Here is the details of the code to plot.

 Clear[x, y, ψ1, ψ2, ψ3, ψ4, eqn, eqnWithInitial,v, j];
    eqn = Thread[
       I D[{ψ1[x, y, t], ψ2[x, y, t], ψ3[x, y, t], ψ4[
            x, y, t]}, 
          t] == {v (-I D[ψ3[x, y, t], x] - D[ψ3[x, y, t], y]) + 
          2 Δ ψ4[x, y, t], 
         v (-I D[ψ4[x, y, t], x] - D[ψ4[x, y, t], y]), 
         v (-I D[ψ1[x, y, t], x] + D[ψ1[x, y, t], y]), 
         v (-I D[ψ2[x, y, t], x] + D[ψ2[x, y, t], y]) + 
          2 Δ ψ1[x, y, t]}];
    eqnWithInitial = 
      Join[eqn, 
       Thread[{ψ1[x, y, 0], ψ2[x, y, 0], ψ3[x, y, 
           0], ψ4[x, y, 0]} == {1, 1, 1, 
           1} (x + I*y) Exp[-(x^2 + y^2)]], 
       Thread[{ψ1[-5, y, t], ψ2[-5, y, t], ψ3[-5, y, 
           t], ψ4[-5, y, t]} == {ψ1[5, y, t], ψ2[5, y, 
           t], ψ3[5, y, t], ψ4[5, y, t]}], 
       Thread[{ψ1[x, -5, t], ψ2[x, -5, t], ψ3[x, -5, 
           t], ψ4[x, -5, t]} == {ψ1[x, 5, t], ψ2[x, 5, 
           t], ψ3[x, 5, t], ψ4[x, 5, t]}]];
    
    v = 1;
    
    Δ = 1;
    
    tMax = 8;
    
    solution = 
      First@NDSolve[
        eqnWithInitial, {ψ1[x, y, t], ψ2[x, y, t], ψ3[x, y,
           t], ψ4[x, y, t]}, {x, -5, 5}, {y, -5, 5}, {t, 0, tMax}, 
        Method -> {"MethodOfLines", 
          "SpatialDiscretization" -> {"TensorProductGrid", 
            "DifferenceOrder" -> "Pseudospectral"}}];
    
    Ψ1[x_, y_, t_] = ψ1[x, y, t] /. solution;
    Ψ2[x_, y_, t_] = ψ2[x, y, t] /. solution;
    Ψ3[x_, y_, t_] = ψ3[x, y, t] /. solution;
    Ψ4[x_, y_, t_] = ψ4[x, y, t] /. solution; 

myrotorz1[x_, y_,t_] = -I/2*(-Conjugate[D[Ψ1[x, y, t], y]]*
D[Ψ2[x, y, t], x] +D[Ψ1[x, y, t],y]*Conjugate[D[[Ψ1[x, y, t],x]]+Conjugate[D[Ψ1[x, y, t], x]]*D[Ψ1[x, y, t], y]-D[Ψ1[x, y, t], x]*Conjugate[Ψ1[x, y, t], y]]);

I would like to plot for myrotorz1[x, y, t] as a function of t but shows an error. Any tips would be greatly appreciated.

Plot[Re[NIntegrate[myrotorz1[x, y, t], {x, -5, 5}, {y, -5, 5}]], {t, 0, 6}]