Junyoung Heo | Searcharxiv

arXiv2026

Nonlinear Schr\"odinger systems with all attractive forces

In this paper we investigate in a systematic way the solution structure of nonnegative solutions for coupled nonlinear Schr\"odinger systems in the all attractive regime. For all frequencies equal case we obtain results on Morse index of synchronized positive vector solutions and nonnegative semi-vector solutions as well as their kernel of linearized systems at these solutions. We establish a variational characterization of synchronized positive vector solutions and as applications we give a new existence result about positive vector solutions for the general systems with arbitrary frequencies. We also examine the synchronization phenomenon of positive vector solutions, and prove that if the interaction matrix has exactly one positive eigenvalue, any positive vector solution is synchronized. Finally we provide examples of domains for which synchronization of positive solutions fails to hold if the interaction matrix has at least two positive eigenvalues. Our results reveal the effect of the spectral information of the interaction matrix of the couplings and geometry of a domain on the solution structure.

math.AP↗

arXiv2025

Optimizing Resource Distribution in a One-Dimensional Logistic Diffusion Model

In this article, we study the optimization of resource distributions in a one-dimensional logistic diffusive model. The goal is to determine a distribution on a bounded one-dimensional domain that maximizes the total population at equilibrium. Previous works have shown that optimal resources are bang-bang, and in one dimension, a sufficiently large dispersal rate forces the optimal resource to be concentrated. For general dispersal rates, however, the analysis becomes more difficult because the equilibrium population may behave irregularly, and the optimal resource may be fragmented. To address this, we introduce a block decomposition that reduces fragmented resources to a collection of concentrated blocks. We then define an advantage function, which measures the gain in the equilibrium population obtained by allocating resources on a fixed interval and is used to analyze the contribution of each block to the total population. This function also allows us to reformulate the optimization problem as a convexity analysis of the advantage function. We prove the superlinearity of this function when the total resource is small enough, and this property leads to an explicit characterization of the optimal control with sufficiently small total resource.

math.AP↗