(*
    Satisfies

    I D[u[x,t],t] + D[u[x,t],x,x] + 2 u[x,t]^2 Conjugate[u[x,t]] == 0
*)
Soliton[a_, b_][x_, t_] := a*Sech[a*(x - 2*b*t)]*Exp[I*(b*x + (a^2 - b^2)*t)]

(*
    Satisfies

    I D[u[x,t],t] + D[u[x,t],x,x] + 2 u[x,t]^2 Conjugate[u[x,t]] == \[Alpha] x u[x,t]
*)
LinearPotentialSoliton[a_, b_, \[Alpha]_][x_, t_] := Soliton[a,b][x + \[Alpha] t^2,t] Exp[-I (\[Alpha] x t + (1/3) \[Alpha]^2 t^3)]