Dark Multi-Soliton Solution of the Nonlinear Schrödinger Equation with Non-Vanishing Boundary

The inverse scattering transform for the nonlinear Schrödinger equation in normal dispersion with non-vanishing boundary values is re-examined using an affine parameter to avoid double-valued functions. An operable algebraic procedure is developed to evaluate dark multi-soliton solutions. The dark two-soliton solution is given explicitly as an example, and is verified by direct substitution. The additional motion of the soliton center is given by its asymptotic behavior.     

A Soliton Perturbation Theory for the MNLS Equation with Corrections in terms of an Auxiliary Spectral Parameter

Based on the lakharov-Shabat equation of the inverse scattering transform for the MNLS equation, a perturbation theory for the MNLS equation with corrections is developed in terms of an auxiliary spectral parameter ζ. All necessary formulae for calculating the scattering data are derived without any assumption beyond the usual conditions for the inverse scattering transform. Based upon these formulae, the effects due to the corrections can be calculated.     

Reduced Maxwell-Bloch equations with anisotropic dipole momentum

We develop the inverse scattering transform for the recently found integrable system of reduced Maxwell-Bloch equations with two components of polarization and with an anisotropic dipole momentum. The model describes few-cycle pulses of optical or other field propagations. We find that the existence of a nontrivial group of symmetry of the corresponding Lax pair leads to a particular form of the inverse scattering transform equations. We show that solutions can be expressed in terms of the solution of a matrix Riemann-Hilbert problem formulated for the complex plane with a nontrivial group of automorphisms.     

Representation of supercharges over the scattering data for super-sine-Gordon system

We determine the form of the superchargesQ ₁ andQ ₂ in terms of the scattering data for the super-sine-Gordon equation. The technique is to ascertain the effect of supertranslation on the scattering data used in the inverse scattering transform. The same result was obtained by Kulish and Tsypleyv from the Riccatti equation by expanding in A.     

Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation

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 Schrödinger operator for a quantum particle in a time-dependent potential.     

The Eigenfunctions and Exact Solutions of Discrete mKdV Hierarchy with Self-Consistent Sources via the Inverse Scattering Transform

Another form of the discrete mKdV hierarchy with self-consistent sources is given in the paper. The self-consistent sources are presented only by the eigenfunctions corresponding to the reduction of the Ablowitz-Ladik spectral problem. The exact soliton solutions are also derived by the inverse scattering transform.     

The method of solution for stimulated Raman scattering and two-photon propagation

A uniquely different inverse scattering transform is presented for solving stimulated Raman Scattering and two-photon propagation. This transform is based on the generalized Zakharov-Shabat problem, and in certain cases one will not have $\text{r = ±q}{}^1$, bound-state eigenvalues may move, and the space dependence of the scattering data may itself depend on a second inverse scattering transform.     

求解DNLS方程的反散射法的基本问题

对DNLS方程, 反散射法应选取k²=p为基本谱参数, 其中k为通常谱参数. 并引入一个新的谱参数及一个规范变换, 由此得到自由Jost解的规范正交系, 这就是反散射法的基本问题. 同时还得到了基于谱参数p的Marchenko方程和Zakharov-Shabat反散射方程. 关键词： DNLS方程 反散射法 自由Jost解 规范正交系     

An Inverse Scattering Transformation for the Modified Nonlinear Schrijdinger Equation

A new Lax pair of the modified nonlinear Schrödinger equation is introduced in terme of the variable of the Fourier transform λ. The Lax pair has no usual symmetries between 12 and 21 elements and avoids the factor λ^1/2. The basic equation of inverse scattering transformation is deduced in the Zakharov-Shabat form as well as in the Marchenko form.     

Nonlinear science abstracts

The class of exactly integrable non-linear evolution equations (NLEE) related to the general first order n × n linear problem is studied. The set of independent scattering data $\text{J}$ is determined and trace identities are obtained. Next we use a special integro-differential operator Λ and obtain the expansions of the potential and its variation in the eigenfunctions of Λ. These enable us to consider the inverse scattering method as a generalized Fourier transform and to write down in a compact form the integrals of motion. We prove that there exist a hierarchy of Hamiltonian structures generated by Λ. At the end we calculate the symplectic form in terms of the scattering data variations.     

Inverse scattering transform for the time dependent Schrödinger equation with applications to the KPI equation

For the direct-inverse scattering transform of the time dependent Schrödinger equation, rigorous results are obtained based on an opertor-triangular-factorization approach. By viewing the equation as a first order operator equation, similar results as for the first ordern x n matrix system are obtained. The nonlocal Riemann-Hilbert problem for inverse scattering is shown to have solution.     

On hierarchies of equations for the Benjamin-Ono equation

Two hierarchies of the Benjamin-Ono equation and their Lax pairs are derived by an algebraic approach. The hierarchies are proved to be the isospectral flows and non-isospectral flows and the corresponding time evolutions of the scattering data are obtained, thus the inverse scattering transform can be applied to solve them.     

The T-matrix for particle with arbitrary permittivity tensor and parallelization of the computational code

Using the T-matrix approach electromagnetic scattering by a non-axisymmetric particle with an arbitrary permittivity tensor is studied. The electromagnetic fields inside the scatterer are expressed by a system of quasi-spherical vector wave functions which are derived by the use of inverse Fourier transform. Using this expansion a solution of the light scattering problem in the framework of the null-field method is obtained. The implemented T-matrix program is parallelized using both OpenMP and MPI parallel paradigms. Numerical scattering results for anisotropic ellipsoids are presented.     

Bethe ansatz equations for the multicomponent nonlinear Schrödinger model with supermatrices

The Bethe ansatz equations for the multicomponent nonlinear Schrödinger model with supermatrices are obtained by using the quantum inverse scattering transform.     

Hamiltonian Formalism of mKdV Equation with Non-vanishing Boundary Values

Hamiltonian formalism of the mKdV equation with non-vanishing boundary valueis re-examined by a revised form of the standard procedure. It is known that the previous papers did not give the final results and involved some questionable points [T.C. Au Yeung and P.C.W. Fung, J. Phys. A 21 (1988) 3575]. In this note, simple results are obtained in terms of an affine parameter and a Galileo transformation is introduced to ensure the results compatible with those derived from the inverse scattering transform.     

Acoustic images of objects moving under an inhomogeneous layer

The possibilities and limitations of the methods of obtaining acoustic images of objects moving under an inhomogeneous scattering layer are considered. It is shown that the previously proposed algorithms of signal processing that select a random phase modulation from the received field make it possible to reconstruct the images of extended objects observed through an inhomogeneous medium in the presence of strong stationary reflectors. The proposed methods are compared with the Zverev's method of an inverse aperture synthesis in a dark field.     

Integrable equations and differential geometry

It is shown that the equations which are integrable by the inverse scattering transform method and this method itself admit a natural interpretation in terms of vector and principal zero-curvature fibre bundles with certain structure groups G.     

A SOLITON PERTURBATION THEORY FOR THE UNSTABLE NONLINEAR SCHR?DINGER EQUATION WITH CORRECTIONS

Based on the Zakharov-Shabat equation of the inverse scattering transform for the unstable nonlinear Schr?dinger equation, for which a perturbation theory with corrections is developed in this paper. All necessary formulae for calculating the scattering data are derived. Based upon these formulae, the effect due to the corrections can be studied. As an example, the correction due to the damping is calculated.