NONRELATIVISTIC ASYMPTOTICS OF SOLITARY WAVES IN THE DIRAC EQUATION WITH SOLER-TYPE NONLINEARITY

We use the perturbation theory to build solitary wave solutions phi(omega) (x)e-(iwt) to the nonlinear Dirac equation in R-n, n >= 1, with the Sole-type nonlinear term f (psi(*) beta psi)beta psi with f(tau) = vertical bar tau vertical bar vertical bar o(vertical bar tau vertical bar(k)), k > 0, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit w less than or similar to m; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schrodinger charge critical (k = 2/n), then one has Q'(w) < 0 for w less than or similar to m with Q(w) being the charge of the corresponding solitary wave; this implies the absence of the degeneracy of zero eigenvalue of the linearization at this solitary wave.