Cauchy problem for a Schr\"odinger-type equation related to the Riemann zeta function

TL;DR

This paper investigates a nonlinear Schr"odinger equation involving the Riemann zeta function, establishing existence, uniqueness, and finite-time extinction of solutions in one dimension.

Contribution

It introduces a novel Schr"odinger equation with the Riemann zeta function and proves global existence, uniqueness, and finite-time vanishing in one dimension.

Findings

Existence of global solutions in $H^1( abla)$

Uniqueness of solutions in the distribution sense

Finite-time extinction in one dimension

Abstract

We study the Cauchy problem in the space $H^1(\Sigma)$ for a nonlinear damped Schr\"odinger equation of the form \begin{equation}\tag{NLS-$\zeta$}\label{nls} i u_t + \Delta u + i \lambda u \, \zeta(|u|+1) = 0, \quad u(0,x) = u_0, \end{equation} where $\zeta$ denotes the Riemann zeta function. We first establish the uniqueness of solutions in the sense of distributions. Then, by considering a regularized problem, we prove the existence of a global solution in $H^1(\Sigma)$, using uniform estimates and compactness arguments. Finally, we show that the limiting solution indeed satisfies the original equation in the weak sense. In the addition we proof that, the one-dimensional case, we show that it becomes zero in finite time.