Clear["@"]

u[x_, 
        t_] := (2  a/q)^(1/2)  Exp[
          I (1/2  c x - (c^2/4 - a) t)]*
        Sech[a^(1/2) (x - c t)]

u[x, t]

u[x, 
        t] /. Complex[0, 
          n_] -> -Complex[0, n]

f[x_] = u[x, 0];

f[x]

g[x_] = Simplify[D[u[x, t], t] /.  t -> 0];

u0[x_, t_] = f[x]

A0[x_] = 
        Simplify[
          u0[x, t]^2*(u0[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n])];

A0[x]

u1[x_, t_] = 
      Simplify[Expand[
          I*(q*Integrate[A0[x], {t, 0, t}] + 
                Integrate[D[u0[x, t], {x, 2}], {t, 0, t}])]]

A1[x_] = 
      Simplify[2 u0[x, t]  * 
            u1[x, t]*(u0[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] ) +  
          u0[x, t]^2*(u1[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n])]

u2[x_, t_] = 
      Simplify[I*(q*Integrate[A1[x], {t, 0, t}] + 
              Integrate[D[u1[x, t], {x, 2}], {t, 0, t}])]

A2[x_] = 
      Simplify[2 u0[x, t]  * 
            u2[x, t]*(u0[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] )  +  
          u1[x, t]^2*(u0[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] ) + 
          2*u0[x, t]  * 
            u1[x, t]*(u1[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] ) +  
          u0[x, t]^2*(u2[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] )]

u3[x_, t_] = 
      Simplify[I*(q*Integrate[A2[x], {t, 0, t}] + 
              Integrate[D[u2[x, t], {x, 2}], {t, 0, t}])]

A3[x_] = 
      Simplify[2 u0[x, t]  * 
            u3[x, t]*(u0[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] )  +  
          2 u1[x, t]  * 
            u2[x, t]*(u0[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] ) + 
          2 u0[x, t]  * 
            u2[x, t]*(u1[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] ) + 
          u1[x, t]^2*(u1[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] ) + 
          2*u0[x, t]  * 
            u1[x, t]*(u2[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] ) +  
          u0[x, t]^2*(u3[x, 
                  t] /. Complex[0, 
                    n_] -> -Complex[0, n] )]

u4[x_, t_] = 
      Simplify[I*(q*Integrate[A3[x], {t, 0, t}] + 
              Integrate[D[u3[x, t], {x, 2}], {t, 0, t}])]

solu[x_, t_] = 
        u0[x, t] + u1[x, t] + u2[x, t] + u3[x, t] + u4[x, t];

schrothe[x_, t_] := 
      Simplify[u[x, t] /. {a -> 1/1000, q -> 1, 
            c -> 1/10}]
p1 = 
      Plot3D[Abs[schrothe[x, t]], {x, -80, 100}, {t, 0, 108}, 
        AxesLabel -> {"x", "t", "u   "}, 
        PlotRange -> All, ColorOutput -> GrayLevel]

schroado[x_, t_] := 
      Simplify[solu[x, t] /. {a -> 1/1000, q -> 1, 
            c -> 1/10}]
p2 = 
      Plot3D[Abs[schroado[x, t]], {x, -80, 100}, {t, 0, 108}, 
        AxesLabel -> {"x", "t", "U   "}, 
        PlotRange -> All, ColorOutput -> GrayLevel]

err[t_] := 
      Norm[Table[
          N[Abs[schroado[x, t]] - Abs[schrothe[x, t]]] , {x, -80, 100, 
            0.1}]]

ploter = 
      Plot[err[t], {t, 0, 108}, PlotRange -> All, 
        PlotStyle -> {{Thickness[0.001]}}, 
        AxesLabel -> {"t", "error"}]