I try to solve an integro-differential equation with a convolution term using this code. Here i used a periodic condition.

L=10; tmax = 2;

NDSolve[{D[u[x, t], t] + u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 2}] + 
D[u[x, t], {x, 4}] + 1/(2 L)*NIntegrate[D[u[xp, t],{x, 3}]*Cot[\[Pi](x - xp)/(2*L)], {xp, -L, L}(*,PrincipalValue\[Rule]True*)] == 0, u[-L, t] == u[L, t],
u[x, 0] == Cos[\[Pi]/L*x]}, u, {x, -L, L}, {t, 0, tmax}]

which gives me

NIntegrate::inumr: "The integrand ... has evaluated to non-numerical values for all sampling points"

The warning is understandable because the function u[xp, t] is still unknow when NIntegrate is evaluated. I also tried Method -> "MethodOfLines", but it still doesn't work.

My problem may closely relate to this, which is linear equation and more like a wave equation in the sense of $u_{tt}$. But my pde is nonlinear and has 1st-derivative in $t$. So I post my problem here for a more general solution. Thank you for any suggestion.

Update:

1. As indicated by bbgodfrey in a comment: it is necessary to convert the convolution to the same range as used in NDSolve. For a spatial periodic solution, I can change the convolution with integral limits of infinite to a NIntegrate over a finite interval.

2. With Convolve one can use PrincipalValue -> True for a principal value integral, however, using NIntegrate how could we calculate the Principal Value (commented in the code)?

3. To use MethodOfLines introduced here

We first create $2M$ equidistant grid points $x_m=(m-M)dx$ with $m=1,2,...,2M$. The x-position of grid points is stored in xtab:

M = 50; dx = 2*L/(2*M); xtab = Table[(m - M) dx, {m, 1, 2*M}];

Then we should discretize the solution of PDE along $x$ into $2M$ solutions of a set of coupled ODEs. u[m][t] denotes the solution of function $u(x,t)$ at point $x_m$. Here, I didn't include the left end-point, since it can be set to be u[0][t]=u[2*M][t] according to the periodicity.

U[t_] = Table[u[m][t], {m, 1, 2*M}];

The spatial derivatives are discretized using 2nd-order central differences, for example:

dUdxx=ListCorrelate[{1, -2, 1}/dx^2, U[t]]

To discretize the integral, we may introduce the mid-points $x_{m+1/2}=(x_m+x_{m+1})/2$ for $m=1,2,...,2M-1$ with $x_{1/2}=(-L+x_1)/2$, which are saved in midxtab. Correspondingly, we define u[m+1/2][t]=(u[m][t]+u[m+1][t])/2.

midxtab = Join[{(-L+(1-M) dx)/2}, Table[((m - M) dx + (m + 1 - M) dx)/2, {m, 1, 2*M-1}]];

Now, we discretized the integral at these mid-points, here dUdxxx[t] represents 3rd-derivative wrt x

usum = dUdxxx[t].Cot[\[Pi](midxtab - xtab)/(2*L)*h;