Plot wave front curvatures K,S for various initial values.

SetDirectory["~/writing/riccati"]
Off[General::spell]
Off[General::spell1]

<< Graphics`Graphics`

G0 = 
{{a + I α, 
            b  +  I β}, 
{b + I β, 
            c + I γ}};
G0  //  MatrixForm

\(\(\( (*\ 
      Define\ G[t]\  = \ \(solution\ to\ Riccati\ equation

        G'\  + \ G^2 = \(0\ 

          
          G \((0)\) = 
            G_  0\)\)
*) \)\(
\)\(G[
        t_] := \ G0 . 
        Inverse[IdentityMatrix[2]\  + \ 
            G0\ t]

 (*\ 
      Gaussian\ and\ mean\ curvature\ of\ F_t*) 

    K[t_] := \ Det[ComplexExpand[Re[G[t]]]]

    S[t_] := \ 
      1/2  Tr[ComplexExpand[
            Re[G[t]]]]

 (*\ 
      eigenvalues\ of\ Re\ G\ *) 

    s1[t_] := S[t] + Sqrt[S[t]^2 - K[t]]\ 

    s2[t_] := 
      S[t] - Sqrt[S[t]^2 - K[t]]

 (*\ 
      eigenvalues\ of\ Im\ G\ *) 

    Ki[t_] := \ Det[ComplexExpand[Im[G[t]]]]

    Si[t_] := \ 1/2  Tr[ComplexExpand[Im[G[t]]]]

    si1[t_] := Si[t] + Sqrt[Si[t]^2 - Ki[t]]\ 

    si2[t_] := Si[t] - Sqrt[Si[t]^2 - Ki[t]]

    \)\)\)

Solve[{KK == λ1 λ2,  
        2 SS == λ1  +  λ2}, {λ1, λ2}]

maxt = 5;
β = 0;  α = 1;  γ = 
      1;

pltOptions = {{Dashing[{0.01, 0.01}], 
            Thickness[0.005]}, {Thickness[
              0.006]}};
\(\(\($TextStyle = {FontFamily -> "Times", FontSize -> 8, \ 
        FontSlant -> "Italic"}\)\(
\)\(
\)
    \) (*\ 
      is\ the\ phase\ front\ within\ the\ area\ of\ \(\(influence\)\(?\)\)\ *) \)
condition[t_] := 
      Module[{}, 
If[(s1[t] s2[t] > 
                0) &&  
(K[t] > 
                1/t^2), 
+1,  
-1
]
]

    
makePlot[aa_, bb_, cc_,  title_,  ticks_] := 
      Module[{}, 
a = aa; b = bb; 
        c = cc; 
Plt1 = 
          Plot[{s1[t], s2[t], si1[t], si2[t],  0.2  condition[t]}, {t, 0, 
              maxt},  


PlotStyle ->  {{Thickness[
                    0.003]}, 
{Thickness[
                    0.003]}, 
{Dashing[{0.01, 0.015}], 
                  Thickness[
                    0.003]}, 
{Dashing[{0.01, 0.015}], 
                  Thickness[0.003]}, 
{Thickness[
                    0.004]}}, 


PlotRange ->  All,  
            PlotLabel -> title,  
Ticks ->  
              ticks
]
]

plotZZ = 
      makePlot[0, 0, 
        0, "a=0, c=0", {{0, 1, 2, 3, 4, 5}, {-0.5, 0, 0.5, 1, 1.5, 2, 
            3, 4, 5}}]

plotPZ = 
      makePlot[1, 0, 
        0, "a=1, c=0", {{0, 1, 2, 3, 4, 5}, {-1, 0, 1, 2, 3, 
            4}}]

plotPP = 
      makePlot[1, 0, 
        1, "a=1, c=1", {{0, 1, 2, 3, 4, 5}, {-1, 0, 1, 2, 3, 
            4}}]

plotPM = 
      makePlot[-1, 0, 
        1, "a=-1, c=1", {{-1, 0, 1, 2, 3, 4, 5}, {-2, -1, 0, 
            1, 2, 3, 4}}]

plotMM = 
      makePlot[-1, 
        0, -1, "a=-1, c=-1", {{-2, -1, 1, 2, 3, 4, 
            5}, {-3, -2, -1, 0, 1, 2, 3, 4}}]

plotMZ = 
      makePlot[-1, 0, 
        0, "a=-1, c=0", {{-2, -1, 0, 1, 2, 3, 4, 
            5}, {-2, -1, 0, 1, 2, 3, 4}}]

DisplayTogetherArray[
{
{plotPP,  
          plotMM, plotMZ}, 
{plotPM,  plotPZ,  
          plotZZ}
}
]

Export["~/writing/riccati/KSpm.eps", %]

\(\(\( (*\ 
      which\ beams\ should\ satisfy\ condition\ at\ \(\(infinity\)\(?\)\)\ *) \)\(
\)\(
\)\(AsymptoticCond[a_, c_] := 
      Sign[a \((1 + c^2)\) + c \((1 + a^2)\)]
 (*\ 
      negative\ means\ that\ for\ big\ t, \ 
      the\ beam\ is\ inside\ the\ 
area\ of\ influence\ 
*) 

    AsymptoticCondNum[a_, c_] := 
      a \((1 + c^2)\) + c \((1 + a^2)\)
 (*\ 
      negative\ means\ that\ for\ big\ t, \ 
      the\ beam\ is\ inside\ the\ 
area\ of\ influence\ 
*) 


    {
{AsymptoticCond[1, 1], 
          AsymptoticCond[\(-1\), \(-1\)], 
          AsymptoticCond[\(-1\), 
            0]}, 
{AsymptoticCond[\(-1\), 1], 
          AsymptoticCond[1, 0], 
          AsymptoticCond[0, 0]}
}\  // \ 
      MatrixForm

    \)\)\)

AsymptoticCondNum[-1, 0.9]
AsymptoticCondNum[-0.1, 0]

maxt = 200;
makePlot[-1, 0, 
      0.9, "a=-1, c=0.9", {{0, 10, 20, 30, 40, 50, 100, 120, 
          160}, {-1, 0, 1, 2, 3, 4}}]
maxt = 40;
makePlot[-0.1, 0, 
      0, "a=-0.1, c=0", {{0, 10, 20, 30, 40, 50}, {-1, 0, 1, 2, 3, 
          4}}]