On some exact solutions of the nonlinear Dirac equation

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

“…The above choice of nonlinearity is motivated from the so-called Soler model in quantum field theory, e.g. λ 2 = 0 and λ 1 = 0 [37,40,68], and BEC with a chiral confinement and/or spin-orbit coupling, e.g. λ 1 = 0 and λ 2 = 0 [25,43,44].…”

“…In the analytical aspect, for the existence and multiplicity of bound states and/or standing wave solutions, we refer to [2,3,15,24,[29][30][31]51] and references therein. Particularly, for the case where d = 1, ε = 1, V (x) ≡ 0, λ 1 = −1 and λ 2 = 0 in the choice of F(Φ), the NLDE (1.1) admits explicit soliton solutions [26,40,45,52,56,60,65,66]. In the numerical aspect, many accurate and efficient numerical methods have been proposed and analyzed, such as the finite difference time domain (FDTD) methods [19,46,59], the time-splitting Fourier spectral (TSFP) methods [10,18,39,49] and the Runge-Kutta discontinuous Galerkin methods [48].…”

Ion Collection by a Hollow Probe in ECR Microwave Plasma under a Divergent Magnetic Field

“…This makes it possible to construct new families of exact solutions using the solution generation technique (see [ 6 ] ) . As is shown in [ 6 ] the formula of generating solutions by final transformations of the four-parameter special conformal group has the form…”

On Symmetry of the Generalized Breit Equation

“…For many applications (for example, for description of the coordinate system where there are solutions in the separating variables), the knowledge of symmetry operators from wider classes than generators of symmetry groups is necessary [8,9]. Here we find the complete set of symmetry operators of the Breit equation in the class M 1 of differential operators of the first order with matrix coefficients, which satisfy conditions (6).…”