Evolutionary Systems and Flows on Solutions Spaces of Finite Type Equations

Abstract

The article is devoted to extending the methods of the theory of finite dimensional dynamics to systems of evolutionary differential equations with several space variables.

As an example, we consider the Boussinesq system with two space variables. Finite dimensional dynamics and parametric family of exact solutions are constructed for this equation.

APPENDIX

Here we provide the Maple code for computing dynamics and solutions of the Boussinesq equations from the previous section.

>restart;

>with(DifferentialGeometry):with(JetCalculus):with(Tools):with(PDETools):

>Preferences("JetNotation", "JetNotation2"):

>DGsetup([x,y], [v,w], M, 3, verbose);

>e1:=diff(u(t, x, y),t)-diff(u(t, x, y),x$ 2)-2*diff(v(t, x, y),x):

e2:=diff(v(t, x, y),t)+diff(v(t, x, y),x$ 2)-2*u(t, x, y)*diff(u(t, x, y),x)

+2*diff(u(t, x, y),y):

>psi1:=v[2,0]+2*w[1,0]; psi_2:=-w[2,0]+2*v[0]*v[1,0]-2*v[0];

>A:=a(x, y): B:=b(x, y): C:=c(x, y): P:=p(x, y): Q:=q(x, y): R:=r(x, y):

>F:=v[2,0]=A, v[1,1]=B,v[0]=C, w[2,0]=P, w[1,1]=Q,w[0]=R:

>F3:={map(TotalDiff,F[1],x),map(TotalDiff,F[1],y),map(TotalDiff,F[2],x),

map(TotalDiff,F[2],y),map(TotalDiff,F[3],x),map(TotalDiff,F[3],y),

map(TotalDiff,F[4],x),map(TotalDiff,F[4],y)}:

>sub:={op(F2),op(F3)}:

>phi1:=eval(psi1,sub); phi2:=eval(psi2,sub);

>omega[1]:=eval(evalDG(dv[0]-v[1,0]*dx-v[0]*dy),sub);

omega[2]:=eval(evalDG(dw[0]-w[1,0]*dx-w[0]*dy),sub);

omega[3]:=eval(evalDG(dv[1,0]-v[2,0]*dx-v[1,1]*dy),sub);

omega[4]:=eval(evalDG(dv[0]-v[1,1]*dx-v[0]*dy),sub);

omega[5]:=eval(evalDG(dw[1,0]-w[2,0]*dx-w[1,1]*dy),sub);

omega[6]:=eval(evalDG(dw[0]-w[1,1]*dx-w[0]*dy),sub);

>Frob:=[]:

for i from 3 by 1 to 6 do Frob:=[op(Frob),

evalDG(ExteriorDerivative(omega[i]) & w omega[1] & w

omega[2] & w omega[3] & w omega[4] & w omega[5] & w omega[6])] end do:

W:=[]:

for i from 1 by 1 to nops(Frob) do W:=[op(W),op(DGinfo(Frob[i],

"CoefficientList", "all"))] end do:

>phi1:=eval(psi1,sub): phi2:=eval(psi2,sub):

S:=evalDG(phi1*D_v[0]+phi2*D_w[0]+TotalDiff(phi1,x)*D_v[1,0]

+TotalDiff(phi2,x)*D_w[1,0]+TotalDiff(phi1,y)*D_v[0]+TotalDiff(phi2,y)*D_w[0]):

>S:=eval(S,sub);

>mu:=evalDG(omega[1] & w omega[2] & w omega[3] & w

omega[4] & w omega[5] & w omega[6]):

>Lee:=alpha->DGinfo(evalDG(LieDerivative(S,alpha) & w mu),

"CoefficientList", "all"):

>L:=W:

for i from 3 to 6 do L:=[op(L),op(Lee(omega[i]))] end do:

>cf:=z->coeffs(collect(z,{v[2, 0], v[1, 1], v[0, 2], w[2, 0], w[1, 1],

w[0]},’distributed’),{v[2, 0], v[1, 1], v[0, 2], w[2, 0], w[1, 1], w[0]});

>N:=[]:

for i from 1 to nops(L) do N:=[op(N),cf(L[i])] end do:

>cfp:=z->coeffs(collect(z,v[0],w[0],v[1, 0], v[0], w[1, 0], w[0],

’distributed’),v[1, 0], v[0], w[1, 0], w[0, 1]);

>G:=L:

for i from 1 to nops(N) do G:=[op(G),cfp(N[i])] end do:

>solution_G :=pdsolve(G);

As a result, we get

$$solution\_G:=\{a(x,y)=0,b(x,y)=0,c(x,y)=0,$$

$$p(x,y)=\_C1,q(x,y)=\_C2,r(x,y)=\_F1(y)\}$$

>solution_G1:=eval(solution_G,{_C1=eta,_C2=delta,_F1(y)=h(y)}):

>F:=eval(F,solution_G1);

>F3:={map(TotalDiff,F[1],x),map(TotalDiff,F[1],y),

map(TotalDiff,F[2],x),map(TotalDiff,F[2],y),

map(TotalDiff,F[3],x),map(TotalDiff,F[3],y),

map(TotalDiff,F[4],x),map(TotalDiff,F[4],y)};

>sub2:={op(F),op(F3)}:

>S:=eval(eval(S,solution_G1),sub2):

>Phi:=Flow(S,t):

>Xi:=simplify(InverseTransformation(Phi),power):

>dF:=convert(F,DGdiff):

>dF:=eval(eval(dF,solution_G1),sub2):

>sol_dF:=pdsolve(dF):

We get

$$sol\_dF:=\{h(y)=diff(\_F2(y),y,y),v(x,y)=\_C3y+\_C1x$$

$$+\_C2,w(x,y)=(1/2)x^{2}\eta+x\delta y+\_C4x+\_F2(y)\}$$

Therefore, we put

>V:=[v[0]-(C1*x+C2*y+C3),w[0] -((1/2)*eta*x2̂

+x*delta*y+g(y)+x*C4)]:

>vec:=[V[1],V[2],TotalDiff(V[1],x),TotalDiff(V[2],x),

TotalDiff(V[1],y),TotalDiff(V[2],y)]:

> PV:=Pullback(Xi,vec):

>resh:=solve(PV,v[0],w[0],v[1,0],w[1,0],v[0],w[0]):

>zam:={u(t,x,y)=rhs(resh[1]),v(t,x,y)=rhs(resh[4])}

This is a required solution. Let us check:

simplify(eval([e1,e2],zam));

We have $[0,0]$.