Clear ["Global`*"]
c = 0.03;
k = 10 π;
w = 10^7;
h0[x_] := Exp[-x^2/0.05];
p[t_, x_] := -D[h[t, x], {x, 2}] - c Exp[2 I k x + 2 I w t]
eq1 = D[h[t, x], t] == 1/3 D[(h[t, x]^3 D[p[t, x], x]), x];
TraditionalForm[eq1];
x0 = -1.5;
x1 = 1.5;
con1 =
  Derivative[0, 1][h][t, x0] == D[h0[x], x] /. x -> x0;
con2 =
  Derivative[0, 1][h][t, x1] == D[h0[x], x] /. x -> x1;
con3 =
  Derivative[0, 3][h][t, x0] == D[h0[x], {x, 3}] /. x -> x0;
con4 =
  Derivative[0, 3][h][t, x1] == D[h0[x], {x, 3}] /. x -> x1;

sol = NDSolve[{eq1, h[0, x] == h0[x], con1, con2, con3, con4},
h, {t, 0, 2.5}, {x, x0, x1} ,
MaxSteps -> 10^5,
Method -> {"PDEDiscretization" -> {"MethodOfLines",
   "SpatialDiscretization" -> {"TensorProductGrid",
     "MinPoints" -> 2500}}},
PrecisionGoal -> 2, MaxStepSize -> 0.01];
Plot[Abs[Evaluate[h[1., x] /. sol[[1, 1]]]], {x, x0, x1},
PlotRange -> All]