Refined energy inequality with application to well-posedness for the fourth order nonlinear Schrodinger type equation on torus

TL;DR

This paper establishes well-posedness for a fourth order nonlinear Schrödinger equation on the torus by developing a refined energy inequality and a modified energy approach to handle derivative nonlinearities.

Contribution

It introduces a novel modified energy method to prove well-posedness for 4NLS with derivative nonlinearities on the torus, overcoming limitations of classical energy methods.

Findings

Proved local and global well-posedness for 4NLS on the torus.

Developed a refined energy inequality for derivative nonlinearities.

Established a priori estimates using the modified energy approach.

Abstract

We consider the time local and global well-posedness for the fourth order nonlinear Schrodinger type equation (4NLS) on the torus. The nonlinear term of (4NLS) contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to (4NLS).