Symmetry and the exact solutions of the nonlinear Galilean-invariant equations for a spinor field (Q1176881)

The author treats here the nonlinear Galilei invariant spinor equation \[ [-i(\gamma_ 0+\gamma_ 4)\partial_ t-i\gamma_ a\partial_{x_ a}+m(\gamma_ 0-\gamma_ 4)]\psi=F(\psi^*,\psi) \leqno (1) \] with the nonlinearity (NL) \(F=[f_ 1+f_ 2(\gamma_ 0+\gamma_ 4)]\psi\), where \[ f_ i=f_ i(\bar\psi\psi,\bar\psi(\gamma_ 0+\gamma_ 4)\psi),\;\bar\psi=(\psi^*)^ T\gamma_ 0.\leqno (2) \] That is, equation (1) with \(F=0\) is invariant by the group \(G(1,3)\) with the bases \(P_ 0=\partial_ t\), \(P_ a=\partial_{x_ a}\), \(M=2im\), \(G_ a=t\partial_{x_ a}+2imx_ a+({1\over 2})(\gamma_ 0+\gamma_ 4)\gamma_ a\), \(J_{ab}=x_ a\partial_{x_ b}-x_ b\partial_{x_ a}-({1\over 2})\gamma_ a\gamma_ b\), and the Galilei transform group with 3 parameters given by \(G_ a\) is \(t'=t\), \(x_ a'=x_ a+tv_ a\), \[ \psi'(x')=\exp\{-2im(v_ ax_ a+(1/2)v_ av_ at)-(1/2)(\gamma_ 0+\gamma_ 4)\gamma_ av_ a\}\psi(x). \] Then he derives the NL(2) as a necessary and sufficient condition for the G(1,3) invariance of the equation (1). In order to give the exact solution of the equation (1) with the NL (2) he sets \(\psi(x)=A(x)\varphi(\omega)\), where \(A(x)\): \(4\times 4\) matrix, \(\varphi(\omega)\): complex valued vector, \(\omega(x)\): real valued scalar function. Here conditions \[ Q_ aA\equiv[\xi_{a0}(x)\partial_ t+\xi_{ab}(x)\partial_{x_ b}+\eta_ a(x)]A=0,\;[\xi_{a0}(x)\partial_ t+\xi_{ab}(x)\partial_{x_ b}]\omega=0,\;a=\overline{1,3} , \] by \(\|\xi_{a\mu}\|\) with rank 3 must hold. He gives \(G(1,3)\)-nonequivalent 3 dimensional subalgebras \(A_ i (i=1,\ldots,16)\), and derives the orginary differential equations of \(\varphi\) and the exact solution \(\psi\) correspondent to each subalgebra. So far as \(A_ i (i=6,\ldots,13)\), first integrals of the ordinary differential equations are given.