Your PDEs system can be solved exactly with *help* of Maple by the following scheme ( I leave out bulky outputs).

with(PDETools);

sys:=[diff(v(s,n),n) + diff(u(s,n),s) +diff(xi(s,n),s) + A*n*diff(c(s),s) = 0,

A1*diff(xi(s,n),n) + diff(v(s,n),s) -c(s)+A2*v(s,n) + A3* c(s) = 0,

diff(u(s,n),s) + 2*A2* u(s,n)=A2*(xi(s,n) + A*n*c(s)) -A1*diff(xi(s,n),s)-A2*n*c(s)];

Here, as I understand, c(s) is a parameter.

From the beginning we need to split the system

cas:=[casesplit(sys,[xi,u,v])];

As a result we obtain system of 4 equations, which is equivalent to initial one. First of all we have to solve the last equation

pp := diff(v(s,n),`$`(s,3)) = (diff(c(s),`$`(s,2))*A1-3*A1*A2*A*diff(c(s),s)-A1*diff(v(s,n),`$`(n,2),s)-2*A2*A1*diff(v(s,n),`$`(n,2))-A1*A*diff(c(s),`$`(s,2))+A1*A2*diff(c(s),s)-A3*diff(c(s),`$`(s,2))*A1-A2*diff(v(s,n),`$`(s,2))*A1+4*A2*diff(v(s,n),`$`(s,2))-diff(c(s),`$`(s,2))+3*A2*A3*diff(c(s),s)+3*A2^2*diff(v(s,n),s)-3*A2*diff(c(s),s)+A3*diff(c(s),`$`(s,2)))/(-1+A1);

for unknown function v(s,n). It is linear non-homogeneous third-order PDE (so, generally speaking, we need *three* initial conditions !). One of exact solutions of this PDE can be obtained

an0:=subs({_c[2]=0,_C1=0,_C2=0,_C3=0,_C4=0,_C5=0

},pdsolve(pp,v, INTEGRATE, build));

Then we will seek the solution in the familiar form

an_W := v(s,n) = w(s,n) + rhs(an0);

which lead to the *homogeneous *linea*r* PDE with *constant* coefficients for w(s,n)

ppp:=collect(numer(factor(pdetest(an_W,pp))),diff);

*ppp := -3*A2^2*diff(w(s,n),s)+2*A2*A1*diff(w(s,n),`$`(n,2))+(A1*A2-4*A2)*diff(w(s,n),`$`(s,2))+A1*diff(w(s,n),`$`(n,2),s)+(-1+A1)*diff(w(s,n),`$`(s,3));*

* *

This equation can be solved by Fourier method or by separation of variables (with similar result). Suppose that

an_w:=w(s,n)=Int(Phi(s,omega)*exp(I*n*omega),omega=-infinity..infinity);

and from

pdetest(an_w,ppp);

after some manipulation (by hand) we find that Phi(s,omega) must fit the following ODE

WW := (-A1*omega^2-3*A2^2)*diff(Phi(s,omega),s)+(A1*A2-4*A2)*diff(Phi(s,omega),`$`(s,2))+(-1+A1)*diff(Phi(s,omega),`$`(s,3))-2*A2*A1*Phi(s,omega)*omega^2;

which can be solved

ans_w:=pdsolve(WW);

Substituting it to an_W we obtain the general solution for v(s,n). Now to produce a complete solution we need to specify 3 arbitrary functions {_F1(omega), _F2(omega), _F3(omega)} in above solution by standard way from general solution and the set of initial conditions .

Then substituting it to cas we get the split system for u and xi, which can be easily solved too .