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We address the topology of the set of singularities of a solution of a Hamilton-Jacobi equation. For this we will apply the idea of the first two authors (Generalized characteristic and Lax-Oleinik operators: global result, preprint, arXiv:1605.07581, 2016.) to use the positive Lax-Oleinik semi-group to propagate singularities. {\it To cite this article: P. Cannarsa, W. Cheng, A. Fathi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

The nonlinear Schrdinger equation (NLS) has been a fundamental model for understanding vortex motion in superfluids. The vortex motion law has been formally derived on various physical grounds and has been around for almost half a century. We study the nonlinear Schrdinger equation in the incompressible fluid limit on a bounded domain with Dirichlet or Neumann boundary condition. The initial condition contains any finite number of degree 1 vortices. We prove that the NLS linear momentum weakly converges to a solution of the incompressible Euler equation away from the vortices. If the initial NLS energy is almost minimizing, we show that the vortex motion obeys the classical Kirchhoff law for fluid point vortices. Similar results hold for the entire plane and periodic cases, and a related complex GinzburgLandau equation. We treat as well the semi-classical (WKB) limit of NLS in the presence of

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In this article, we study wave dynamics in the fractional nonlinear Schrödinger equation with a modulated honeycomb potential. This problem arises from recent research interests in the interplay between topological materials and nonlocal governing equations. Both are current focuses in scientific research fields. We first develop the Floquet-Bloch spectral theory of the linear fractional Schrödinger operator with a honeycomb potential. Especially, we prove the existence of conical degenerate points, i.e., Dirac points, at which two dispersion band functions intersect. We then investigate the dynamics of wave packets spectrally localized at a Dirac point and derive the leading effective envelope equation. It turns out the envelope can be described by a nonlinear Dirac equation with a varying mass. With rigorous error estimates, we demonstrate that the asymptotic solution based on the effective envelope equation approximates the true solution well in the weighted-H^s space.