Thanks to everybody for the comments.

What I am using as a "toy" system is just the simplest form of the equations that I came to from my original system.

As jacubi suggests, to overcome the problem I could simply change variables to make the system a "time evolving" one that maple will like. However, it is a rather complicated way of solving a problem that would require a very simple numerical discretization.

The current system that I have to solve is written in a moving frame where it is possible to apply the SVEA (slowly varying moving approximation) condition of nonlinear optics.

To answer to Robert's comment, you are quite right I changed the initial condition to "the right one" (u(x) = exp(x)) however the problem still remains.

> pdsolve(PDE);
              /                                  /  t  \ 
             { v(x, t) = _C1 exp(_c[1] x) _C2 exp|-----|,
              \                                  \_c[1]/ 

                                                       /  t  \\
               u(x, t) = _C1 _c[1] exp(_c[1] x) _C2 exp|-----| }
                                                       \_c[1]//
 

From this solution I just plugged in the initial and boundary conditions that correspond to _C1=1 and _C2=1:

> IBC := {u(x, 0) = exp(x), v(0, t) = exp(t)}:

However Maple gives the error that I mentioned before. Yes, I guess this is not the standard PDEs that Maple expect as for the second equation there is no time evolution. . Unfortunately the problem that I have to solve (relativistic nonlinear optics) has a similar structure and gives the same error.