X1[f_,A_,U_, G_]=Integrate[1/Sqrt[2G(A-2 Sin[1/2f]^2)/(U-1)],f]
X2[f_,A_,U_,G_]=Integrate[1/-Sqrt[2G(A-2 Sin[1/2f]^2)/(U-1)],f]

h[y_,A_,U_,G_]= f/.Solve[X1[f,A,U, G] == y,f][[1]]

(*f0 := 2ArcSin[Sqrt[1/2]]*)

X (f)

Manipulate[Plot[{X1[f,A,U, G],X2[f,A,U, G]},{f, -2Pi,2 Pi}, PlotRange->{{-2Pi, 2Pi}, {-10,10}},AxesLabel-> {"f","X"}],{A,-1,3},{U,0,2},{{G,1},-1,1,2}]

Fall 1 : 0 < A < 2, (U^2)> 1

Manipulate[Plot[{h[x,A,U, 1]},{x,-20,20}, PlotRange ->{{-10,10},{-2Pi, 2Pi}},PlotPoints -> 200, PlotStyle-> Thickness[0.005],AxesLabel-> {"X","f"}],{A,0.1,2},{U,1.1,2}]

Fall 2 : 0 < A < 2, (U^2)< 1

Manipulate[Plot[h[x-Sqrt[U] t+10,A,U, -1],{x,-10,10}, PlotRange ->{{-10,10},{-2Pi, 2Pi}},PlotStyle-> Thickness[0.005],AxesLabel-> {"X","f"}],{A,0.1,2},{U,0,0.9},{t,0,50}]

Manipulate[Show[ParametricPlot3D[{{x,Sin[h[x-Sqrt[U] t,A,U, -1]],-Cos[h[x-Sqrt[U] t,A,U, -1]]},{x,0,0}},{x,0,2Pi}, PlotRange ->{{0,2Pi},{-1,1},{-1,1}},PlotStyle-> {{Red,Thickness[0.01]},{Gray,Thickness[0.001]}},AxesLabel-> {"X","f"}],Graphics3D[{Blue,Table[Arrow[{{x,0,0},{x,Sin[h[x-Sqrt[U] t,A,U, -1]],-Cos[h[x-Sqrt[U] t,A,U, -1]]}}],{x,0,2Pi,0.2}]}],ImageSize->600],{A,0.1,2},{U,0,0.9},{t,0,50}]

Fall 3 : A < 0 , (U^2)< 1

Manipulate[Plot[h[x-Sqrt[U] t+10,A,U, 1],{x,-10,10}, PlotRange ->{{-10,10},{-2Pi, 2Pi}},PlotStyle-> Thickness[0.005],AxesLabel-> {"X","f"}],{A,-1,0},{U,0,0.9},{t,0,50}]

Manipulate[Show[ParametricPlot3D[{{x,Sin[h[x-Sqrt[U] t,A,U, 1]],-Cos[h[x-Sqrt[U] t,A,U, 1]]},{x,0,0}},{x,0,2Pi}, PlotRange ->{{0,2Pi},{-1,1},{-1,1}},PlotStyle-> {{Red,Thickness[0.01]},{Gray,Thickness[0.001]}},AxesLabel-> {"X","f"}],Graphics3D[{Blue,Table[Arrow[{{x,0,0},{x,Sin[h[x-Sqrt[U] t,A,U, 1]],-Cos[h[x-Sqrt[U] t,A,U, 1]]}}],{x,0,2Pi,0.2}]}],ImageSize->600],{A,-1,0},{U,0,0.9},{t,0,50}]

Fall 4 :  A > 2, (U^2)>1

Manipulate[Plot[h[x-Sqrt[U]t+10,A,U, 1],{x,-10,10}, PlotRange ->{{-10,10},{-2Pi, 2Pi}},PlotStyle-> Thickness[0.005],AxesLabel-> {"X","f"}],{A,2.1,3},{U,1.1,2},{t,0,50}]

Manipulate[Show[ParametricPlot3D[{{x,Sin[h[x-Sqrt[U] t,A,U, 1]],-Cos[h[x-Sqrt[U] t,A,U, 1]]},{x,0,0}},{x,0,2Pi}, PlotRange ->{{0,2Pi},{-1,1},{-1,1}},PlotStyle-> {{Red,Thickness[0.01]},{Gray,Thickness[0.001]}},AxesLabel-> {"X","f"}],Graphics3D[{Blue,Table[Arrow[{{x,0,0},{x,Sin[h[x-Sqrt[U] t,A,U, 1]],-Cos[h[x-Sqrt[U] t,A,U, 1]]}}],{x,0,2Pi,0.2}]}],ImageSize->600],{A,2.2,3},{U,1.1,2},{t,0,5}]

Fall 5 :  A = 2, (U^2)>1

Manipulate[Plot[h[x-Sqrt[U] t+10,2,U, 1],{x,-10,10}, PlotRange ->{{-10,10},{-2Pi, 2Pi}},PlotStyle-> Thickness[0.005],AxesLabel-> {"X","f"}],{U,1.1,2},{t,0,30}]

Manipulate[Show[ParametricPlot3D[{{x,Sin[h[x-Sqrt[U] t,2,U, 1]],-Cos[h[x-Sqrt[U] t,2,U, 1]]},{x,0,0}},{x,0,2Pi}, PlotRange ->{{0,2Pi},{-1,1},{-1,1}},PlotStyle-> {{Red,Thickness[0.01]},{Gray,Thickness[0.001]}},AxesLabel-> {"X","f"}],Graphics3D[{Blue,Table[Arrow[{{x,0,0},{x,Sin[h[x-Sqrt[U] t,2,U, 1]],-Cos[h[x-Sqrt[U] t,2,U, 1]]}}],{x,0,2Pi,0.2}]}],ImageSize->600],{U,1.1,2},{t,0,7}]

Fall 6 :  A = 0, (U^2)<1

X[f_,A_, U_]= Integrate[1/Sqrt[2(-2  Sin[1/2f]^2)/(U-1)],f]

Logarithmengesetze: (kennt Mathematica nicht?)

(((Sqrt[2]) (Log[Tan[(f/4)]]) Sin[(f/2)])/(Sqrt[((-1+Cos[f])/(-1+U))]))

Umformen: G ist Faktor für +/- 
(((Sqrt[2]) (Sqrt[1-U])(Log[Tan[(f/4)]]) Sin[(f/2)])/(Sqrt[1-Cos[f]])) = (Sqrt[1-U])(Log[G Tan[(f/4)]]

X[f_,A_,U_, G_] =  (Sqrt[1-U])Log[G Tan[(f/4)]]

F[y_,A_, U_, G_]=  f/.Solve[X[f,A,U, G] == y,f][[1]]

Manipulate[Plot[{X[f,A,U, G], -X[f,A,U, G]}, {f,-2Pi,2 Pi}, PlotRange ->{{-2Pi, 2Pi},{-10,10}}],{U,0,1},{{G,1},-1,1,2}]

Manipulate[Plot[{F[x-Sqrt[U] t+10,0,U, G],F[-(x-Sqrt[U] t+10),0,U, G]}, {x,-10,10}, PlotRange ->{{-10,10},{-2Pi, 2Pi}}],{U,0,1},{{G,1},-1,1,2},{t,0,60}]

Manipulate[Show[ParametricPlot3D[{{x,Sin[F[x-Sqrt[U] t,0,U, 1]],-Cos[F[x-Sqrt[U] t,0,U, 1]]},{x,0,0}},{x,0,2Pi}, PlotRange ->{{0,2Pi},{-1,1},{-1,1}},PlotStyle-> {{Red,Thickness[0.01]},{Gray,Thickness[0.001]}},AxesLabel-> {"X","f"}],Graphics3D[{Blue,Table[Arrow[{{x,0,0},{x,Sin[F[x-Sqrt[U] t,0,U, 1]],-Cos[F[x-Sqrt[U] t,0,U, 1]]}}],{x,0,2Pi,0.2}]}],ImageSize->600],{U,0,0.9},{t,0,30}]

Ableitung ist ein hügelförmiges Soliton:

Abl[x_, U_]= 4 Sech[x/Sqrt[1-U]]/Sqrt[1-U]

Manipulate[Plot[{Abl[x- Sqrt[U] t+ 10,U],-Abl[x- Sqrt[U] t +10,U]}, {x,-2Pi, 2Pi}, PlotRange ->{{-2Pi, 2Pi},{-10,10}}],{U,0,1},{t,0,60}]

Wechselwirkung zweier Solitonen:

Ψ1[x_, t_, U_] = 4 ArcTan[ U Sinh[x/Sqrt[1-U^2]]/Cosh[U t/Sqrt[1-U^2]]]

Manipulate[Plot[Ψ1[x,t,Sqrt[U]],{x,-10, 10}, PlotRange->{{-10,10},{-2Pi, 2Pi}}],{t,-32,32},{U,0.1,0.9}]

Manipulate[Show[ParametricPlot3D[{{x,Sin[Ψ1[x,t,3/4]],-Cos[Ψ1[x,t,3/4]]},{x,0,0}},{x,-10,10}, PlotRange ->{{-10,10},{-1,1},{-1,1}},PlotStyle-> {{Red,Thickness[0.005]},{Gray,Thickness[0.001]}},AxesLabel-> {"X","f"}],Graphics3D[{Blue,Arrowheads[0.02],Table[Arrow[{{x,0,0},{x,Sin[Ψ1[x,t,3/4]],-Cos[Ψ1[x,t,3/4]]}}],{x,-10,10,0.5}]}],ImageSize->600],{t,-16,16}]

Ψ3[x_, t_, U_] = 4 ArcTan[U Cosh[x/Sqrt[1-U^2]]/Sinh[U t/Sqrt[1-U^2]]]

Manipulate[Plot[Ψ3[x,t,3/4],{x,-10, 10}, PlotRange->{{-10,10},{-2Pi, 2Pi}}],{t,-16,16}]

Manipulate[Show[ParametricPlot3D[{{x,Sin[Ψ3[x,t,3/4]],-Cos[Ψ3[x,t,3/4]]},{x,0,0}},{x,-10,10}, PlotRange ->{{-10,10},{-1,1},{-1,1}},PlotStyle-> {{Red,Thickness[0.005]},{Gray,Thickness[0.001]}},AxesLabel-> {"X","f"}],Graphics3D[{Blue,Arrowheads[0.02],Table[Arrow[{{x,0,0},{x,Sin[Ψ3[x,t,3/4]],-Cos[Ψ3[x,t,3/4]]}}],{x,-10,10,0.5}]}],ImageSize->600],{t,-16,16}]

Wechselwirkung hügelförmiger Solitonen:

Ψx[x_,t_,U_] = 4 U Cosh[x/Sqrt[1-U^2]]Cosh[U t/Sqrt[1-U^2]]/(Sqrt[1-U^2](U^2Sinh[x/Sqrt[1-U^2]]^2+Cosh[U t/Sqrt[1-U^2]]^2))

Manipulate[Plot[Ψx[x,t,Sqrt[U]],{x,-10, 10}, PlotRange->{{-10,10},{0, 10}}],{t,-32,32},{U,0.1,0.9}]

Ψx3[x_,t_,U_ ]= D[Ψ3[x,t,U],x]

Manipulate[Plot[Ψx3[x,t,Sqrt[3/4]],{x,-10, 10}, PlotRange->{{-10,10},{-2Pi, 2Pi}}],{t,-16,24}]

Phasenverschiebung :

Ψx1[x_,t_,U_] = D[Ψ1[x,t,U],x]

Ψx2[x_,t_,U_]= D[F[x,0,U,1],x]

Manipulate[Plot[{Ψ1[x,t,Sqrt[U]],F[x-Sqrt[U]t   ,0,U,1],F[x-Sqrt[U]t ,0,U,1]-2Pi},{x,-10, 10}, PlotRange->{{-10,10},{-2Pi, 2Pi}},PlotStyle->{Blue,Red,Red}],{t,-20,20},{U,0.1,0.9}]

Manipulate[Plot[{Ψx1[x,t,Sqrt[0.1]],Ψx2[x-Sqrt[0.1]t,t,0.1]},{x,-10, 10}, PlotRange->{{-10,10},{-2Pi, 2Pi}},PlotStyle->{Blue,Red,Red}],{t,-10,10}]

Breather:

Ψ2[x_,w_,v_, t_]=4 ArcTan[w^-1 Sqrt[1-w^2] Sin[Sqrt[1-v^2]^-1w (t-v x)]/Cosh[Sqrt[1-v^2]^-1Sqrt[1-w^2] (x-v t)]]

Manipulate[Plot[ Ψ2[x,w,v, t], {x,-8,8}, PlotRange -> {{-8,8},{-10,10}}],{w,0.1,1, 0.1},{{v,0},-0.1,0.1},{t,-128,128}]

Manipulate[Show[ParametricPlot3D[{{x,Sin[Ψ2[x,w,v, t]],-Cos[Ψ2[x,w,v, t]]},{x,0,0}},{x,-10,10}, PlotRange ->{{-10,10},{-1,1},{-1,1}},PlotStyle-> {{Red,Thickness[0.005]},{Gray,Thickness[0.001]}},AxesLabel-> {"X","f"}],Graphics3D[{Blue,Arrowheads[0.02],Table[Arrow[{{x,0,0},{x,Sin[Ψ2[x,w,v, t]],-Cos[Ψ2[x,w,v, t]]}}],{x,-10,10,0.5}]}],ImageSize->600],{w,0.05,0.2,0.025},{{v,0},-0.1,0.1},{t,-128,128}]

FullSimplify[D[Ψ1[x,t,U],{t,2}]- D[Ψ1[x,t,U],{x,2}] + Sin[Ψ1[x,t,U]] == 0]