Showing papers on "Inverse scattering transform published in 1984"
TL;DR: In this paper, various numerical methods are employed in order to approximate the nonlinear Schrodinger equation, namely: (i) the classical explicit method, (ii) hopscotch method, implicit-explicit method, Crank-Nicolson implicit scheme, (v) the Ablowitz-Ladik scheme, split step Fourier method (F. Tappert), and (vii) pseudospectral (Fourier) method.
597 citations
Book•
11 Aug 1984
TL;DR: Inverse Methods for Reflector Imaging as discussed by the authors, the Dirac Delta Function, Fourier Transforms, and Asymptotics are used for direct scattering problems, and the Wave Equation in Two and Three dimensions.
Abstract: First-Order Partial Differential Equations. The Dirac Delta Function, Fourier Transforms, and Asymptotics. Second-Order Partial Differential Equations. The Wave Equation in One Space Dimension. The Wave Equation in Two and Three Dimensions. The Helmholtz Equation and Other Elliptic Equations. More on Asymptotic Techniques for Direct Scattering Problems. Inverse Methods for Reflector Imaging. Each chapter includes references. Index.
439 citations
TL;DR: In this paper, a nonlinear evolution equation for one dependent variable in two spatial dimensions integrable by the inverse scattering transform method is presented, which is a special case of the one in this paper.
322 citations
TL;DR: In this article, the inverse problem associated with a rather general system of n first-order equations in the plane is linearized by utilizing a Riemann-Hilbert problem; similarly, a DBAR problem is used when the system is elliptic.
Abstract: The inverse problem associated with a rather general system of n first‐order equations in the plane is linearized. When the system is hyperbolic, this is achieved by utilizing a Riemann–Hilbert problem; similarly, a ‘‘∂’’ (DBAR) problem is used when the system is elliptic. The above result can be employed to linearize the initial value problem associated with a variety of physically significant equations in 2+1, i.e., two spatial and one temporal dimensions. Concrete results are given for the n‐wave interaction in 2+1 and for various forms of the Davey–Stewartson equations. Lump solutions (solitons in 2+1) of the latter equation are given a definitive spectral characterization and are obtained through a linear system of algebraic equations.
237 citations
TL;DR: In this article, the inverse scattering method was used for the calculation of correlation functions in completely integrable quantum models with the R-matrix of XXX-type, including the Bose-gas and the Heisenberg XXX-model.
Abstract: The inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with theR-matrix of XXX-type. These models include the one-dimensional Bose-gas and the Heisenberg XXX-model. The algebraic questions of the problem are considered.
227 citations
TL;DR: In this article, a method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred is presented, which can be regarded as arising from Bianchi identities expressing the commutativity of Backlund transformations.
Abstract: In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of Backlund transformations. Applying an appropriate continuum limit we first obtain integrable nonlinear differential-difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations. As special cases we treat the difference-difference versions and the differential-difference versions of the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrodinger equation, the isotropic classical Heisenberg spin chain, and the complex and real sine-Gordon equation.
207 citations
Book•
01 Dec 1984
TL;DR: A survey of the Vocal Tract Inverse Problem: Theory, Computations and Experiments; Convergence of Discrete Inversion Solutions; Inversion of Band Limited Reflection Seismograms; Some Recent Results in Inverse Scattering Theory; Well-Posed Questions and Exploration of the Space of Parameters in Linear and Nonlinear Inversion; The Seismic Reflection Inverse problem; Migration Methods: Partial but Efficient Solutions to the Seismics InverseProblem; Relationship Between Linearized InverseScattering and Seismical Migration;
Abstract: : Contents: A Survey of the Vocal Tract Inverse Problem: Theory, Computations and Experiments; Convergence of Discrete Inversion Solutions; Inversion of Band Limited Reflection Seismograms; Some Recent Results in Inverse Scattering Theory; Well-Posed Questions and Exploration of the Space of Parameters in Linear and Nonlinear Inversion; The Seismic Reflection Inverse Problem; Migration Methods: Partial but Efficient Solutions to the Seismic Inverse Problem; Relationship Between Linearized Inverse Scattering and Seismic Migration; Project Review on Geophysical and Ocean Sound Speed Profile Inversion; Acoustic Tomography; Inverse Problems of Acoustic and Elastic Waves; Finite Element Methods with Anisotropic Diffusion for Singularly Perturbed Convection Diffusion Problems; Adaptive Grid Methods for Hyperbolic Partial Differential Equations; Some Simple Stability Results for Inverse Scattering Problems; Inverse Scattering for Stratified, Isotropic Elastic Media Using the Trace Method; A Layer-Stripping Solution of the Inverse Problem for a One- Dimensional Elastic Medium; On Constructing Solutions to an Inverse Euler- Bernoulli Beam Problem; Far Field Patterns in Acoustic and Electromagnetic Scattering Theory; Renaissance Inversion; On the Equilibrium Equations of Poroelasticity; GPST-A Versatile Numerical Method for Solving Inverse Problems of Partial Differential Equations; and Applications of Seismic Ray-Tracing Techniques to the Study of Earthquake Focal Regions.
121 citations
TL;DR: On demontre l'existence d'operateurs de diffusion definis partout for certaines equations de Klein-Gordon non lineaires as discussed by the authors, l'operateur de diffusion
Abstract: On demontre l'existence d'operateurs de diffusion definis partout pour certaines equations de Klein-Gordon non lineaires
97 citations
TL;DR: In this paper, the inverse scattering problem is formulated as a nonlinear optimization problem, and a linearized model is proposed to solve the problem in a constrained manner, based on the ideas of a priori assumptions and compactness arguments.
Abstract: The inverse scattering problem we are considering in this paper is to determine the shape of a sound-soft, bounded, connected obstacle from a knowledge of the time-harmonic incident wave with frequency in the resonance region and the far field pattern of the scattered wave. After some introductory remarks, we begin by describing the method of integral equations and the null-field method for solving the direct scattering problem. This enables us to define an operator T mapping the boundary of the scattering obstacle and the incident field onto the far field pattern. The inverse scattering problem is to invert this operator. In order to do this, we first must examine the range of T, i.e. to characterize the class of far field patterns, and to establish the existence of T' on the range of T, i.e. to show the uniqueness of the solution to the inverse scattering problem. This analysis shows that for a given measured far field pattern, in general no solution exists to the inverse scattering problem, and if a solution does exist, it does not depend continuously on the measured data, i.e. the problem is improperly posed. This motivates us to consider a linearized model and to examine various methods for studying linearized improperly posed problems based on the ideas of a priori assumptions and compactness arguments. We then consider a simple model problem that focuses on the nonlinear character of the inverse scattering problem and, motivated by our study of the linearized model, reformulate the inverse scattering problem as a problem in constrained optimization. We conclude by considering the numerical solution of this nonlinear optimization problem. "Approach your problems from the right end and begin with the answers. Then, one day, perhaps you will find the final question." From "The Hermit Clad in Crane Feathers" in The Chinese Maze Murders, by R. Van Gulik.
77 citations
TL;DR: In this paper, a new method of stability investigation for solutions of nonlinear equations integrable with the help of the inverse scattering transform (IST) is presented for periodic nonlinear waves in weakly dispersive media with respect to transverse perturbations.
TL;DR: The completeness relation for the system of squared solutions of the discrete analog of the Zakharov-Shabat problem is derived in this article, which allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results.
Abstract: The completeness relation for the system of ‘‘squared’’ solutions of the discrete analog of the Zakharov–Shabat problem is derived It allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results These include: (i) the expansion of the potential and its variations over the system of ‘‘squared’’ solutions, the expansion coefficients being the scattering data and their variations, respectively; thus the interpretation of the inverse scattering transform (IST) as a generalized Fourier transform becomes obvious; (ii) compact expressions for the trace identities through the operator Λ, for which the ‘‘squared’’ solutions are eigenfunctions; (iii) brief exposition of the spectral theory of the operator Λ; (iv) direct calculation of the action‐angle variables based on the symplectic form of the completeness relation; (v) the generating functional of the M operators in the Lax representation; (vi) the quantum ve
TL;DR: In this article, direct linearization is used to find broad classes of solutions to nonlinear evolution equations in 1+1 and 2+1 dimensions, which can be solved through the inverse scattering transform.
Abstract: The paper deals with the direct linearization, an approach used to generate particular solutions of the partial differential equations that can be solved through the inverse scattering transform. Linear integral equations are presented which enable one to find broad classes of solutions to certain nonlinear evolution equations in 1+1 and 2+1 dimensions.
TL;DR: In this paper, a linearizing integral transform is proposed which relates solutions of a spectral problem associated with a class of integrable partial difference equations to any given solution of the spectral problem.
TL;DR: In this article, the authors formulate stationary axially symmetric (SAS) fields in the framework of harmonic mappings of Riemannian manifolds and show that the configuration space of the fields is a symmetric space.
Abstract: We formulate stationary axially symmetric (SAS) Einstein–Maxwell fields in the framework of harmonic mappings of Riemannian manifolds and show that the configuration space of the fields is a symmetric space. This result enables us to embed the configuration space into an eight‐dimensional flat manifold and formulate SAS Einstein–Maxwell fields as a σ‐model. We then give, in a coordinate free way, a Belinskii–Zakharov type of an inverse scattering transform technique for the field equations supplemented by a reduction scheme similar to that of Zakharov–Mikhailov and Mikhailov–Yarimchuk.
TL;DR: In this article, it was shown that the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the Benjamin-Ono equation for any δ > 0.
Abstract: The intermediate long wave equation is a physically important singular integrodifferential equation containing a parameter, referred to here as δ. For δ → ∞ it reduces to the Benjamin–Ono equation. It has been recently shown that the inverse scattering transform schemes of the above equations have certain significant differences. Here it is shown that for δ → ∞, the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the Benjamin–Ono equation.
TL;DR: In this paper, the inverse scattering problem is solved for the two-dimensional time-independent Schrodinger equation, where the potential is reconstructed from the scattering amplitude, which is assumed to be known for all energies and angles.
Abstract: The inverse scattering problem is solved for the two‐dimensional time‐independent Schrodinger equation. That is, the potential is reconstructed from the scattering amplitude, which is assumed to be known for all energies and angles.
TL;DR: In this article, the inverse spectral and scattering problem of the Zakharov-Shabat (ZS) operator is studied and the similarity transformation between ZS operators is examined when their potentials have the common nonvanishing asymptotic values Q± at the infinity.
Abstract: The inverse spectral and scattering problem of the Zakharov–Shabat (ZS) operator is studied The similarity transformation between ZS operators is examined when their potentials Q(x) have the common nonvanishing asymptotic values Q± at the infinity The Marchenko equation is derived from the Parseval equation We give the necessary as well as the sufficient condition of the scattering data for the potential of the specified class
TL;DR: In this article, a systematic method for deriving Backlund transformations for singular integral equations is presented, which leads in a natural way to the Backlund transformation for corresponding (integrable) nonlinear partial differential equations.
Abstract: A systematic method for deriving Backlund transformations for singular (linear) integral equations is presented. The method leads in a natural way to the Backlund transformations for the corresponding (integrable) nonlinear partial differential equations. By repeated use of the method a hierarchy of singular integral equations and Backlund transformations can be derived, corresponding to so-called multi-modified partial differential equations. Specific examples include the Korteweg-de Vries equation and the first, second, and third modified Korteweg-de Vries equation; the Nonlinear Schrodinger equation and the Anisotropic Heisenberg Spin Chain; the sine-Gordon equation and the modified sine-Gordon equation.
TL;DR: In this paper, a method for solving the inverse problem for a class of multidimensional first-order systems is given, which yields equations which the scattering data must satisfy; these equations are natural candidates for characterizing admissible scattering data.
Abstract: A method for solving the inverse problem for a class of multidimensional first-order systems is given. The analysis yields equations which the scattering data must satisfy; these equations are natural candidates for characterizing admissible scattering data. The results are used to solve the multidimensional N-wave resonant interaction equations.
TL;DR: In this paper, a Backlund transformation, an associated linear scattering problem, and a method for finding the conservation laws for the modified intermediate long wave equation were given, which reduces to the modified Korteweg-deVries equation in the shallow water limit.
Abstract: In this paper, we give a Backlund transformation, an associated linear scattering problem, and a method for finding the conservation laws for the so‐called modified intermediate long wave equation. This equation reduces to the modified Korteweg–deVries equation in the shallow water limit.
TL;DR: In this article, a hierarchy of nonlinear evolution equations associated with a generalization of the Schrodinger spectral problem is derived, and it is shown that each equation is Hamiltonian and that their flows commute.
Abstract: A hierarchy of nonlinear evolution equations associated with a generalization of the Schrodinger spectral problem is derived. It is shown that each equation is Hamiltonian and that their flows commute. The spectral equation is examined and certain difficulties in the inverse problem are pointed out.
TL;DR: In this article, the spherical Raman-Nath differential equation is solved using a generalisation of an operatorial approach successfully applied to differential-difference equations which are particular cases of the one discussed here.
Abstract: The authors discuss the solution of the spherical Raman-Nath differential equation. This type of equation appears in diverse physical problems, one of which is stimulated Compton scattering. The solution technique exploits the generalisation of an operatorial approach successfully applied to differential-difference equations which are particular cases of the one discussed here.
TL;DR: In this paper, the T-function defined as a limiting transformation of the Riemann theta function leading to a system of algebraic dispersion equations, yields solutions of some nonlinear partial differential equations in the form of solitons on a background of quasi-periodic waves.
Abstract: The direct employment of the T-function defined as a limiting transformation of the Riemann theta function leading to a system of algebraic dispersion equations, yields solutions of some nonlinear partial differential equations in the form of solitons on a background of quasi-periodic waves. Pure solitons and pure quasi-periodic solutions appear as particular cases. The method is illustrated by the example of the KdV equation and is also compared with the Hirota bilinear operator technique.
TL;DR: In this article, it was shown that the methods using the Heisenberg equation (the so-called Lax representation) are not suitable to obtain the equations studied here, and a comparison of that method with other methods for finding nonlinear evolution equations integrable by the inverse problem method is given.
Abstract: Several new nonlinear evolution equations integrable by the inverse problem method are obtained. The method applied in finding these equations is believed to be essentially new. The comparison of that method with other methods for finding nonlinear evolution equations integrable by the inverse problem method is given. In particular, it is shown that the methods using the Heisenberg equation (the so-called Lax representation) are not suitable to obtain the equations studied here.Bibliography: 23 titles.
TL;DR: The existence of an infinite sequence of scattering frequencies for the equation □ u + qu = 0 is established in this paper, where q is a real valued potential which may assume negative values.
TL;DR: The large energy asymptotic behavior of scattering solutions of the three-dimensional time-dependent Schrodinger equation is investigated in this article, where the second term of the expansion leads to the "miracle" of Newton's three-dimension inverse scattering theory.
Abstract: The large‐energy asymptotic behavior of scattering solutions of the three‐dimensional time‐dependent Schrodinger equation is investigated. The second term of the expansion leads to the ‘‘miracle’’ of Newton’s three‐dimensional inverse scattering theory.
TL;DR: In this article, the authors provided a solution of the nonlinear partial differential equation ∇2p=k2sinh p previously obtained by Plint and Breig with the objective of confirmation of expected temperature dependence of scattering of light by certain crystals.
Abstract: The paper provides solution of the nonlinear partial differential equation ∇2p=k2 sinh p previously obtained by Plint and Breig [J. Appl. Phys. 35, 2745 (1964)] with the objective of confirmation of expected temperature dependence of scattering of light by certain crystals. The solution is of interest both for the experimental problem and as a solution of a nonlinear partial differential equation without customary linearizing assumptions. This is done using Adomian’s decomposition method [G. Adomian, Stochastic Systems (Academic, New York, 1983)]. This has been shown elsewhere in Adomian’s work and numerous following papers to provide a rapidly convergent series solution which is accurate and computable even if strong nonlinearities or stochastic behavior is involved.
TL;DR: In this paper, conditions for the existence of localized stationary states of a general class of nonlinear Dirac equations are obtained and the nonlinearities considered are such that, in the field equation, they approach zero faster than the field itself.
Abstract: Conditions for the existence of localized stationary states of a general class of nonlinear Dirac equations are obtained The nonlinearities considered are such that, in the field equation, they approach zero faster than the field itself