The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

Rational solutions for three semi-discrete modified Korteweg–de Vries-type equations

In this paper, we consider three semi-discrete modified Korteweg–de Vries type equations which are the nonlinear lumped self-dual network equation, the semi-discrete lattice potential modified Korteweg–de Vries equation and a semi-discrete modified Korteweg–de Vries equation. We derive several kinds of exact solutions, in particular rational solutions, in terms of the Casorati determinant for these three equations, respectively. For some rational solutions, we present the related asymptotic analysis to understand their dynamics better.

Regular and chaotic solutions in BEC for tilted bichromatical optical lattice

We investigate a Bose–Einstein condensate held in a 1D tilted bichromatical optical lattice potential by constructing its Poincaré sections in phase space. We explore dynamic of the system based on the relations between the system parameters and the solution behaviors. It is demonstrated that the system exhibits shock-wave like dynamic. The power spectrum graphs, bifurcation and Lyapunov exponents of BEC system are also presented.