g[a_]:=(a+1/a)/2;
v[a_]:=(1/a-a)/(1/a+a);
a1=.5;
onesol[x_,t_,a_]:=4 ArcTan[Exp[ g[a] (x-v[a] t+6)]];
Animate[Plot[onesol[x,t,a1],{x,-10,10}],{t,0,15},AnimationRunning->False]

twosol[x_,t_]:=4 ArcTan[(a2+a1)/(a1-a2) Sinh[(-g[a2](x-v[a2] t-x2)+g[a1](x-v[a1] t-x1))/2]/Cosh[(g[a2] (x-v[a2] t-x2)+g[a1] (x-v[a1] t-x1))/2]];

a1=.5;a2=.1;x1=-20;x2=-50;L=70;
v[a1]
v[a2]

Plot[twosol[x,0],{x,-L,L},PlotRange->Full]

Animate[Plot[twosol[x,t],{x,-L,2L},PlotRange->{-10,10}],{t,0,150},AnimationRunning->False]

Periodic (Breather)

onep[x_,t_]:=4 ArcTan[Tan[a] Sin[Cos[a] (t Cosh[b]-x Sinh[b])-c]/Cosh[Sin[a] (x Cosh[b]-t Sinh[b]-q Cosh[b])]];

Simplify[D[onep[x,t],t,t]-D[onep[x,t],x,x]+Sin[onep[x,t]]]

b=.5;
c=0;
a=Pi/4;
q=.3;
Plot[{onep[x,0],onep[x,2 Pi/Cos[a]Cosh[b]]},{x,-10,10}]

Animate[Plot[onep[x,t],{x,-10,15},PlotRange->{-50,55}],{t,0,2 Pi/Cos[a]Cosh[b]},AnimationRunning->False]