Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.

We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interaction g(2)/κ+1(̅ΨΨ)(κ+1) and with mass m. Using the exact analytic form for rest frame solitary waves of the form Ψ(x,t)=ψ(x)e(-iωt) for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results.

Relativistic explicit correlation: coalescence conditions and practical suggestions.

To set up the general framework for relativistic explicitly correlated wave function methods, the electron-electron coalescence conditions are derived for the wave functions of the Dirac-Coulomb (DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), modified Dirac-Coulomb (MDC), and zeroth-order regularly approximated (ZORA) Hamiltonians. The manipulations make full use of the internal symmetries of the reduced two-electron Hamiltonians such that the asymptotic behaviors of the wave functions emerge naturally. The results show that, at the coalescence point of two electrons, the wave functions of the DCG Hamiltonian are regular, while those of the DC and DCB Hamiltonians have weak singularities of the type r(12)(ν) with ν being negative and of O(α(2)).