On the Inverse Scattering Transform for the Kadomtsev-Petviashvili Equation

The initial value problem of the Kadomtsev-Petviashvili equation for one choice of sign in the equation has been recently investigated in the literature. Here we consider the other choice of sign. We introduce suitable eigenfunctions which though bounded are not analytic in the spectral parameter. This, in contrast to the known case, prevents us from formulating the inverse problem as a nonlocal Riemann-Hilbert boundary value problem. Nevertheless a suitable formulation is given and a formal solution is constructed via a linear integral equation.     

Inverse Scattering Transform for the Multi‐Component Nonlinear Schrödinger Equation with Nonzero Boundary Conditions

The Inverse Scattering Transform (IST) for the defocusing vector nonlinear Schrödinger equations (NLS), with an arbitrary number of components and nonvanishing boundary conditions at space infinities, is formulated by adapting and generalizing the approach used by Beals, Deift, and Tomei in the development of the IST for the N ‐wave interaction equations. Specifically, a complete set of sectionally meromorphic eigenfunctions is obtained from a family of analytic forms that are constructed for this purpose. As in the scalar and two‐component defocusing NLS, the direct and inverse problems are formulated on a two‐sheeted, genus‐zero Riemann surface, which is then transformed into the complex plane by means of an appropriate uniformization variable. The inverse problem is formulated as a matrix Riemann‐Hilbert problem with prescribed poles, jumps, and symmetry conditions. In contrast to traditional formulations of the IST, the analytic forms and eigenfunctions are first defined for complex values of the scattering parameter, and extended to the continuous spectrum a posteriori.     

Lectures on the Inverse Scattering Transform

In 1967 a new method of mathematical physics was discovered by Gardner, Greene, Kruskal, and Miura. Their significant results have received much attention, and numerous developments have occurred within the decade. In this paper a survey of some of these developments is made. By and large the emphasis is on relating the key ideas via typical examples.     

Inverse Scattering for the 1-D Helmholtz Equation

We prove a uniqueness result for Nevanlinna functions. and this result is then used to give an elementary proof of the uniqueness in the inverse scattering problem for the equation \( u'' + \frac{k^2}{c^2}u=0 \) on \({\mathbb R}\). Here \(c\) is a real positive measurable function that is bounded from below by a positive constant, and is close to \(1\) at \(\pm \infty \).     

Inverse Spectral Transform for the q-Deformed Volterra Equation

We use the inverse spectral transform to study a q-deformation of the Volterra equation. The q-deformed time dependence of the spectral data is computed, and the one-soliton solution is explicitly constructed.     

Inverse Spectral Transform for the Modified Kadomtsev-Petviashvili Equation

The 2 + 1-modified Kadomtsev-Petviashvili (mKP) equation is studied by the inverse-spectral-transform method. The initial-value problems for the mKP-1 and mKP-11 equations are solved by the nonlocal Riemann-Hilbert and techniques for initial data decaying sufficiently rapidly at infinity. The lump solutions for the mKP-I equation are found explicitly. Wide classes of the exact solutions for the mKP equation—namely, the rational solutions, including the plane lumps for the mKP-I equation; solutions with functional parameters; the plane solitons; and breathers—are constructed by the use of the method based on the nonlocal . The Miura transformation between the mKP and KP equations is discussed.     

The Inverse Scattering Transform for the Benjamin-Ono Equation—A Pivot to Multidimensional Problems

The initial value problem associated with the Benjamin-Ono equation is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation and solution of an associated nonlocal Riemann-Hilbert problem in terms of initial scattering data. Solitons are given a definitive spectral characterization. Pure soliton solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on [INLINEEQUATION], [INLINEEQUATION].     

Inverse Scattering Transform for the Coulomb Plasma with Negative Temperature

The boundary problem for the two-dimensional elliptic sinh-Gordon equation is studied. The exact solutions of these equations are found and trace identities are suggested. An application of this problem to the model of the Coulomb plasma with negative temperature is considered. Bibliography: 10 titles.     

Inverse Scattering Transform for the Focusing Ablowitz‐Ladik System with Nonzero Boundary Conditions

The purpose of this paper is to construct the inverse scattering transform for the focusing Ablowitz‐Ladik equation with nonzero boundary conditions at infinity. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann‐Hilbert problem on a doubly connected curve in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz‐Ladik potential.     

Forced Nonlinear Evolution Equations and the Inverse Scattering Transform

The Korteweg—de Vries and nonlinear Schrödinger equations with an external forcing of distribution type are considered. The reflection coefficient is found to satisfy a nonlinear equation of a certain characteristic form which also appears in the semi-infinite problem.     

The S-Matrix Structure and the Inverse Scattering Transform in the J-Matrix Approach

Using the analytic properties of the S-matrix, we obtain a system of inverse scattering transform equations for nonlocal potentials with Laguerre form factors. Coulomb repulsion can be present in the system.     

A Remark on the Inverse Scattering Problem for the Perturbed Hill Equation

Mathematical Notes - A criterion is given for the Lie algebra of infinitesimal holomorphic automorphisms of the direct product of two germs of real analytic generic CR manifolds to be equal to the...     

Nonlinear Integral Equation of Inverse Scattering Problems of Wave Equation and Iteration

In this paper, we reduce the 1-D, 2-D and 3-D inverse scattering problems of the wave equation into the nonlinear integral equation. The iteration for solving the above integral equations has been considered.     

On the Inverse Scattering of the Time-Dependent Schrödinger Equation and the Associated Kadomtsev-Petviashvili (I) Equation

The Kadomtsev-Petviashvili equation, a two-spatial-dimensional analogue of the Korteweg-deVries equation, arises in physical situations in two different forms depending on a certain sign appearing in the evolution equation. Here we investigate one of the two cases. The initial-value problem, associated with initial data decaying sufficiently rapidly at infinity, is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation of a nonlocal Riemann-Hilbert problem in terms of scattering data expressible in closed form in terms of given initial data. The lump solutions, algebraically decaying solitons, are given a definite spectral characterization. Pure lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t. Many of the above results are also relevant to the problem of inverse scattering for the so-called time-dependent Schrödinger equation.     

Inverse Spectral Transform for the Nonlinear Evolution Equation Generating the Davey-Stewartson and Ishimori Equations

A 2 + 1-dimensional nonlinear differential equation integrable by the inverse-spectral-transform method with the quartet operator representation is proposed. This GL(2, C)-valued chiral-field-type equation is the generating (prototype) equation for the Davey-Stewartson and Ishimori equations. It coincides with the nonlinear equation for the Davey-Stewartson eigenfunction ψ_DS. The initial-value problem for this equation is solved by the techniques for the and the nonlocal Riemann-Hilbert problem. The classes of exact solutions with the functional parameters and exponential-rational solutions are constructed by the method. The static lump solution in the case α = i and the exponentially localized solution at α = i are found. Other similar examples of nonlinear integrable equations in 2 + 1 and 1 + 1 dimensions are discussed.     

Radon Transform for Solving an Inverse Scattering Problem in a Planar Layered Acoustic Medium

A two-dimensional inverse scattering problem in a layered acoustic medium occupying a half-plane is considered. Data is the scattered wavefield from a surface point source measured on the boundary of the half-plane. On the basis of the Radon transform, an algorithm is constructed that recovers the velocity and the acoustic impedance of the medium from the scattering data. An analytical solution is presented for an inverse scattering problem, and several inverse scattering problems are solved numerically.     

An Inverse Problem for the Matrix Schrödinger Equation

By modifying and generalizing some old techniques of N. Levinson, a uniqueness theorem is established for an inverse problem related to periodic and Sturm-Liouville boundary value problems for the matrix Schrödinger equation.     

The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems

A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering. The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator. A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed. The relationship of the scattering theory and Backlund transformations is brought out. In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems.