I’m attempting to resolve set of PDEs as proven under. As a result of a[x_, t_] := aL*Exp[-(x - t)^2/L^2] Sin[x - t] when x < -xmax and t<0, I put a[x,0] and D/Dt(a[x,0]) into ICs.

There seem a number of error messages, considered one of which is:

NDSolveValue::eerr: Warning: scaled native spatial error estimate of 208.66558109964893at t = 39.666155299563634 within the route of impartial variable x is far better than the prescribed error tolerance. Grid spacing with 135 factors could also be too massive to realize the specified accuracy or precision. A singularity could have shaped or a smaller grid spacing will be specified utilizing the MaxStepSize or MinPoints technique choices.

I modified worth of presiciongoal and accuracygoal, however not solved. Thus, I’ve three equations:
(1) Is the ICs right for a[x,t]?
(2) How one can keep away from the errors?
(3) When initional situation of n and ni are modified to n[x,0]==0, ni[x,0]==0, there’s one other error message, hinting stiffness at round t=4. How one can remedy it?

Wants["DifferentialEquations`NDSolveProblems`"];
Wants["DifferentialEquations`NDSolveUtilities`"];
L = 10; b = 10; [Mu] = 1/1800; aL = 1;
[Gamma][x_, t_] := ((1 + a[x, t]^2)/(1 - u[x, t]^2))^(1/2); 
[Gamma]i[x_, t_] := ((1 + [Mu]*a[x, t]^2)/(1 - ui[x, t]^2))^(1/2);
equ = {
   D[a[x, t], {x, 2}] - D[a[x, t], {t, 2}] - (4/b^2) a[x, t] - 
     n[x, t]/[Gamma][x, t]*a[x, t] == 0,
   D[[Gamma][x, t] u[x, t], t] == -eField[x, t] - 
     D[[Gamma][x, t], x],
   D[[Gamma]i[x, t] ui[x, t], t] == -[Mu]*eField[x, t] - 
     D[[Gamma]i[x, t], x],
   D[eField[x, t], t] == n[x, t] u[x, t] - ni[x, t] ui[x, t],
   D[n[x, t], t] + D[n[x, t]*u[x, t], x] == 0 ,
   D[ni[x, t], t] + D[ni[x, t]*ui[x, t], x] == 0 
   };
ic = {
   a[x, 0] == aL*E^(-x^2/L^2) Sin[x],
   By-product[0, 1][a][x, 
     0] == (-E^(-(x^2/L^2)) Cos[x] + 1/L E^(-(x^2/L^2)) x Sin[x])*aL,
   n[x, 0] == 10, ni[x, 0] == 10,
   u[x, 0] == 0,  ui[x, 0] == 0, eField[x, 0] == 0
   };
xmax = 40; tmax = 100;
{asol, nsol, nisol, usol, uisol, eFsol} = 
 NDSolveValue[{equ, ic}, {a, n, ni, u, ui, eField}, {x, -xmax, 
   xmax}, {t, 0, tmax}]

Thanks!