Figure 2.nb

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

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

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

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(* Define G[t] = solution to Riccati equationG ' + G^2 = 0 G (0) = G_0 ... si1[t_] := Si[t] + Sqrt[Si[t]^2 - Ki[t]] si2[t_] := Si[t] - Sqrt[Si[t]^2 - Ki[t]] 

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${{λ1SS - (-KK + SS^2)^(1/2), λ2SS + (-KK + SS^2)^(1/2)}, {λ1SS + (-KK + SS^2)^(1/2), λ2SS - (-KK + SS^2)^(1/2)}}$

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maxt = 5 ; β = 0 ; α = 1 ; γ = 1 ;  pltOptions = {{Dashing[{0.01, 0.01} ... 371;PlotRange All, PlotLabeltitle, Ticks ticks] ]

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[Graphics:HTMLFiles/Figure 2_12.gif]

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[Graphics:HTMLFiles/Figure 2_15.gif]

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[Graphics:HTMLFiles/Figure 2_18.gif]

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[Graphics:HTMLFiles/Figure 2_21.gif]

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[Graphics:HTMLFiles/Figure 2_24.gif]

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[Graphics:HTMLFiles/Figure 2_27.gif]

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

[Graphics:HTMLFiles/Figure 2_30.gif]

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(* which beams should satisfy condition at infinity ? *)AsymptoticCond[a_, c_] ... AsymptoticCond[-1, 1], AsymptoticCond[1, 0], AsymptoticCond[0, 0]} } // MatrixForm

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

[Graphics:HTMLFiles/Figure 2_40.gif]

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[Graphics:HTMLFiles/Figure 2_42.gif]

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Created by Mathematica (March 10, 2006)