Phase Transition for Discrete Non Linear Schr\"odinger Equation in Three and Higher Dimensions
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 lim...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.2212.00276 |