Exact solutions of the nonlinear Dirac equation in terms of Bessel, Gauss and Legendre functions and Chebyshev-Hermite polynomials (Q1813744)

Substitutions are proposed which reduce the nonlinear Dirac equation \([i\gamma_ \mu\delta^ \mu-F(\bar\psi\psi)]\psi=0\) to second order ordinary differential equations \[ a_ 1(\omega)\partial^ 2_ \omega u+a_ 2(\omega)\partial_ \omega u+a_ 3(\omega)u=0, \] the explicit form of which depends essentially on the form of the function \(F\). Exact solutions of the Dirac equation are constructed in terms of the Bessel, Gauss, Legendre functions and Chebyshev-Hermite polynomials.