Asymptotic expansion of small solutions to the Landau–Ginzburg type equations

We obtain the large time asymptotic expansion of small solutions to the Landau–Ginzburg type equations $\[\cases{u_{t}-\alpha u_{xx}+\beta |u|^{2}u=0,\quad x\in {\mathbf{R}},\ t>0,\cr \noalign{\vskip3pt}u(0,x)=\phi (x),\quad x\in {\mathbf{R}},\cr}\]$ where α,β∈C,Rα>0, R(β/ $\sqrt{2|\alpha |^{2}+\alpha ^{2}}$ )≥0, under the condition that the initial data ϕ are sufficiently small in a suitable weighted norm and the mean value ∫ϕ(x) dx≠0.