Something seems missing. In which way you say  "using evalc
even produces values beyond Pi."?

Mapleprimes is an excellent place for group psychotherapy and elaborate frustrations about Maple usage experiences.

Inconsistencies in command syntax are so generalized that it might be healthier to deal with them with a bit of humour (though occasionally I have posted about some of them).

Reasons for these inconsistencies include, I think, the long history of Maple (almost 30 years) traversing generations of developers, and its rather individualistic development "model".

With time, hopefully, some inconsistencies will be removed. But likely, many more will will be added...

Mapleprimes is an excellent place for group psychotherapy and elaborate frustrations about Maple usage experiences.

Inconsistencies in command syntax are so generalized that it might be healthier to deal with them with a bit of humour (though occasionally I have posted about some of them).

Reasons for these inconsistencies include, I think, the long history of Maple (almost 30 years) traversing generations of developers, and its rather individualistic development "model".

With time, hopefully, some inconsistencies will be removed. But likely, many more will will be added...

The option UseAssumptions was introduced in Maple 13.

If you want a program that does what you want, without asking, you should write it yourself...

The option UseAssumptions was introduced in Maple 13.

If you want a program that does what you want, without asking, you should write it yourself...

It is clear that the generic algorithms, in their default setting, are not able to solve this system. And even trying to obtain the determining system for its Lie symmetry generators (which is an algorithmic computation):

DeterminingPDE(sysl);

does not produce any output in a reasonable time. 

So, most likely, some expertise in this field of soliton solutions would be required to guide the system towards a solution. There are many tools that could be used (see ?PDEtools and ?PDEtools,Library ).

It is clear that the generic algorithms, in their default setting, are not able to solve this system. And even trying to obtain the determining system for its Lie symmetry generators (which is an algorithmic computation):

DeterminingPDE(sysl);

does not produce any output in a reasonable time. 

So, most likely, some expertise in this field of soliton solutions would be required to guide the system towards a solution. There are many tools that could be used (see ?PDEtools and ?PDEtools,Library ).

No, fname as it stands originally writes OK in the user directory. The only change that is needed is the replacement of the commented line below with the following one:

  #    cat("\"",kernelopts('bindir'),"\\mint","\""),
     cat(kernelopts('bindir'),"/mint"),

This way I get (in Maple 13 Classic) the same output as shown in the examples above.

So, it may be better to begin from the begining. You may try:

with(PDEtools):
TWSolutions({NNV,Aux1,Aux2}); 

This way I get 8 solutions as polynomials of tanh. See also other options from ?PDEtools[TWSolutions] .

So, it may be better to begin from the begining. You may try:

with(PDEtools):
TWSolutions({NNV,Aux1,Aux2}); 

This way I get 8 solutions as polynomials of tanh. See also other options from ?PDEtools[TWSolutions] .

My first thought about this system of PDEs is that problems might arise because of the unspecified functions p(x,t) and q(y,t) in the coefficients. However, the solutions you have provided show that this system has some symmetry. The problem being whether  pdsolve is able to catch it.

So for removing possible complications, I have chosen values for these two functions so that your solutions become constants:

sys2 := (`@`(simplify, eval))(sysl, [q(y, t) = y, p(x, t) = x]):

sol2 := eval([a[0](x, y, t) = -2*(diff(p(x, t), x))*
(diff(q(y, t), y)), a[1](x, y, t) = 0, a[2](x, y, t) = 
2*(diff(p(x, t), x))*(diff(q(y, t), y)), b[0](x, y, t) = 
(diff(p(x, t), t)+diff(p(x, t), `$`(x, 3))-
2*(diff(p(x, t), x))^3)/(3*(diff(p(x, t), x))), b[1](x, y, t) = 
2*(diff(p(x, t), `$`(x, 2))), b[2](x, y, t) = 
2*(diff(p(x, t), x))^2, c[0](x, y, t) = (diff(q(y, t), t)+
diff(q(y, t), `$`(y, 3))-2*(diff(q(y, t), y))^3)/
(3*(diff(q(y, t), y))), c[1](x, y, t) = 2*(diff(q(y, t), 
`$`(y, 2))), c[2](x, y, t) = 2*(diff(q(y, t), y))^2], 
[q(y, t) = y, p(x, t) = x]):

[a[0](x, y, t) = -2, a[1](x, y, t) = 0, a[2](x, y, t) = 2, 
b[0](x, y, t) = -2/3, b[1](x, y, t) = 0, b[2](x, y, t) = 2, 
c[0](x, y, t) = -2/3, c[1](x, y, t) = 0, c[2](x, y, t) = 2]

However, neither pdsolve, nor InvariantSolutions provide a solution for this much simpler system in a short time by these commands:

pdsolve(sys2, {a[0](x, y, t), a[1](x, y, t), a[2](x, y, t), 
b[0](x, y, t), b[1](x, y, t), b[2](x, y, t), c[0](x, y, t), 
c[1](x, y, t), c[2](x, y, t)});

PDEtools[InvariantSolutions](sys2, {a[0](x, y, t), 
a[1](x, y, t), a[2](x, y, t), b[0](x, y, t), b[1](x, y, t), 
b[2](x, y, t), c[0](x, y, t), c[1](x, y, t), c[2](x, y, t)});

May be that the next step is trying to investigate the symmetries of this system. And also tracing pdsolve, in the hope of finding what goes wrong. Another question is whether that solution is unique.

My first thought about this system of PDEs is that problems might arise because of the unspecified functions p(x,t) and q(y,t) in the coefficients. However, the solutions you have provided show that this system has some symmetry. The problem being whether  pdsolve is able to catch it.

So for removing possible complications, I have chosen values for these two functions so that your solutions become constants:

sys2 := (`@`(simplify, eval))(sysl, [q(y, t) = y, p(x, t) = x]):

sol2 := eval([a[0](x, y, t) = -2*(diff(p(x, t), x))*
(diff(q(y, t), y)), a[1](x, y, t) = 0, a[2](x, y, t) = 
2*(diff(p(x, t), x))*(diff(q(y, t), y)), b[0](x, y, t) = 
(diff(p(x, t), t)+diff(p(x, t), `$`(x, 3))-
2*(diff(p(x, t), x))^3)/(3*(diff(p(x, t), x))), b[1](x, y, t) = 
2*(diff(p(x, t), `$`(x, 2))), b[2](x, y, t) = 
2*(diff(p(x, t), x))^2, c[0](x, y, t) = (diff(q(y, t), t)+
diff(q(y, t), `$`(y, 3))-2*(diff(q(y, t), y))^3)/
(3*(diff(q(y, t), y))), c[1](x, y, t) = 2*(diff(q(y, t), 
`$`(y, 2))), c[2](x, y, t) = 2*(diff(q(y, t), y))^2], 
[q(y, t) = y, p(x, t) = x]):

[a[0](x, y, t) = -2, a[1](x, y, t) = 0, a[2](x, y, t) = 2, 
b[0](x, y, t) = -2/3, b[1](x, y, t) = 0, b[2](x, y, t) = 2, 
c[0](x, y, t) = -2/3, c[1](x, y, t) = 0, c[2](x, y, t) = 2]

However, neither pdsolve, nor InvariantSolutions provide a solution for this much simpler system in a short time by these commands:

pdsolve(sys2, {a[0](x, y, t), a[1](x, y, t), a[2](x, y, t), 
b[0](x, y, t), b[1](x, y, t), b[2](x, y, t), c[0](x, y, t), 
c[1](x, y, t), c[2](x, y, t)});

PDEtools[InvariantSolutions](sys2, {a[0](x, y, t), 
a[1](x, y, t), a[2](x, y, t), b[0](x, y, t), b[1](x, y, t), 
b[2](x, y, t), c[0](x, y, t), c[1](x, y, t), c[2](x, y, t)});

May be that the next step is trying to investigate the symmetries of this system. And also tracing pdsolve, in the hope of finding what goes wrong. Another question is whether that solution is unique.

I think that this is an issue of priorities. Is LaTeX relevant for the main Maplesoft market?

The level of verbosity and usefulness of the error messages is quite multiform throughout the system. E.g. this is more useful in telling what is wrong:

dsolve(diff(f(x),x)=0,x,x=-1..1);

Error, (in ODEtools/info) found wrong extra arguments: {x, x = -1 .. 1}
 

Inserting such values for the functions into the system for l[0]=-1:

(`@`(simplify, eval))(sysl, 
[a[0](x, y, t) = -2*(diff(q(y, t), y))*(diff(p(x, t), x)), 
a[1](x, y, t) = 0, a[2](x, y, t) = 
2*(diff(q(y, t), y))*(diff(p(x, t), x)), 
b[0](x, y, t) = 
(diff(p(x, t), t)+diff(p(x, t), `$`(x, 3))-
2*(diff(p(x, t), x))^3)/(3*(diff(p(x, t), x))), 
b[1](x, y, t) = diff(p(x, t), `$`(x, 2)), 
b[2](x, y, t) = 2*(diff(p(x, t), x))^2, 
c[0](x, y, t) = 
(diff(q(y, t), t)+diff(q(y, t), `$`(y, 3))-
2*(diff(q(y, t), y))^3)/(3*(diff(q(y, t), y))), 
c[1](x, y, t) = diff(q(y, t), `$`(y, 2)), 
c[2](x, y, t) = 2*(diff(q(y, t), y))^2]);

some equations are satisfied (as 0=0) but other ones do not. Check whether their expressions and their translation from TeX notation are correct.