Abstract The discrete nonlinear Schrödinger equation on $${\mathbb Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , $$d \ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is an example of a dispersive nonlinear wave system. Being a Hamiltonian system that conserves also the $$\ell ^2({\mathbb Z}^d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -norm, the well-posedness of the corresponding Cauchy problem follows for square-summable initial data. In this paper, we prove that the well-posedness continues to hold for initial data that can grow towards infinity, namely anything that has at most a certain power law growth far away from the origin. The growth condition is loose enough to guarantee that, at least in dimension $$d=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , initial data sampled from any reasonable equilibrium distribution of the defocusing DNLS satisfies it almost surely.

Aleksis Vuoksenmaa

Journal of Statistical Physics

《统计物理学杂志》在统计物理学的所有领域以及与物理系统中的集体现象有关的相关领域发表原创和特邀评论论文。

The Journal of Statistical Physics publishes original research articles and invited reviews in all areas of statistical physics, with a particular emphasis on topics related to collective phenomena in physical systems.

《Journal of Statistical Physics》致力于发表统计物理学各个领域的原创研究论文和特邀评论，特别关注与物理系统中的集体现象相关的研究。

Category	Quartile
物理与天体物理	3
物理与天体物理, 物理数学物理	3

Dynamics of the Infinite Discrete Nonlinear Schrödinger Equation

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PHYSICS, MATHEMATICAL	3