On The Cauchy Problem for the elliptic Zakharov-Schulman system in dimensions 2 and 3

TL;DR

This paper establishes local well-posedness for the Zakharov-Schulman system in 2 and 3 dimensions within certain Sobolev spaces, advancing understanding of its initial value problem.

Contribution

It proves local well-posedness for the Zakharov-Schulman system in dimensions 2 and 3 for initial data in Sobolev spaces with minimal regularity.

Findings

Well-posedness in Sobolev spaces for n=2,3

Initial data regularity threshold s ≥ n/4

Extension of well-posedness theory to elliptic operators

Abstract

We prove that the Cauchy problem associated to the Zakharov-Schulman system $iu_t+L_1u=uv$, $L_2v=L_3(|u|^2)$ is locally well-posed for given initial data in Sobolev spaces $H^s(R^n)$, $s\geq n/4$, for n =2,3. Here, L_j denote second order operators, with L_1 non-degenerate and L_2 elliptic.