Narrow Results

We use a lattice formulation to study the spectra of the Dirac and the Dirac–Kähler operators on the 2-sphere. This lattice formulation uses differentiation matrices which yield exact values for the derivatives of polynomials, preserving the Leibniz rule in subspaces of polynomials of low degree and therefore, this formulation can be used to study the fermion–boson symmetry on the lattice. In this context, we find that the free Dirac and Dirac–Kähler operators on the 2-sphere exhibit fermionic as well as bosonic-like eigensolutions for which the corresponding eigenvalues and the number of states are computed. In the Dirac case these solutions appear in doublets, except for the bosonic mode with zero eigenvalue, indicating the possible existence of a supersymmetry of the square of the Dirac operator.

By considering a discrete isospectral problem [H.-X. Yang et al., Phys. Lett. A338, 117 (2005)], integrable positive and negative lattice equations are derived, from which the modified (2+1)-dimensional Toda lattice is obtained. The method of enlarging spectral problems to construct the integrable couplings for lattice soliton equations is extended to higher-dimensional systems. Illustrating by examples, the positive and negative integrable couplings of the resulting lattice hierarchy and three classes of integrable couplings of (2+1)-dimensional mToda lattice are discussed.

Based on the relation between the m-function and the spectral function we construct an inverse quantum scattering procedure at fixed energy which can be applied to spherical radial potentials vanishing beyond a fixed radius a. To solve the Gelfand–Levitan–Marchenko integral equation for the transformation kernel, we determine the input symmetrical kernel by using a minimum norm method with moments defined by the input set of scattering phase shifts. The method applied to the box and Gauss potentials needs further practical developments regarding the treatment of bound states.

Based on the Tu scheme [G.-Z. Tu, J. Math. Phys.30 (1989) 330], we construct a counterpart of the Boiti–Pempinelli–Tu soliton hierarchy from a matrix spectral problem associated with the Lie algebra $\mathrm{\text{so}}(3,ℝ)$, and formulate Hamiltonian structures for the resulting soliton equations by means of the trace identity. We then show that the newly presented equations possess infinitely many commuting symmetries and conservation laws. Finally, we derive the well-known combined KdV-mKdV equation from the new hierarchy.

In this paper, we analyze a Virtual Element Method (VEM) for solving a non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a $C^1$-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-self-adjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.