Dear experts

I try to solve the partial integro-differential equation as following. but I recieved error. I would appreciate it if help me to solve this PIDE.

to have an idea about the equation and boundary conditions you can imagin a tank of water of length l and depth h. eta is considered as initial surface wave. then the PIDE shows the time evolution of the surface wave. (here eta and u(x,t) are the same). inface I try to use approximate solution eta for  initial condition

thank you in advance.

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restart:
with(plots):
with(PDEtools):

h:=1e-2:
H:=1e-3:
a:=H/2:
l:=1:
epsilon:= (a/l)^(2/3);

vel:= sqrt(9.81*h)*(1+epsilon/2-3/20*epsilon^2+3/56*epsilon^3);

X:=x-5:

eta:=h*( epsilon*(sech(X-vel*t))^2#-3/4*epsilon^2*(sech(X-vel*t))^2*(tanh(X-vel*t))^2 + epsilon^3*(5/8*(sech(X-vel*t))^2-101/80*(sech(X-vel*t))^4)*(tanh(X-vel*t))^2

eta:=h*( epsilon*(sech(X-vel*t))^2-3/4*epsilon^2*(sech(X-vel*t))^2*(tanh(X-vel*t))^2 + epsilon^3*(5/8*(sech(X-vel*t))^2-101/80*(sech(X-vel*t))^4)*(tanh(X-vel*t))^2

dx_eta:=diff(eta,x):

pdie:= 2*diff(u(x,t),t) + 3*diff(u(x,t),x) * u(x,t) +(1/3.0)*diff(u(x,t),x$3)
      -1/sqrt(Pi)* int(diff(u(x,tau),x)/sqrt(tau), tau= 0..t) = 0;
bounds:= {u(0,t)=0
        , u(10,t)=0
        , u(x,0)= eval(eta,t=0)
        , D[1](u)(0,t)=eval(dx_eta,x=0)#0.2738612788e-1*sech(13.69306394+27.38612788*t)^2*tanh(13.69306394+27.38612788*t)
         }:

#Generate IC for D[2](u)(x,0):
bounds:= bounds union{simplify@isolate}(
   convert(value(eval(convert(eval(pdie, {t= 0}), Diff), bounds)), D),
   D[2](u)(x,0)
);

#Convert original PDIE to a true PDE (i.e., without integrals):
pde:= diff(pdie, t);

#Solve PDE system:
psol:= pdsolve(pde, bounds, numeric, optimize=true):
psol:-plot3d(u(x,t), x= 0..10, t= 0..0.2, grid=[150,150],orientation=[-90,0,0],axes=boxed);

Error, (in pdsolve/numeric/plot3d) unable to compute solution for t>HFloat(0.0):
Newton iteration is not converging

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Dear Christian Wolinski 844 , First of all, thank you for your code. It is very interesting for me.

Could you please explain me the last two lines?

map(`@`(evalf, subs), sols, map(`@`(abs, lhs - rhs), EQ));
  fnormal(%, Digits - 4);