

Abstract: Certain explicit solutions to the Korteweg–de Vries equation in the first quadrant of the xt-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schrodinger equation. In the reflectionless case such solutions reduce to pure N-soliton solutions. An illustrative example is provided.

Inverse Scattering Transform for the Camassa-Holm equation

Abstract: An Inverse Scattering Method is developed for the Camassa-Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are explicitly constructed in terms of the scattering data. The main difference with respect to the standard Inverse Scattering Transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.

A family of approximations spanning the Born and Rytov scattering series

Abstract: A new hybrid scattering series is derived that incorporates as special cases both the Born and Rytov scattering series, and includes a parameter so that the behavior can be continuously varied between the two series. The parameter enables the error to be shifted between the Born and Rytov error terms to improve accuracy. The linearized hybrid approximation is derived as well as its condition of validity. Higher order terms of the hybrid series are also found. Also included is the integral equation that defines the exact solution to the forward scattering problem as well as its Frechet derivative, which is used for the solution of inverse multiple scattering problems. Finally, the linearized hybrid approximation is demonstrated by simulations of inverse scattering off of uniform circular cylinders, where it is shown that the hybrid approximation achieves smaller error than either the Born or Rytov approximations alone.

An integral equation approach for 2.5-dimensional forward and inverse electromagnetic scattering

Abstract: SUMMARY We present 2.5-D forward and inverse algorithms for modelling low-frequency electromagnetic scattering problems. These algorithms are intended to be used for interpretation of large-scale electromagnetic geophysical data. The algorithms are based on an integral equation approach. To solve the forward problem, a standard conjugate gradient normal residual method is employed, while the inverse problem is solved with the so-called multiplicative regularized contrast source inversion method. Inversion results with low-frequency electromagnetic data for single- and cross-well configurations are presented. Furthermore the advantages of combining the long-offset single-well and cross-well data are discussed.

Computational method for the quantum Hamilton-Jacobi equation: one-dimensional scattering problems.

Abstract: One-dimensional scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism First, the pole structure of the quantum momentum function for scattering wave functions is analyzed The significant differences of the pole structure of this function between scattering wave functions and bound state wave functions are pointed out An accurate computational method for the quantum Hamilton-Jacobi equation for general one-dimensional scattering problems is presented to obtain the scattering wave function and the reflection and transmission coefficients The computational approach is demonstrated by analysis of scattering from a one-dimensional potential barrier We not only present an alternative approach to the numerical solution of the wave function and the reflection and transmission coefficients but also provide a computational aspect within the quantum Hamilton-Jacobi formalism The method proposed here should be useful for general one-dimensional scattering problems

A well-posed integral equation formulation for three-dimensional rough surface scattering

Abstract: We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage–Werner type integral equation formulation of this problem has been proposed, based on an ansatz as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green’s function. Moreover, it has been shown in the three-dimensional case that thi si ntegral equation is uniquely solvable in the space L 2 ðGÞ when the scattering surface G does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, k ,f orkO0, if the coupling parameter h is chosen proportional to the wave number. In the case when G is a plane, we show that the choice hZk=2 is nearly optimal in terms of minimizing the condition number.

Inverse scattering transform for the Toda hierarchy with quasi-periodic background

Abstract: We provide a rigorous treatment of the inverse scattering trans- form for the entire Toda hierarchy in the case of a quasi-periodic nite-gap background solution.

Improved Fokker-Planck Equation for Resonance-Line Scattering

Abstract: A new Fokker-Planck equation is developed for treating resonance-line scattering, which is especially relevant to the treatment of Lyα in the early universe. It is a "corrected" form of the equation of Rybicki & Dell'Antonio that now obeys detailed balance, so the approach to thermal equilibrium is properly described. The new equation takes into account the energy changes due to scattering off moving particles, the recoil term of Basko, and stimulated scattering. One result is a surprising unification of the equation for resonance-line scattering and the Kompaneets equation. An improved energy exchange formula due to resonance-line scattering is derived. This formula is compared to previous formulas of Madau and coworkers and Chen & Miralda-Escude.

The point source method for inverse scattering in the time domain

Abstract: Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time-harmonic scattering, or nearly time-harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non-time-harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time-dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi-frequency data. Copyright (C) 2006 John Wiley & Sons, Ltd.

Inverse scattering via nonlinear integral equations for a Neumann crack

Abstract: In this paper we present a new method for solving the time-harmonic acoustic inverse scattering problem for a sound-hard crack in . Using the integral equation method to solve the inverse scattering problem, one obtains a Fredholm integral equation of the first kind. Instead of applying regularized Newton's method directly to this integral equation, we derive an equivalent system of two nonlinear integral equations for the inverse problem. In this setting, not only can the regularized Newton's method still be used to solve the inverse problem numerically, but also has the advantage of removing the need to solve a related direct problem at every iteration.

On Solution of the Dullin-Gottwald-Holm Equation

Abstract: In this paper, we study the inverse scattering problem for a new nonlinear disper- sive shallow water wave equation, called Dullin-Gottwald-Holm equation. We used the inverse scattering transform method, first, to establish the inverse scattering equation of the DGH equa- tion and a series of solving equations, and then, to represent the known one-soliton solution in a simple parametric form by using the scattering data, last, to present a few examples of the one-soliton solution profile.

On the space-time Monopole equation

Abstract: The space-time monopole equation is obtained from a dimension reduction of the anti-self dual Yang-Mills equation on $\R^{2,2}$. A family of Ward equations is obtained by gauge fixing from the monopole equation. In this paper, we give an introduction and a survey of the space-time monopole equation. Included are alternative explanations of results of Ward, Fokas-Ioannidou, Villarroel and Zakhorov-Mikhailov. The equations are formulated in terms of a number of equivalent Lax pairs; we make use of the natural Lorentz action on the Lax pairs and frames. A new Hamiltonian formulation for the Ward equations is introduced. We outline both scattering and inverse scattering theory and use Backlund transformations to construct a large class of monopoles which are global in time and have both continuous and discrete scattering data.

Integrable nonlinear ladder system with background-controlled intersite resonant coupling

Abstract: A new spectral problem on one-dimensional lattices is found allowing consistently to support the zero-curvature representation for a wide class of integrable nonlinear ladder systems. The modified recurrence technique for obtaining an infinite set of conservation laws is developed and some basic conserved quantities are explicitly derived. The eigenvalue problems associated with the limiting spectral operator for the special case of rapidly vanishing boundary conditions on Schrodinger-type fields and finite background condition on a concomitant field are solved and the domains of analyticity of Jost functions are presented both analytically and graphically. This particular example shows that the original auxiliary spectral problem is basically of fourth order and must generate a set of four distinct Jost functions that have to be involved in the procedure of inverse scattering transform. Moreover, there exists a critical background value of accompanying field which separates two principally different possibilities in the organization of analyticity domains of Jost functions. This crossover should inevitably lead to qualitative rearrangements in the structure of model solutions. Thus already in the limit of low-amplitude excitations we strictly observe the loss of stability regarding the linear spectrum of Schrodinger subsystem just above the critical background value of practically unexcited concomitant field, whereas in the stability region the structure of linear spectrum is essentially controlled by the magnitude of background level via effective modification of both intersite resonant coupling and self-site coupling.

Inverse scattering problem for two-dimensional Schrödinger operator

Abstract: This work deals with the inverse scattering problem for two-dimensional Schrödinger operator. The following problem is studied: To estimate more accurately first nonlinear term from the Born series which corresponds to the scattering data with all energies and all angles in the scattering amplitude. This estimate allows us to conclude that the singularities and the jumps of the unknown potential can be obtained exactly by the Born approximation. Especially, for the potentials from L p -spaces the approximation agrees with the true potential up to the continuous function.

Evolution of the Scattering Coefficients of the Camassa–Holm Equation, for General Initial Data

Abstract: We consider the Camassa-Holm equation for general initial data, particularly when the potential in the scattering problem of the Lax pair, m + K, becomes negative over a finite region. We show that the direct scattering problem of the eigenvalue problem of the Lax pair for this equation may be solved by dividing the spatial infinite interval into a union of separate intervals. Inside each of these intervals, the initial potential is uniformly either positive or negative. Due to this, one can define Jost functions inside each interval, each of which will have a uniform asymptotic form. We then demonstrate that one can obtain the t-evolution of the scattering coefficients of the scattering matrix of each interval. In the process, we also demonstrate that the evolution of the zeros of m + K can be given entirely in terms of limits of the scattering coefficients at singular points.

Factorization of the nonlinear Schrödinger equation and applications

Abstract: We consider factorizations of the stationary and non-stationary Schrodinger equation in which are based on appropriate Dirac operators. These factorizations lead to a Miura transform which is an analogue of the classical one-dimensional Miura transform but also closely related to the Riccati equation. In fact, the Miura transform is a nonlinear Dirac equation. We give an iterative procedure which is based on fix-point principles to solve this nonlinear Dirac equation. The relationship to nonlinear Schrodinger equations like the Gross–Pitaevskii equation is highlighted. §Dedicated to Richard Delanghe on the occasion of his 65th birthday.

Monte Carlo neutrino oscillations

Abstract: We demonstrate that the effects of matter upon neutrino propagation may be recast as the scattering of the initial neutrino wave function. Exchanging the differential, Schrodinger equation for an integral equation for the scattering matrix $S$ permits a Monte Carlo method for the computation of $S$ that removes many of the numerical difficulties associated with direct integration techniques.

Inverse scattering problem for the Schrödinger-type equation with a polynomial energy-dependent potential

Abstract: In the present paper the inverse scattering problem for the Schrodinger-type equation with a polynomial energy-dependent potential is investigated on the entire real line (−∞; + ∞). The scattering data of the scattering problem are defined and the properties of the data are studied. The system of the main integral equations of the scattering problem is derived and the potential functions are recovered.

Scattering transform for nonstationary Schrödinger equation with bidimensionally perturbed N-soliton potential

Abstract: In the framework of the extended resolvent approach the direct and inverse scattering problems for the nonstationary Schrodinger equation with a potential being a perturbation of the N-soliton potential by means of a generic bidimensional smooth function decaying at large spaces are introduced and investigated. The initial value problem of the Kadomtsev-Petviashvili I equation for a solution describing N wave solitons on a generic smooth decaying background is then linearized, giving the time evolution of the spectral data.

Direct integral method, complete discrimination system for polynomial and applications to classifications of all single traveling wave solutions to nonlinear differential equations:a survey

Abstract: Complete discrimination system for polynomial and direct integral method were discussed systematically. In particularly, we pointed out some mistaken viewpoints. Combining with complete discrimination system for polynomial, direct integral method was developed to become a powerful method and was applied to a lot of nonlinear mathematical physics equations. All single traveling wave solutions to theses equations can be obtained. As examples, we gave all traveling wave solutions to some equations such as mKdV equation, Sine-Gordon equation, Double Sine-Gordon equation, triple Sine-Gordon equation, Fujimoto-Watanabe equation, coupled Harry-Dym equation and coupled KdV equation and so on. At the end, we give the partial answers to Fan's a problem.