out kk;
off nat;
%********************************************************
% EXAMPLE (16 08 97)
% 1 - soliton solutions of mKdV eq. 
% as curvature of moving curve in 2D euclidean space
%********************************************************


%for all xxx let cos(xxx)**2 + sin(xxx)**2 = 1;
%for all xxx let sinh(xxx) = sqrt(cosh(xxx)**2 - 1);
for all xxx let sinh(xxx)**2 = cosh(xxx)**2 - 1;
%for all xxx let sinh(xxx)**2 + 1 = cosh(xxx)**2;

%------- a curve in normal parametrization ---------------
%        s - normal parameter
%        p - parameter of evolution
%--------------------------------------------------------- 

xs:=  (2/a)*1/cosh(a*s - (a**3)*p - cx);
%us:=  s - (2/a)*sinh(a*s - (a**3)*p - cx)/cosh(a*s - (a**3)*p - cx) 
%        - (2/a)*sinh((a**3)*p + cx)/cosh((a**3)*p + cx) + cu;

us:=  s - (2/a)*sinh(a*s - (a**3)*p - cx)/cosh(a*s - (a**3)*p - cx) 
        - (2/a)*sinh(cx)/cosh(cx) + cu;


%------- tangent vector to the curve ---------------

ttx:=df(xs,s)/sqrt(df(xs,s)**2 + df(us,s)**2);
ttu:=df(us,s)/sqrt(df(xs,s)**2 + df(us,s)**2);

dtt:=sqrt( ttx**2 + ttu**2  );

%------- normal vector to the curve ---------------
nnx:=df(ttx,s)/sqrt(df(ttx,s)**2 + df(ttu,s)**2);
nnu:=df(ttu,s)/sqrt(df(ttx,s)**2 + df(ttu,s)**2);


%------- tangent curvature of the curve ---------------

kaptt:=sqrt( df(ttx,s)**2 + df(ttu,s)**2  );

kaptt:=sqrt( (( ttx*df(ttu,s) - df(ttx,s)*ttu  )/(dtt**3))**2 );


%------------------ mkdv test  ---------------------------


df(kaptt,p) + (3/2)*kaptt**2*df(kaptt,s) + df(kaptt,s,3);

%------- symmetry generator ---------------

vvx:=df(xs,p);
vvu:=df(us,p);
df(kaptt,s);
kaptt**2/2;

c1:=-kaptt**2/2;
c2:=-df(kaptt,s);

genx:=c1*ttx + c2*nnx;
genu:=c1*ttu + c2*nnu;
vvx;
vvu;
genx-vvx;
genu-vvu;

%***************************************************************
%                 x,u coordinates (pierwszy lisc)
%***************************************************************
% a*s - (a**3)*p - cx>=0 -----> sh():=sqrt(ch()**2 - 1)

cosh(a*s - (a**3)*p - cx):=  (2/a)*1/x;
sinh(a*s - (a**3)*p - cx):=  sqrt(cosh(a*s - (a**3)*p - cx)**2 - 1);

ttx;
ttu;
nnx;
nnu;
kaptt;
genx;
genu;
%---- generator z drugiego liscia, a*s - (a**3)*p - cx<0 
gen2x:=-genx;
gen2u:=genu;

eeex:=1;
eeeu:=ttu/ttx;
%----------commutators----------------

procedure agen(fff); rrr:=genx*df(fff,x) + genu*df(fff,u);
procedure agen2(fff); rrr:=gen2x*df(fff,x) + gen2u*df(fff,u);
procedure att(fff); rrr:=ttx*df(fff,x) + ttu*df(fff,u);
procedure ann(fff); rrr:=nnx*df(fff,x) + nnu*df(fff,u);
procedure aeee(fff); rrr:=eeex*df(fff,x) + eeeu*df(fff,u);
%----------------------
agen(eeex) - aeee(genx);
agen(eeeu) - aeee(genu);

agen(ttx) - att(genx);
agen(ttu) - att(genu);

agen2(ttx) - att(gen2x);
agen2(ttu) - att(gen2u);

ann(ttx) - att(nnx);
ann(ttu) - att(nnu);
%---------- pde ----------------

agen(kaptt);
att(kaptt);

c1*att(kaptt);
c2*kaptt**2;
att(att(c2));

agen(kaptt) - c1*att(kaptt) - c2*kaptt**2 - att(att(c2));

%***************************************************************
%                 x,u coordinates (drugi lisc)
%***************************************************************
% a*s - (a**3)*p - cx<0 -----> sh():=-sqrt(ch()**2 - 1)

cosh(a*s - (a**3)*p - cx):=  (2/a)*1/x;
sinh(a*s - (a**3)*p - cx):=  -sqrt(cosh(a*s - (a**3)*p - cx)**2 - 1);

ttx;
ttu;
nnx;
nnu;
kaptt;
genx;
genu;
%---- generator z pierwszego liscia, a*s - (a**3)*p - cx>=0 
gen1x:=-genx;
gen1u:=genu;

eeex:=1;
eeeu:=ttu/ttx;
%----------commutators----------------

procedure agen(fff); rrr:=genx*df(fff,x) + genu*df(fff,u);
procedure agen1(fff); rrr:=gen1x*df(fff,x) + gen1u*df(fff,u);
procedure att(fff); rrr:=ttx*df(fff,x) + ttu*df(fff,u);
procedure ann(fff); rrr:=nnx*df(fff,x) + nnu*df(fff,u);
procedure aeee(fff); rrr:=eeex*df(fff,x) + eeeu*df(fff,u);
%----------------------
agen(eeex) - aeee(genx);
agen(eeeu) - aeee(genu);

agen(ttx) - att(genx);
agen(ttu) - att(genu);

agen1(ttx) - att(gen1x);
agen1(ttu) - att(gen1u);

ann(ttx) - att(nnx);
ann(ttu) - att(nnu);

%---------- pde ----------------

agen(kaptt);
att(kaptt);

c1*att(kaptt);
c2*kaptt**2;
att(att(c2));

agen(kaptt) - c1*att(kaptt) - c2*kaptt**2 - att(att(c2));
%************************************************************
shut kk;
bye;
end;


shut kk;
bye;
end;