pInitialData = ParametricPlot3D[{{t, 0, 0},{t, 0, Sech[t]^2}}, {t, -2, 2}, PlotStyle -> Tube[0.02]]

pSolution = ParametricPlot3D[{t + Sech[t] tau, tau, Sech[t]^2}   , {t, -2, 2}, {tau,-2,2}, Mesh ->{10, 1}, PlotStyle -> Opacity[.8]]

Show[pInitialData,pSolution, pInitialData, PlotRange -> All, AxesLabel ->{x,y,z}]

pInitialData2 = ParametricPlot3D[{{0, t, 0},{0,t, Sech[t]^2}}, {t, -2, 2}, PlotStyle -> Tube[0.02]]

pSolution2 = ParametricPlot3D[{Sech[t]^2 tau, t + tau, Sech[t]^2}   , {t, -2, 2}, {tau,-2,2}, Mesh ->{10, 1}, PlotStyle -> Opacity[.8]]

Show[pInitialData2,pSolution2, PlotRange -> All, AxesLabel ->{x,y,z}]

pInitialData3 = ParametricPlot3D[{{0, t, 0},{0,t, t}}, {t, -2, 2}, PlotStyle -> Tube[0.05]]

pSolution3 = ParametricPlot3D[{t tau, t + tau,t}   , {t, -2, 2}, {tau,-2,2}, Mesh ->{10, 1}, PlotStyle -> Opacity[.8]]

Show[pInitialData3,pSolution3, PlotRange -> All,  AxesLabel ->{x,y,z}]

pInitialData4 = ParametricPlot3D[{{t, 2t, 0},{t,2t, 1}}, {t, -2, 2}, PlotStyle -> Tube[0.05]]

pSolution4 = ContourPlot3D[z == x y/(x y + 2 x - y), {x, -2, 2}, {y, -4, 4}, {z, -2,2}, ContourStyle -> Opacity[.8]]

Show[pInitialData4,pSolution4, PlotRange -> All,  AxesLabel ->{x,y,z}]