Soliton resolution for a coupled generalized nonlinear Schr{o}dinger equations with weighted Sobolev initial data

In this work, we employ the $bar{partial}$ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(mathbb{R})}$. The large time asymptotic behavior of the solution $u(x,t)$ and $v(x,t)$ are derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})={(x,t)inmathbb{R}^{2}: x=x_{0}+vt, ~x_{0}in[x_{1},x_{2}], ~vin[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the cgNLS equations which contains the soliton term confirmed by $|mathcal{Z}(mathcal{I})|$-soliton on discrete spectrum and the $t^{-frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-frac{3}{4}})$.