Question: Graphical projection question: Projecting ODE trajectory solutions onto velocity manifolds (u(x,t), v(x,t))

Help needed to graphically map (x(t), t) trajectories onto velocity manifolds (u, v)

MRB: 16/6/19

The velocity manifolds u(x,t), v(x,t) are depicted in the figure below:

The above velocity plot above shows classical pairs of negative (red) low and high velocity, and pairs of positive (blue) low and high velocity.  At any given instant there are fast and slow propagating solutions, from which classical trajectories on (x,t) are derived below for each of the red and blue velocity pulse solutions.  The cut away regions of the velocity manifolds represent complex  and

We now solve the velocity ODE to plot a (x,t) trajectory on the red velocity manifold:

If the set of dependent variables is specified as a list, those variables appear in the same order in the output.  Now extract the solution points at given times

Here is a solution at a specified time:

Here is the the solution for the (x,t) trajectories:

We now plot trajectories on (x,t) space:

We do not generate trajectories in the region  for which regions of the manifolds are no longer real-valued.  The above trajectories are solutions of the above coupled Burger's equations for (u(x,t), v(x,t)). We wish plot these trajectories on their  [t,x(t),D(x(t)] manifolds.

The following is an example of how to project curves onto a surface, assuming an analytic form for the curves.  However, we wish to project the above two ODE solution trajectory curves onto the surfaces u(x,t) and v(x,t) which are solutions of two PDEs.

Below is an example of projecting a specified curve onto a surface  Our requirement is to project a curve which is an integral solution of the ODE (4) above onto one of the velocity manifolds u(x,y), v(x,y).  Below is an example of an algebraically specified trajectory being projected onto a manifold (surface):

QUESTION: We wish instead to project the ODE  (rather than algebraic) solution curves onto the surface.   Please someone provide MAPLE code to do that?