<< Graphics`PlotField`

    
Clear[x, u, v, t, xnaughtw, xnaughtu,  xnaughtv, ProjectBacku, 
      ProjectBackv, ProjectBackw, OrderedPair]

      
{ic1[x_], ic2[x_]} = {Exp[-x^2], 
        Exp[-x^2]}
InitCond[x_] = {ic1[x], 
        ic2[x]}

 (*Input flux function*) 
{ff1[u_],  ff2[v_]} = {u, -v}
FluxFunc[u_, v_] = {ff1[u], 
        ff2[v]}

ProjectBacku[x_, t_] := 
      xnaughtu /. 
        FindRoot[
          x - xnaughtu - t*ff1'[ic1[xnaughtu]], {xnaughtu, 
            0}]
ProjectBackv[x_, t_] := 
      xnaughtv /. 
        FindRoot[
          x - xnaughtv - t*ff2'[ic2[xnaughtv]], {xnaughtv, 
            0}]
OrderedPair[x_, t_] := {ic1[ProjectBacku[x, t]], 
        ic2[ProjectBackv[x, 
            t]]}

ParametricPlot[
      InitCond[x],  {x, -5, 5}]
  
Show[
      Table[ListPlot[Table[OrderedPair[x, t], {x, -3, 3,  .01}], 
          ImageSize -> {550, 400}], {t, 0, 
          2,  .1}]]
PlotVectorField[
      FluxFunc[u, v],  {u, 0, 1}, {v, 0, 
        1}]