Approximate analytic solutions to coupled nonlinear Dirac equations

Abstract

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalarscalar self interactions g 2 1 2 (ψψ) 2 + g 2 2 2 (φφ) 2 + g 2 3 (ψψ)(φφ) as well as vector-vector interactions of the form g 2 1 2 (ψγµψ)(ψγ µ ψ) + g 2 2 2 (φγµφ)(φγ µ φ) + g 2 3 (ψγµψ)(φγ µ φ). Writing the two components of the assumed solitary wave solution of these equation in the form ψ = e −iω 1 t {R1 cos θ, R1 sin θ}, φ = e −iω 2 t {R2 cos η, R2 sin η}, and assuming that θ(x), η(x) have the same functional form they had when g3=0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 2 3 /g 2 2 and g 2 3 /g 2 1 . In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.

Study of instability of the Fourier split-step method for the massive Gross–Neveu model

“…10 we plot the growth rate of the "noise floor" versus L. The theoretical values are inferred from the spectral radius of Φ(L), which was found by the above analysis, via the relation (see (45a)): growth rate = ln ρ(Φ(L)) /L . (46) The agreement between our analysis and direct numerics is seen to be quite good. 44), (46).…”

“…(46) The agreement between our analysis and direct numerics is seen to be quite good. 44), (46). Circles are the result of SSM simulation, where we measured the amplitude of the Fourier harmonic at k = k max /2 and followed the procedure described at the end of Section 5.…”

“…Finally, we note that results of our study should alert researchers who numerically study new models similar to the Gross-Neveu model (see, e.g., [46,47]) about possible NI that may occur in simulations of those models. Our analysis may also serve as a prototype for that of the Zakharov equations, describing interaction of short and long waves [48].…”

“…In that case, Eqs. (44) and (46) predict that the NI growth rate peaks to about 2.9 · 10 −4 , i.e., slightly lower than the spectral edge NI, near L = 40π + 3.1. Simulations by the SSM support this analytical result.…”

“…Note different vertical scales in the three panels. Solid lines correspond to the analytical solution of (44), (46). Circles are the result of SSM simulation, where we measured the amplitude of the Fourier harmonic at k = k max /2 and followed the procedure described at the end of Section 5.…”