Off[Solve::ifun]


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

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

Screw-shaped solutions (stable)
Solution for pendulum problem

U=(v^2)

With[{U=0.5,t=0,A=-1},Plot[h[x-U t,A,U, 1],{x,0,2Pi}]]

With[{U=0.5,t=0,A=-1},
Show[
Graphics3D[{
Red,PointSize[.02],Table[Point[{x,Sin[h[x-U t,A,U, 1]],-Cos[h[x-U t,A,U, 1]]}],{x,0,2π,0.1}],
RGBColor[.33,.6,.7],Thickness[.005],Table[Line[{{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,0.1}],
RGBColor[0.686,0.694,0.671],Thickness[.02],Line[{{0,0,0},{2π,0,0}}]}],
ImageSize->600]]

Bäcklund transformation

With[{a=2,t=0},Plot[4ArcTan[Exp[a x +(t/a)]],{x,-10,10}]]

Soliton solutions

u[x_,t_,v_,w_]:=Which[
w=="soliton",4*ArcTan[Exp[(x-v*t)/(1-v^2)]],
w=="antisoliton",4*ArcTan[Exp[-(x-v*t)/(1-v^2)]],
w=="soliton-soliton collision",4*ArcTan[v*Sinh[x/(1-v^2)]/Cosh[v*t/(1-v^2)]],
w=="soliton-antisoliton collision",4*ArcTan[Sinh[v*t/(1-v^2)]/(v*Cosh[x/(1-v^2)])],
w=="breather",4*ArcTan[Sin[v*t/(1-v^2)]/(v*Cosh[x/(1-v^2)])]
]

Manipulate[
Graphics3D[{
Table[{
RGBColor[.33,.6,.7],Thickness[.005],Line[{{0,x,0},{2*Sin[u[x,t,v,w]],x,-2*Cos[u[x,t,v,w]]}}],
RGBColor[0.686,0.694,0.671],Thickness[.02],Line[{{0,-7,0},{0,7,0}}],
Red,PointSize[.02],Point[{2*Sin[u[x,t,v,w]],x,-2*Cos[u[x,t,v,w]]}]},{x,-7,7,r}]},SphericalRegion->True,PlotRange->{{-2,2},{-7,7},{-2,2}},ImageSize->{800,600}, ViewPoint->{3, 2.4,3}
],
{{t,-20,"time"},-20,20,0.4},{{v,.5,"velocity parameter"},0.1,0.84},{{r,.32,"distance between the pendulums"},.2,.5},{{w,"soliton","solution"},{"soliton","antisoliton","soliton-soliton collision","soliton-antisoliton collision","breather"}},
SaveDefinitions->True]