Dynamics of the Infinite Discrete Nonlinear Schrödinger Equation

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.