The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. The minimal two species periodic assembly is the one with the least energy per lattice cell area. There is a parameter $b$ in $[0,1]$ and the type of the lattice associated with the minimal assembly varies depending on $b$. There are several threshold defined by a number $B=0.1867...$ If $b \in [0, B)$, the minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, the minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, the minimal assembly is associated with a rhombic lattice with an acute angle in $[\frac{\pi}{3}, \frac{\pi}{2})$. Only when $b=1$, this rhombic lattice is the hexagonal lattice. None of the other values of $b$ yields the hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where the hexagonal lattice is ubiquitously observed.

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We present and analyze two numerical methods for the logarithmic Schrödinger equation (LogSE) consisting of a regularized splitting method and a regularized conservative Crank–Nicolson finite difference method (CNFD). In order to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a regularized logarithmic Schrödinger equation (RLogSE) with a small regularized parameter 0<ε≪1 is adopted to approximate the LogSE with linear convergence rate O(ε). Then we use the Lie–Trotter splitting integrator to solve the RLogSE and establish its error bound O(τ^{1/2}ln(ε^{−1})) with τ>0 the time step, which implies an error bound at O(ε+τ^{1/2}ln(ε^{−1})) for the LogSE by the Lie–Trotter splitting method. In addition, the CNFD is also applied to discretize the RLogSE, which conserves the mass and energy in the discretized level. Numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the LogSE.

Based on a global estimate of the heat kernel, some important inequalities such as Poincaré inequality and log-Sobolev inequality, furthermore a tight logarithm Sobolev inequality are presented on graphs, just under a positive curvature condition CDE'(n,K) with some K > 0. As consequences, we obtain exponential integrability of integrable Lipschitz functions and moment bounds at the same assumption on graphs.