In[]:=
func[x_,y_,t_]=With[{ξ=a1x+a2y+a3t+a4},U[ξ]]​​pde=β1*D[ψ[x,y,t],{t,1}]+β2*
2
ψ[x,y,t]
*D[ψ[x,y,t],{x,1}]-D[ψ[x,y,t],{x,3}]-β3*D[ψ[x,y,t],{x,1}]*Integrate[D[ψ[x,y,t],{y,1}],x]-β4*ψ[x,y,t]*D[ψ[x,y,t],{y,1}]+β5*Integrate[D[ψ[x,y,t],{y,2}],x]0;​​​​Assuming[{{x,y,t,a1,a2,a3,a4}∈Reals},pde/.ufunc//Simplify]/.a1x+a2y+a3t+a4ξ
Out[]=
U[a4+a3t+a1x+a2y]
Out[]=
β5∫
(0,2,0)
ψ
[x,y,t]x+β1
(0,0,1)
ψ
[x,y,t]+β2
2
ψ[x,y,t]
(1,0,0)
ψ
[x,y,t]β4ψ[x,y,t]
(0,1,0)
ψ
[x,y,t]+β3∫
(0,1,0)
ψ
[x,y,t]x
(1,0,0)
ψ
[x,y,t]+
(3,0,0)
ψ
[x,y,t]