Question: How to write vector fileds corresponding to infinitesimal transformations ?

Dear All

It is well known that the package "PDEtools" is helpful in finding infinitesimal transformations for PDEs which I illustrate as follow:


DepVars := [u(x, y, t)]

[u(x, y, t)]


declare(u(x, y, t)):

u(x, y, t)*`will now be displayed as`*u


U := diff_table(u(x, y, t)):

PDE1 := U[t, x]+(3/2)*u(x, y, t)*U[x, x]+(3/2)*U[x]^2+(1/4)*U[x, x, x, x]+(3/4)*sigma*U[y, y] = 0:

G := [seq(xi[j](x, y, t, u), j = [x, y, t]), seq(eta[j](x, y, t, u), j = [u])]:


eta(x, y, t, u)*`will now be displayed as`*eta


xi(x, y, t, u)*`will now be displayed as`*xi


DetSys := DeterminingPDE(PDE1, G, integrabilityconditions = false):


{eta[u](x, y, t, u) = (1/9)*(-2*(diff(diff(diff(_F1(t), t), t), t))*y^2-4*(diff(diff(_F2(t), t), t))*y+6*sigma*(-(3/2)*(diff(_F1(t), t))*u+(1/2)*(diff(diff(_F1(t), t), t))*x+diff(_F3(t), t)))/sigma, xi[t](x, y, t, u) = (3/2)*_F1(t)+_C1, xi[x](x, y, t, u) = (1/6)*(-2*(diff(diff(_F1(t), t), t))*y^2-4*(diff(_F2(t), t))*y+3*sigma*((diff(_F1(t), t))*x+2*_F3(t)))/sigma, xi[y](x, y, t, u) = (diff(_F1(t), t))*y+_F2(t)}


The set (4) gives infinitesimal transformations. How we can write  vector fields corresponding to arbitrary constant C1and arbitrary functions "F1(t), F2(t), F3(t) "?"" 




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