p = .011;
ky = 10;
c = 27;
[CapitalOmega] = 2800/p;
L = p/0.3;
v = p^2* [CapitalOmega]/L;
A0 = 1;
xf=3;
[CapitalTheta][x_] := -(3 c (1 + p^2 ky^2)/(2 p^2*v*ky)) (Sech[
      x c/(2 p*Sqrt[c^2 - ky^2 A0^2])]^2);

pde1 = D[A[t, x] +
  p^2 (ky^2 A[t, x] +
     2 A0*[CapitalTheta][x]*D[[CapitalTheta]1[t, x], x] -
     D[A[t, x], x, x]), t] -
c*D[A[t, x] +
   p^2 (ky^2 A[t, x] +
      2 A0*[CapitalTheta][x]*D[[CapitalTheta]1[t, x], x] -
      D[A[t, x], x, x]), x] +
p^2 ([CapitalTheta][x])^2 (D[A[t, x], t] -
   c*D[A[t, x],
     x]) == ((v p^2 ky)/(1 + (p^2 ky^2))) (2*([CapitalTheta][x])*
   D[A[t, x], x] + A0*D[[CapitalTheta]1[t, x], x, x]);


pde2 = A0 (D[[CapitalTheta]1[t, x], t] -
       c*D[[CapitalTheta]1[t, x], x]) -
    p^2 (D[[CapitalTheta][x], x] (D[A[t, x], t] - c*D[A[t, x], x]) +
       D[A0 D[[CapitalTheta]1[x], x, x] +
         2 [CapitalTheta][x] D[A[t, x], x], t] -
       c*D[A0 D[[CapitalTheta]1[x], x, x] +
          2 [CapitalTheta][x] D[A[t, x], x],
         x]) == ((v p^2 ky)/(1 + (p^2 ky^2))) (2 A0 [CapitalTheta][
         x] D[[CapitalTheta]1[t, x], x] - D[A[t, x], x, x]) -
    p^2 ky A0 D[vy[t, x], x, x];


pde3 = (1/(p^4 [CapitalOmega]^2)) (D[vy[t, x], t] -
      c*D[vy[t, x], x]) ==
   ky A0^2 D[[CapitalTheta]1[t, x], x, x] +
    D[2 ky A0 A[t, x] [CapitalTheta][x], x];

pde = {pde1, pde2, pde3};

ic = {A[0, x] == 10^(-5),
   vy[0, x] == 0, [CapitalTheta]1[0, x] == 0};


bc = {A[t, xf] == 0, (D[A[t, x], x] /. x -> xf) ==
    0, (D[A[t, x], x] /. x -> 0) ==
    0, (D[[CapitalTheta]1[t, x], x] /. x -> xf) ==
    0, (D[[CapitalTheta]1[t, x], x] /. x -> 0) ==
    0, [CapitalTheta]1[t, xf] == 0,
   vy[t, xf] == 0, (D[vy[t, x], x] /. x -> 0) == 0};

begintime = 0; endtime = 80;
points@x = 200; points@t = 25;
m = 200;
difforder = 8;
domain@x = {0, 3}; domain@t = {begintime, endtime};
(grid@# = Array[# &, points@#, domain@#]) & /@ {x, t};


ptoafunc =
  pdetoae[{A, [CapitalTheta]1, vy}[t, x], grid /@ {t, x},
   difforder];
del = #[[2 ;; -2]] &;
ae = Map[del, Most /@ ptoafunc@pde, {2}];
aeic = del /@ ptoafunc@ic;
aebc = ptoafunc@bc;
var = Outer[#[#2, #3] &, {A, [CapitalTheta]1, vy}, grid@t, grid@x];
{barray, marray} =
   CoefficientArrays[Flatten@{ae, aeic, aebc},
    Flatten@var]; // AbsoluteTiming
Block[{p = .011,
    ky = 10,
    c = 27,
    [CapitalOmega] = 2800/p,
    L = p/0.3,
    v = p^2* [CapitalOmega]/L,
    A0 = 1},
   sollst = LinearSolve[N@marray, -barray]]; // AbsoluteTiming
solmatlst = ArrayReshape[sollst, var // Dimensions];
solfunclst = ListInterpolation[#, grid /@ {t, x}] & /@ solmatlst;
plot[f_, style_, t_] :=
 Plot[solfunclst[t, x] // Through // f //
   Evaluate, {x, domain@x} // Flatten // Evaluate,
  PlotStyle -> style]

{17.5364, Null}