Title: Phase Transition for Discrete Non Linear Schrödinger Equation in Three and Higher Dimensions Show Chinese title

Title: 三维及更高维离散非线性薛定谔方程的相变

Abstract: We analyze the thermodynamics of the focusing discrete nonlinear Schr\"odinger equation in dimensions $d\ge 3$ with general nonlinearity $p>1$ and under a model with two parameters, representing inverse temperature and strength of the nonlinearity, respectively. We prove the existence of limiting free energy and analyze the phase diagram for general $d,p$. We also prove the existence of a continuous phase transition curve that divides the parametric plane into two regions involving the appearance or non-appearance of solitons. Appropriate upper and lower bounds for the curve are constructed. We also look at the typical behavior of a function chosen from the Gibbs measure for certain parts of the phase diagram. Show Chinese abstract

Abstract: 我们分析了在维度$d\ge 3$下具有通用非线性项$p>1$的聚焦离散非线性薛定谔方程的热力学性质，并在一种包含两个参数的模型下进行分析，这两个参数分别表示逆温度和非线性的强度。 我们证明了极限自由能的存在性，并对一般的$d,p$分析了相图。 我们还证明了存在一条连续相变曲线，该曲线将参数平面分为两个区域，分别涉及孤子的出现或不出现。 构建了该曲线的适当上下界。 我们还研究了在相图某些部分中从吉布斯测度中选取的函数的典型行为。