out bb11; off nat; %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; for all xxx let asinh(sinh(xxx)) = xxx; for all xxx let sinh(asinh(xxx)) = xxx; %------- a curve in normal parametrization --------------- xs:= asinh(s + cx); us:= cosh(asinh(s + 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 ); %------- 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 ); %------- 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); %*************************************************************** % x,u coordinates %*************************************************************** s + cx:=sinh(x); xs; us; ttx; ttu; kaptt; nnx; nnu; kaptt; %genx:=ttx*df(-kaptt,x)*nnx + (-(kaptt**2)/2)*ttx; %genu:=ttx*df(-kaptt,x)*nnu + (-(kaptt**2)/2)*ttu; genx:=a*sinh(x)/cosh(x); genu:=1 + a*sinh(x)**2/cosh(x); c1:=(1/cosh(x) + a)*sinh(x); c2:=1/cosh(x); genx:=c1*ttx + c2*nnx; genu:=c1*ttu + c2*nnu; eeex:=1; eeeu:=ttu/ttx; procedure agen(fff); rrr:=genx*df(fff,x) + genu*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); %----------commutators---------------- agen(eeex) - aeee(genx); agen(eeeu) - aeee(genu); agen(ttx) - att(genx); agen(ttu) - att(genu); 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 bb11; bye; end;