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The spectrum of a Schrödinger operator with a perfect honeycomb lattice potential has special points, called Dirac points, where the lowest two branches of the spectrum touch. Deformations can result in the merging and disappearance of the Dirac points, and the originally intersecting dispersion relation branches separate. Corresponding to these deformations, nonlinear envelope equations are derived and their dynamics are studied. In the region where Dirac points exist, a maximally balanced equation is derived which has limits to a nonlinear Schrödinger--Kadomtsev--Petviashvili (NLSKP)-type equation and its dispersionless reduction. When the Dirac points disappear and a gap opens, a different maximally balanced equation is derived which has the NLSKP equation and a one-dimensional nonlocal evolution equation as limits. When the gap is sufficiently wide, a nonlinear Dirac equation with nonzero mass and a nonlinear Schrödinger focusing-defocusing system are found. The latter two equations admit nonlinear localized modes. Typical dynamical behaviors of the effective envelope equations are presented.

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The Cauchy problem for the homogeneous real/complexMongeAmp`ere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of K¨ahler metrics equipped with the Mabuchi metric. This Cauchy problem is believed to be ill-posed and a basic problem is to characterize initial data of (weak) solutions which exist up to time T . In this article, we use a quantization method to construct a subsolution of the HCMA on a general projective variety, and we conjecture that it solves the equation for as long as the unique solution exists. The subsolution, called the “quantum analytic continuation potential,” is obtained by (i) Toeplitz quantizing the Hamiltonian flow determined by the Cauchy data, (ii) analytically continuing the quantization, and (iii) taking a certain logarithmic classical limit. We then prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) the quantum analytic continuation potential coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel [29], it is proved that the Legendre transform potential ceases to solve the HCMA once it ceases to be smooth. The results here and in the sequels in particular characterize the initial data of smooth geodesic rays.

We prove the vanishing of the Koszul homology group$H$_{μ}(Kos($M$)_{μ}), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.