δ=0.022;
sol=NDSolve[{(∂_t)u[x,t]+u[x,t](∂_x)u[x,t]+δ^2(∂_(x,x,x))u[x,t]==0,u[0,t]==u[2,t],u[x,0]==Cos[x Pi]},u[x,t],{t,0,50},{x,0,2},MaxStepSize->0.01,StartingStepSize->0.001,MaxSteps->10^5]

Manipulate[Plot[u[x,t]/.sol/.t->τ,{x,0,4},PlotRange->{{0,4},{-2,4}},ImageSize->Full,PlotPoints->100,PlotStyle->Thick],{τ,0,20,AnimationRate->0.1,Appearance->"Open"}]