Clear[xnaught, ic1,  ic2, ff1, ff2, InitCond, FluxFunc, Solu, x, 
      X]
<< Graphics`PlotField`
{ic1[x_], ic2[x_]} = {Exp[-x^2], 
        Exp[-x^2]}
InitCond[x_] = {ic1[x], ic2[x]}
{ff1[u_],  ff2[v_]} = {u, -v}
X[x_, t_] = {x + t*ff1'[ic1[x]], 
        x + t*ff2'[ic2[x]]}
Solu[x_, t_] = 
      Transpose[{X[x, t], InitCond[x]}]
Show[
      Table[ParametricPlot[{First[Solu[x, t]], Last[Solu[x, t]]}, {x, -5, 
            5}, PlotRange -> {0, 1}, Background -> GrayLevel[0],  
          ImageSize -> {550,  400}], {t, 0, 
          5,  .2}]]