If we have an equation for the generalized Bloch sphere i.e.,

\partial_{t}(u^{2} + v^{2} + w^{2}) = 0,

where u, v and w are functions of x and t and the initial conditions u=v=0, w=-1. Then how to plot this equation on maple?

Is there any way to write the expression into the ratio of simple determinants? i..e,

h1=-i(c1*exp(A)-c2*exp(-A));

h2=c2*exp(A)+c1*exp(-A);

A=(i*a/2)*(x+b*t);

c1=sqrt(a+2*lambda)/a;

c2=sqrt(a-2*lambda)/a;

How to simplify the product of h1 and h2 in terms of trignometric functions?

h1*h2=?

If

p_{x} = a*p + u*q;    (1)

q_{x} = -conjugate(u)*p - a*q;  (2)     # where a is complex parameter, p_{x} mean derivative of p w.r.t x

Define, u=p^2+conjugate(q)^2,  (3)  

Now take the derivative of (3) w.r.t x and by using (1) and (2), we get

 u_{x}=2(a*p^2 - conjugate(a)*conjugate(q)^2) + 2*(p^2 + conjugate(q)^2)*(p*q - conjugate(p)*conjugate(q)).  (4)

How to calculate the results (4) on maple?

123.mw

I want to calculate (partialH/partialq). But I encountered an error when I evaluate it.

restart;
with(PDEtools);
with(linalg);
alias(q = q(x, t), p = p(x, t));
                              q, p
H := lambda*p*q+conjugate(lambda)*conjugate(p)*conjugate(q)+(1/2*(p^2+conjugate(q)^2))*(conjugate(p)^2+q^2);
                                      
diff(H, q(x, t));

maple shows error:

Error, invalid input: diff received q(x, t), which is not valid for its 2nd argument
How to fix this issue?