3
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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.

enter image description here

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?

This question on StackOverflow.

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}
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  • $\begingroup$ Are the boundary conditions periodic? $\endgroup$ – nicoguaro Apr 27 at 19:04
  • $\begingroup$ The boundary conditions are periodic $\endgroup$ – yun shi Apr 29 at 2:25
  • $\begingroup$ 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. $\endgroup$ – nicoguaro Apr 29 at 14:29

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