Reviews in Mathematical Physics

We present examples in three symmetry classes of topological insulators in one or two dimensions where the proof of the bulk-edge correspondence is particularly simple. This serves to illustrate the mechanism behind the bulk-edge principle without the overhead of the more general proofs which are available. We also give a new formula for the ${\mathbb{Z}}_2$-index of our time-reversal invariant systems inspired by Moore and Balents.

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney’s approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest $2$-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by $(k+1,k+2)$ curves which are of cusp type for $k=2n+1$, $n=1,2,3,\dots .$ Regularization of these singularities by deformation of plane into plane mappings into surface $S^{2+k}(\subset {ℝ}^{2+k})$ to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems.

A uniform construction of quasi-periodic solutions to the negative-order Jaulent–Miodek (nJM) hierarchy is presented by using a family of backward Neumann type systems. From the backward Lenard gradients, the nJM hierarchy is put into the zero-curvature setting and the bi-Hamiltonian structure displaying its integrability. The nonlinearization of Lax pair is generalized to the nJM hierarchy such that it can be reduced to a sequence of backward Neumann type systems, whose involutive solutions yield finite parametric solutions of the nJM hierarchy. The negative $N$-order stationary JM equation is given to specify a finite-dimensional invariant subspace for the nJM flows. With a spectral curve determined by the Lax matrix, the nJM flows are linearized on the Jacobi variety of a Riemann surface. Finally, the Riemann–Jacobi inversion is applied to Abel–Jacobi solutions of the nJM flows, by which some quasi-periodic solutions are obtained for the nJM hierarchy.

We consider the multi-particle tight-binding Anderson model and prove that its lower spectral edge is non-random under some mild assumptions on the inter-particle interaction and the random external potential. We also adapt to the low energy regime the multi-particle multi-scale analysis initially developed by Chulaevsky and Suhov in the high disorder limit, if the marginal probability distribution of the i.i.d. random variables is log-Hölder continuous and we obtain the spectral exponential and strong dynamical localization near the bottom of the spectrum.