New potential symmetries for some evolution equations

In this paper we derive new potential symmetries that seem not to be recorded in the literature. These potential symmetries are determined by considering a generalized potential system, rather than the natural potential system or a general integral variable. An inhomogeneous diffusion equation, a porous medium equation and the Fokker-Planck equation have been considered as application of this procedure.     

On the nuclear polarization potential for heavy-ion scattering

The real polarization potential ΔV due to transfer reactions is studied using a microscopic semiclassical formalism. It is found that ΔV is typically of the same order of magnitude as the corresponding absorptive potential W. Different types of possible energy dependences for ΔV and W are explored qualitatively. Specific calculations for ¹⁶O + ²⁰⁸Pb and ¹⁶O + ⁶⁰Ni scattering show features similar to those which have been deduced empirically.     

On some operators in multichannel scattering

We investigate the strong limit of an operator valued sequence used in other form in the nonrelativistic theory of multichannel scattering, and also some of its consequences.     

On the theory of complex potential scattering

We study the effect of the addition of a complex potential λV_sep to an arbitrary Schrödinger operator H = H₀ + V on the singularities of the S matrix, as a function of λ. Here V_sep is a separable interaction, and λ is a complex coupling parameter. The paths of these singularities are determined to a great extent by certain saddle points in the momentum (or energy) plane. We explain certain critical phenomena recently reported in the literature. Associated with these saddles are branch-type singularities in the complex λ plane, which are dynamical in origin. Some examples are discussed in detail.     

On the left-hand cut in potential scattering

Partial wave $\text{N}\text{D}$ equations in potential scattering are solved for the exponential, Hulthén and Morse potentials. The driving terms are taken to be either the contributions of a finite number of Born terms or the total contributions of only the nearest singularities (first n poles). For repulsive potentials one observes ghosts, anomalous bound states or resonances if the order of approximation is small with respect to the potential strength. The origin and meaning of these unphysical phenomena are explained. For attractive potentials such anomalies occur only at very large potential strengths if at all. Input-equivalent Bargmann potentials are employed to determine the quality and nature of the approximate $\text{N}\text{D}$ solutions. Rough criteria for the validity of approximations within the $\text{N}\text{D}$ approach are formulated.     

Multichannel nonlinear scattering for nonintegrable equations

We consider a class of nonlinear Schrödinger equations (conservative and dispersive systems) with localized and dispersive solutions. We obtain a class of initial conditions, for which the asymptotic behavior (t±) of solutions is given by a linear combination of nonlinear bound state (time periodic and spatially localized solution) of the equation and a purely dispersive part (decaying to zero with time at the free dispersion rate). We also obtain a result ofasymptotic stability type: given data near a nonlinear bound state of the system, there is a nonlinear bound state of nearby energy and phase, such that the difference between the solution (adjusted by a phase) and the latter disperses to zero. It turns out that in general, the time-period (and energy) of the localized part is different fort+ from that fort–. Moreover the solution acquires an extra constant asymptotic phasee ^{iy ±}.This research was supported in part by grants from the National Science FoundationThe results of this paper were announced in a lecture (June, 1988) on which the Proceedings article [Sof-We] is based     

Variable amplitude equations for one-dimensional scattering

Nonlinear first-order equations, similar to Calogero's equations, are derived for the forward and backward one-dimensional scattering amplitudes. In particular, the even potential case yields two uncoupled equations for the even and odd parity phase shifts. The present approach provides a fast and accurate means for the numerical solution of one-dimensional scattering problems. It also has many analytic merits, some of which are discussed. The connection between one-dimensional and three-dimensional high-energy scattering is reviewed. It is demonstrated that in the one-dimensional case, a slightly modified WKB wavefunction provides an excellent approximation to the exact wavefunction in the shortwave limit. In this limit, additivity of phase shifts for nonoverlapping static potentials is satisfied.     

On a method for solving the inverse problem in potential scattering

A method for solving the inverse problem in the non-relativistic elastic scattering theory, using the analytic and asymptotic properties of the scattering amplitude is proposed and the influence of the discontinuity parameters of the scattering amplitude on the properties of the resulting potentials is discussed. The case with spherically symmetric forces and without bound states is considered. The possibility for solving the inverse problem by this method, leading to the singular repulsive potentials is mentioned.     

Fredholm approximants for potential scattering

An approximate method based on Fredholm approximants is used to solve Lippmann-Schwinger equation fors-andp-wave Yukawa ands-wave exponential potential scattering problems. The method is then applied to a problem ofπ-π scattering i nvolving the use of a well known equivalent potential. The equation for partial wave amplitudes is solved to generate the first threeπ-π resonances,ρ, f ₀, andg, in a self-consistent manner.     

On the equivalence of some exact master equations

In this note we show the equivalence of two procedures for obtaining exact, convolutionless master equations.     

On some new forms of lattice integrable equations

Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon — type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings.     

On the nonlinear evolution equations connected with multidimensional scattering problems

A multidimensional first-order matrix scattering problem is considered. The expression for the scattering matrix in terms of the potential is obtained. It is shown that only a small class of nonlinear evolution equations (isospectral deformations) is connected with the scattering problem under consideration.     

Integral equations for N-particle scattering

Integral equations are derived for the N-particle transition operators. The equations couple together only transition operators between two-body channels. The kernel of the equations becomes connected after a single iteration. Transition operators involving channels with three or more particles can be obtained by quadratures from the solution of the equations. It is also shown that the N-particle equations can be reduced to multichannel two-body equations by the use of the quasiparticle method.     

Diffraction equations for weak scattering reflective surface

In order to investigate the surface roughness dependence of speckle patterns, the complex-amplitude distribution of the speckle field should be obtained first. In previous studies, most investigators have treated this problem using the Fresnel or Fraunhofer diffraction equation. But for a weakly scattering reflective surface, when the observation plane is not parallel to the object plane, the Fresnel and Fraunhofer diffraction equations become inapplicable. Therefore, a reflective surface diffraction model (RSDM) is formed. When the difference between the RSDM and the transmission aperture diffraction model (TADM) is considered, then a general diffraction equation is presented. Considering the variations of the near-field approximation caused by coordinate system rotation, the near-field diffraction equation is derived. By introducing the far-field approximation, the far-field diffraction equation is obtained. The physical meanings of factors in the new equations are interpreted. Comparisons between the Fresnel and Fraunhofer diffraction equations and the newly derived ones show that the former are just the special cases of the latter. Finally, an application of these new diffraction equations is proposed.     

On the validity of some new acoustic scattering approximations

Two new approximations for predicting the elastic scattering of plane acoustic waves by a weak scatterer are proposed. The approximations have been obtained by drawing an analogy between acoustic and light scattering problems. The validity of these approximations has been examined numerically for the exactly soluble case of scattering by a homogeneous sphere. Results show that for small angle scattering the proposed approximations have a considerably larger domain of validity in comparison to the extensively used Born approximation.     

On the validity of some new acoustic scattering approximations

Abstract

Two new approximations for predicting the elastic scattering of plane acoustic waves by a weak scatterer are proposed. The approximations have been obtained by drawing an analogy between acoustic and light scattering problems. The validity of these approximations has been examined numerically for the exactly soluble case of scattering by a homogeneous sphere. Results show that for small angle scattering the proposed approximations have a considerably larger domain of validity in comparison to the extensively used Born approximation.