K:=Integrate[1/Sqrt[1-k^2 Sin[θ]^2],{θ,0,Pi/2}]
e:=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]-e/K),rn[t_]]]

a:=1
b:=1
m:=1
λ:=1

Manipulate[ListPlot[Table[{c,-(1/b)Log[1-(-(((Sqrt[1-(k^2)]) EllipticE[(k^2)])/(EllipticK[((k^2)/(-1+(k^2)))]))+JacobiDN[(1/(Sqrt[1-(k^2)]))2 EllipticK[((k^2)/(-1+(k^2)))] ((n/λ)+(((Sqrt[a b]) (Sqrt[1-(k^2)]) t)/(2 EllipticK[((k^2)/(-1+(k^2)))] (Sqrt[-1+(((Sqrt[1-(k^2)]) EllipticE[(k^2)])/(EllipticK[((k^2)/(-1+(k^2)))]))+(1/((JacobiSN[((2 EllipticK[((k^2)/(-1+(k^2)))])/((Sqrt[1-(k^2)]) λ)),k])^2))])))),k]^2)/(1-(((Sqrt[1-(k^2)]) EllipticE[(k^2)])/(EllipticK[((k^2)/(-1+(k^2)))]))-(1/((JacobiSN[((2 EllipticK[((k^2)/(-1+(k^2)))])/((Sqrt[1-(k^2)]) λ)),k])^2)))]/.n->c},{c,-20,20}],PlotRange->{-0.4,0.2},PlotStyle->Directive[PointSize[0.02],Purple],TextStyle->{FontWeight->Bold,FontSlant->"Plain",FontSize->20},AxesStyle->Thickness[0.005],AxesLabel->{n,r}],{k,0.001,0.999},{t,-200,200}]