Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_3 \times {\Bbb R}$ and $G_4$ satisfy in Ricci Soliton equation. Among the left-invariant Lorentzian metrics on $H_3 \times {\Bbb R}$, ~ $g_{\lambda}^{+}$ is a shrinking while $g_{\lambda}^{+}$ and $g_{\mu}$ are expanding and also $g_0^1, g_0^2, g_0^3$ have Ricci solitons. We exhibit among the non-isometric left invariant Lorentz metric on the group $G_4$ only $g_1^\lambda, g_2^\lambda$ have Lorentz Ricci solitons and $g_2^\lambda$ is a shrinking.