This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging to solve such a large-scale GEP efficiently. We combine the null-space free method with the inexact shift-invert residual Arnoldi method and MINRES linear solver to solve the GEP with a matrix dimension as large as 5,308,416. The eigenstructure is heavily determined by the chirality parameter $\gamma$. We show that all the eigenvalues are real and finite for a small chirality $\gamma$. For a critical value $\gamma = \gamma^*$, the GEP has $2 \times 2$ Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when $\gamma$ increases from $\gamma^*$, the $2 \times 2$ Jordan block will first split into a complex conjugate eigenpair, then rapidly collide \tr{with} the real axis and bifurcate into \tr{positive (resonance) and negative eigenvalues with modulus} smaller than the other existing positive eigenvalues. The resonance band also exhibits an anticrossing interaction. Moreover, the electric and magnetic fields of the resonance modes are localized inside the structure, with only a slight amount of field leaking into the background (dielectric) material.

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \mathcal{A} to an appropriate q-polynomial algebra. In the case that \mathcal{A} is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \mathcal{A} is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.

Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a periodic potential. Using multiscale perturbation methods, the envelope equations of solitary waves near Bloch bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of the second-order dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch modes, the envelope equations lead to a host of solitary wave structures, including reduced-symmetry solitons, dipole-array solitons, vortex-cell solitons, and so on—many of which have not been reported before to our knowledge. It is also shown analytically that the centers of envelope solutions can be positioned at only four possible locations at or between potential peaks. Numerically, families of these solitary waves are directly computed both near and far away from the band edges. Near the band edges, the numerical solutions spread over many lattice sites, and they fully agree with the analytical solutions obtained from the envelope equations. Far away from the band edges, solitary waves are strongly localized, with intensity and phase profiles characteristic of individual families.

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