Calculate the Bloch wave

The eigenvalue problem $$\frac{d^2u}{dx^2}+2i k\frac{du}{dx}-[k^2-6\sin(x)^2]u(x)=-\mu u(x)$$

gives the first five eigenvalues with $$k=0$$ or $$k=1$$ which are $$2.06$$, $$2.26$$, $$5.16$$, $$6.81$$, and $$7.74$$.

And those eigenvalues I have calculated, but the corresponding eigenvector that I got did not agree with Fig.2 in the paper Z. Shi and J. Yang, "Solitary waves bifurcated from Bloch-band edges in two-dimensional periodic media," Phys. Rev. E, vol. 75, no. 5, pp. 056602, May 2007.

In this paper the author tells us the vector at the edge is all real, but I calculate it to be complex. Why is there an inconsistency with the paper?

PRE75.056602

Clear["Global*"]
Lx = \[Pi];
k0x = 2 \[Pi]/Lx;
V = 6 Sin[x]^2;
kx = 0;
{eig0, funs} = NDEigensystem[{-D[u[x],{x,2}]-2I kx D[u[x],x]+(kx^2+V)u[x],
u[0]== u[\[Pi]]}, u[x], {x,0,\[Pi]}, 3,
Method->{"VectorNormalization"->True,
"PDEDiscretization"->{"FiniteElement",
{"MeshOptions"->{"MaxCellMeasure"->0.01}}}}]

{2.06318, 6.81429, 7.74678},

kx=1;
{eig2,fun2} = NDEigensystem[{-D[u[x],{x,2}]-2I kx D[u[x],x]+(kx^2+V)u[x],
u[0]== u[\[Pi]]}, u[x], {x,0,\[Pi]}, 3,
Method->{"PDEDiscretization"->{"FiniteElement",
{"MeshOptions"->{"MaxCellMeasure"->0.01}}}}]

{2.26673, 5.16594, 12.0926}
`
• Are the boundary conditions periodic? – nicoguaro Apr 27 at 19:04
• The boundary conditions are periodic – yun shi Apr 29 at 2:25
• I think that it would be good to add that information to your question. That being said, I solved your problem using Finite Differences and obtained (about) the same eigenvalues and complex eigenvectors. – nicoguaro Apr 29 at 14:29