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I want to plot the roots of the complex equation of a1. I have tried the following code but this is not giving me any result. I need bistable plot. How it is possible.?

del = -1.5;
g0 = 4.8;
del0 = 1.5;
ome = 40;
A1=0;
B1=1;
k1 = 0.1;
kex = 0.1;
kL = (k1+kex)/2-del0*(k1-kex)/(2*ome);
Gma = 0.5;
Solve[{I*del*a1 + I*g0*(1 - del/ome)*A1*Re[b1]*a1 + 
 I*P0*(1 - del0/(2*ome))/Sqrt[2] + 
 I*P0*g0*B1*Re[b1]/(Sqrt[2]*ome) - 
 kL/2*a1 - (k1 - kex)*g0/ome*B1*Re[b1]*a1 == 
0, -I*ome*b1 + I*g0*(1 - del0/ome)*A1*Abs[a1]^2/2 + 
 I*P0*g0*B1*Re[a1]/(Sqrt[2]*ome) - Gma*b1/2 == 0}, {a1, b1}];
Plot[{Evaluate[Abs[a1]^2 /. %]}, {P0, 0, 3}, Frame -> True,    
FrameLabel -> {Style["P0", Bold, 20], Style[" N", Bold, 20]}, 
FrameTicksStyle -> Directive[FontSize -> 20], 
PlotStyle -> {Thickness[0.0005], Thickness[0.011]}]
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  • $\begingroup$ What are A1,B1,b What does it mean Re[a1], Re[b1]? $\endgroup$ – OkkesDulgerci Feb 9 at 12:41
  • $\begingroup$ @OkkesDulgerci oh sorry, A1=0,B1=1; b is b1 and Re[a1] is Real of a1, and Re[b1] is Real of b1. because a1 and b1 are complex. $\endgroup$ – vini Feb 9 at 12:49
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Edit:

ClearAll["Global`*"]
A1 = 0;
B1 = 1;
del = -1.5 // Rationalize;
g0 = 4.8 // Rationalize;
del0 = 1.5 // Rationalize;
ome = 40 // Rationalize;
k1 = 0.1 // Rationalize;
kex = 0.1 // Rationalize;
kL = (k1 + kex)/2 - del0*(k1 - kex)/(2*ome);
Gma = 0.5 // Rationalize;

{a1, b1} = {a1, b1} /. 
  Flatten@Solve[{I*del*a1 + I*g0*(1 - del/ome)*A1*Re[b1]*a1 + 
       I*P0*(1 - del0/(2*ome))/Sqrt[2] + 
       I*P0*g0*B1*Re[b1]/(Sqrt[2]*ome) - 
       kL/2*a1 - (k1 - kex)*g0/ome*B1*Re[b1]*a1 == 
      0, -I*ome*b1 + I*g0*(1 - del0/ome)*A1*Abs[a1]^2/2 + 
       I*P0*g0*B1*Re[a1]/(Sqrt[2]*ome) - Gma*b1/2 == 0}, {a1, b1}]

enter image description here

xVal = Range[0, 3, 0.01];
a1 = a1 /. P0 -> xVal;
pts0 = a1 Conjugate[a1] // Chop;
pts = Transpose[{xVal, pts0}];


ListLinePlot[pts, Frame -> True, 
     FrameLabel -> {Style["P0", Bold, 20], Style[" N", Bold, 20]}, 
     FrameTicksStyle -> Directive[FontSize -> 20], PlotLegends -> {"a1"}]

enter image description here

Original Answer

    ClearAll["Global`*"] 
        A1 = 0;
        B1 = 1;
        del = -1.5;
        g0 = 4.8;
        del0 = 1.5;
        ome = 40;
        k1 = 0.1;
    kex = 0.1;
    kL = (k1 + kex)/2 - del0*(k1 - kex)/(2*ome);
    Gma = 0.5;
    sol = Values@
      Flatten@Solve[{I*del*a1 + I*g0*(1 - del/ome)*A1*b1*a1 + 
           I*P0*(1 - del0/(2*ome))/Sqrt[2] + I*P0*g0*B1*b1/(Sqrt[2]*ome) -
            kL/2*a1 - (k1 - kex)*g0/ome*B1*b1*a1 == 
          0, -I*ome*b1 + I*g0*(1 - del0/ome)*A1*Abs[a1]^2/2 + 
           I*P0*g0*B1*a1/(Sqrt[2]*ome) - Gma*b1/2 == 0}, {a1, b1}]

$\left\{\text{a1}\to -\frac{(3854.71\, -24.092 i) \text{P0}}{1. \text{P0}^2-(8331.6\, -329.861 i)},\text{b1}\to -\frac{8.17708 \text{P0}^2}{1. \text{P0}^2-(8331.6\, -329.861 i)}\right\}$

    a1 = sol[[1]] /. P0 -> Subdivide[3, 100];
    pts0 = a1 Conjugate[a1] // Chop;
pts = Transpose[{Subdivide[3, 100], pts0}];



     ListLinePlot[pts, Frame -> True, 
 FrameLabel -> {Style["P0", Bold, 20], Style[" N", Bold, 20]}, 
 FrameTicksStyle -> Directive[FontSize -> 20], PlotLegends -> {"a1"}]

enter image description here

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  • $\begingroup$ I am using mathematica version 9.0 . Why my plot is not coming for the same code? $\endgroup$ – vini Feb 9 at 14:41
  • $\begingroup$ I am not sure, I using M12. Does it produce any error? $\endgroup$ – OkkesDulgerci Feb 9 at 15:00
  • 1
    $\begingroup$ Values was introduced in v10. Replace it with {a1, b1} /. $\endgroup$ – Bob Hanlon Feb 9 at 15:58
  • $\begingroup$ @OkkesDulgerci Have you plotted the (a1 dagger a1) ? beacuse in y-axis it is a1 dagger a1 $\endgroup$ – vini Feb 9 at 17:19
  • $\begingroup$ Let's assume a1=2i+P0 how you wanna plot it? How you define a1*a1? $\endgroup$ – OkkesDulgerci Feb 9 at 18:43

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