Inverse Scattering Transform in Squared Spectral Parameter for DNLS Equation under Vanishing Boundary Conditions

After a transformation, the inverse scattering transform for the derivative nonlinear Schr6dinger （DNLS） equation is developed in terms of squared spectral parameter. Following this approach, we obtain the orthogonality and completeness relations of free Jost solutions, which is impossibly constructed with usual spectral parameter in the previous works. With the help these relations, the Zakharov-Shabat equations as well as Marchenko equations of IST are derived in the standard way.     

Demonstration of Inverse Scattering Transform for DNLS Equation

Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as |λ| →∞, the usual inverse scattering transform (IST) must be revised. Beside the Kaup and Newell's approach, we propose a simple revision in constructing the equations of IST, where the usual Zakharov-Shabat kern is revised by multiplying λ-2 or λ-1. To justify the revision we show that the Jost solutions obtained do satisfy the pair of compatibility equations.     

Demonstration of Inverse Scattering Transform for DNLS Equation

Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as ｜λ｜→∞, the usual inverse scattering transform （IST） must be revised. Beside the Kaup and Newell＇s approach, we propose a simple revision in constructing the equations of IST, where the usual Zakharov-Shabat kern is revised by multiplying λ^-2 or λ^-1. To justify the revision we show that the Jost solutions obtained do satisfy the pair of compatibility equations.     

New approach to the solution of the scattering of spinless particles by central potentials

The explicit forms of the regular solutions, of the Jost solutions and functions for the radial Schrödinger equation, which describe the scattering of spinless particles by central potentials, are found. The regular solutions are derived from the iterative solution of the integral equation which their suitably modified Laplace transforms fulfil. Two general classes of potentials are used each of them being expressed by the corresponding inverse Laplace transform. As such forms of the regular solutions are related to those of the Jost solutions, the Jost solutions (along with the Jost functions) are written directly. The regions of the complex angular momenta and wave numbers, to which they can be analytically continued, are specified. Some testing relations are also derived.Dedicated to Academician Václav Votruba on the occasion of his seventieth birthday.     

Relations between the wave solutions and the jost solutions off the energy shell for local central potentials

Using the Green function techniques we express the wave solutions of the radial inhomogeneous Schrödinger equation by means of the on-shell Jost and regular solutions. Making use of their boundary behaviour atr = andr = 0 we reexpress them alternatively in terms of the off-shell Jost and regular solutions. Relations among the different generalized (fully off the energy shell) Jost functions are derived and the radial matrix elements of the transition and reaction (reactance) operators are given in terms of these Jost functions. The relations reflect the principle of detailed balance.     

On- and off-shell Jost functions and their integral representations

By judicious exploitation of the transpose operator relation in conjunction with the differential equations of special functions of mathematical physics, integral representations of the on- and off-shell Jost functions are derived from the particular integrals of the inhomogeneous Schrödinger equation. Using the particular integral of the inhomogeneous Schrödinger equation, exact analytical expressions for the Coulomb and Coulomb plus Yamaguchi off-shell Jost solutions are constructed in the maximal reduced form. As a case study, the limiting behaviours and the on-shell discontinuities of the Coulomb plus Yamaguchi Jost solutions are verified numerically.     

Multiple Pole Solutions of the Modified Nonlinear Schrödinger Equation

A general formula for the N-tuple polesoliton solutions of the modified nonlinear Schrödinger equation, which corresponds to a nonzero pole of order N of the Jost solution to the corresponding Lax-pair equations, is derived.     

An eighth-order KdV-type equation in (1+1) and (2+1) dimensions: multiple soliton solutions

In this work we study an eighth-order KdV-type equations in (1+1) and (2+1) dimensions. The new equations are derived from the KdV6 hierarchy. We show that these equations give multiple soliton solutions the same as the multiple soliton solutions of the KdV6 hierarchy except for the dispersion relations.     

Explicit forms of the off-shell Jost solutions for general central potentials

We derive explicit forms of the regular solutions and the Jost solutions off the energy shell, which satisfy the inhomogeneous Schrödinger equation. The used forms of the Yukawa-like and Gauss-like potentials are related to the two known integral representations of the Hankel functions. The explicit form of the introduced fully off-shell Jost functions enables us to write it in the alternative integral forms, which contain the Jost solutions or the regular solution.     

Demonstration of Soliton Solutions of DNLS Equation by Liouville Theorem

In the inverse scattering transform (IST), the reflectionless Jost solutions are combined by their analytic properties in the complex spectrum parameter plane, and then can be shown to satisfy the two Lax equations indeed by Liouville theorem. So the corresponding soliton solutions certainly satisfy the nonlinear equation by compatibility condition. Especially the multi-soliton solutions of DNLS equation can be demonstrated in this way. PACS Numbers: 05.45.Yv, 02.30.-f, 11.10.Ef     

A Reduced System of Linear Algebraic Equations for Giving Soliton Solutions of the Generalized ZS/AKNS Systems

A simple method for finding soliton solutions of the generaked ZS/AKNS systems whose Lax pairs are matrices with high orders is considered. An explicit expreesion of transformation between the Jost solution relating to the (n-1)-soliton solution and that relating to the n-soliton solution is found. A reduced system of N algebraic equations for giving N soliton solutions is deduced, it has an identical form no matter how high the order of matrices of the Lax pain is.     

A relationship between Gel'fand-Levitan and Marchenko kernels

A new integral equation which relates the output kernels of the Gel'fand-Levitan and Marchenko inverse scattering equations in a continuous range of their variables is specified. Structural details of this integral equation are studied when theS-matrix is a rational function, and the output kernels are separable in terms of Bessel, Hankel and Jost solutions.     

Relativistic generalization of dynamic equation for Jost function

A system of dynamic equations for the simultaneous determination of the scattering amplitude and the relativistic Jost function is derived from the Mandelstam representation and the unitarity condition. The relativistic generalization of the de Alfaro, Regge and Rossetti dynamic equation for the Jost function is given.     

Negative-order KdV equations in (3+1) dimensions by using the KdV recursion operator

We develop a variety of negative-order Korteweg-de Vries (KdV) equations in (3+1)-dimensions. The recursion operator of the KdV equation is used to derive these higher dimensional models. The new equations give distinct solitons structures and distinct dispersion relations as well. We also determine multiple soliton solutions for each derived model.     

Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties.We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed.     

Application of the Riemann method to the Bethe-Salpeter equation

The Bethe-Salpeter equation describing the interaction of two scalar particles via the exchange of a third scalar particle with mass 0 is in configuration space a hyperbolic partial differential equation of fourth order which will be studied with the help of the Riemann method. This method yields two Volterra equations the solutions of which are special solutions of the Bethe-Salpeter equation. The wave function is a superposition of the special solutions. For the coefficients one gets a system of two integral equations. The Fredholm determinant of the system is the generalization of the nonrelativistic Jost function.     

Green's Function Method for Perturbed sine-Gordon Equation

The usual Green's function method is introduced to show the completeness of the squared Jost solutions of the multi-soliton case in this work. Since sine-Gordon equation contains second derivative in time, the known procedure with generalized Marchenko equation to show completeness relation of the eigenfunctions of linearized equation is unavailable. And the explicit expressions of Jost solutions are not necessary here. Thus a general method of direct perturbation method for the perturbed sine-Gordon equation is developed.     

A new derivation of Judds isolated exact solutions for E ⊗ e and Γ8 ⊗ τ2 Jahn-Teller systems

The Longuet-Higgins recurrence relations for the dynamical J-T problem are equivalent to a system of two ordinary linear first order differential equations. After Laplace transformation, recurrence relations for power series are found, which, in contradistinction to the original ones, are very simple. On the baselines they allow for terminating series, which give Judds isolated exact solutions plus the solution which asymptotically approaches the baseline for infinite coupling strength.     

A DIRECT PERTURBATION THEORY OF THE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

An exact direct perturbation theory of nonlinear Schrodinger equation with corrections is developed under the condition that the initial value of the perturbed solution is equal to the value of an exact multisoliton solution at a particular time. After showing the squared Jost functions are the eigenfunctions of the linearized operator with a vanishing eigenvalue,suitable definitions of adjoint functions and inner product are introduced. Orthogonal relations are derived and the expansion of the unity in terms of the squared Jost functions is naturally implied. The completeness of the squared Jost functions is shown by the generalized Marchenko equation. As an example,the evolution of a Raman loss compensated soliton in an optical fiber is treated.     

Conjectured exact solution for the dynamical Jahn-Teller-effect in E × ϵΓ8 × γ2 and Γ8 × (γ2 × ϵ) systems

The Longuet Higgins recurrence relations of the dynamical J-T problem are equivalent to a system of two ordinary linear first order differential equations whose solutions are required to belong to the space of entire functions. This requirement determines the energy eigenvalues. A solution of the differential equations in terms of Neumann series is given, whose coefficients are determined by simple recurrence relations. In order to select the energy eigenvalues, a sequence with index m of conditions between the coefficients is studied. It is conjectured that the solutions of this sequence converge as m → ∞ and give the correct eigenvalues. Numerical studies show that the convergence is rapid and the Thorson-Moffitt eigenvalues are obtained with m = 4.