K=Assuming[k∈Reals && 0⩽k⩽1,Integrate[1/Sqrt[1-k^2 Sin[θ]^2],{θ,0,Pi/2}]]
e=Assuming[k∈Reals && 0⩽k⩽1,Integrate[Sqrt[1-k^2 Sin[θ]^2],{θ,0,Pi/2}]]

ν=Assuming[k∈Reals && 0⩽k⩽1,Sqrt[a b/m]*1/(2 K) *(1/(JacobiSN[2 K/λ,k]^2)-1+e/K)^(-1/2)]

sol=Assuming[k∈Reals && 0⩽k⩽1,Solve[Exp[-b rn[t_]]-1==m/(a b) (2K ν)^2 (JacobiDN[2(n/λ+ν t)K,k]^2-e/K),rn[t_]]]

Clear[a]
Clear[b]
m:=1
Clear[n]
Clear[k]
λ:=1

Manipulate[Plot[rn[t]/.sol/.k->k0/.a->a0/.b->b0/.n->n0,{t,-50,50},PlotStyle->{{Blue,Thick}},AxesLabel->{t,r},TextStyle->{FontWeight->Bold,FontSlant->"Plain",FontSize->20},AxesStyle->Thickness[0.005]],{k0,0.001,0.999},{a0,1,2},{b0,1,2},{n0,-10,10,1}]