Logarithmic Schrödinger Equations in Infinite Dimensions

Abstract

We study the logarithmic Schrödinger equation with finite range potential on ${\mathbb{R}}^{{\mathbb{Z}}^d}$. Through a ground-state representation, we associate and construct a global Gibbs measure and show that it satisfies a logarithmic Sobolev inequality. We find estimates on the solutions in arbitrary dimension and prove the existence of weak solutions to the infinite-dimensional Cauchy problem.

arXiv comment31 pages

References in corpus (12)

• Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences
• Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent
• Universal dynamics for the defocusing logarithmic Schrodinger equation
• Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation
• Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity
• High speed excited multi-solitons in nonlinear Schrödinger equations
• Orbital stability of Gausson solutions to logarithmic Schrödinger equations
• Existence of multi-solitons for the focusing Logarithmic Non-Linear Schrodinger Equation
• Logarithmic Schr{ö}dinger equation with quadratic potential
• The focusing logarithmic Schrödinger equation: analysis of breathers and nonlinear superposition
• Nonuniqueness and nonlinear instability of Gaussons under repulsive harmonic potential
• Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schr{ö}dinger equation

Cited by in corpus (2)

• Sound propagation in cigar-shaped Bose liquids in the Thomas-Fermi approximation: A comparative study between Gross-Pitaevskii and logarithmic models
• The log-Sobolev inequality for spin systems of higher order interactions