Parameters

λ=4;
H=0.4;
w=10λ;

The sine-Gordon equation

SG=ϕ''[z]-(1/(λ^2)) Sin[ϕ[z]] == 0;

Numerical integration

s=NDSolve[{SG,ϕ'[0]==H,ϕ'[w]==H},ϕ,{z,0,w}, Method->{"Shooting"}]

Plots

Plot[Evaluate[ϕ[z]/.s],{z,0,w},PlotRange->Full]

Evaluate[ϕ'[0]/.s]

Evaluate[ϕ'[w]/.s]

Plot[Evaluate[ϕ'[z]/.s],{z,0,w},PlotRange->Full]

Plot[{((Evaluate[Sin[ϕ[z]]/.s])/(λ^2)),Evaluate[ϕ''[z]/.s]},{z,0,w},PlotRange->Full]