Local well-posedness of the initial value problem for a fourth-order nonlinear dispersive system on the real line

TL;DR

This paper proves local well-posedness for a class of fourth-order nonlinear dispersive PDE systems on the real line, using energy methods, gauge transformations, and approximation techniques.

Contribution

It establishes the well-posedness of a complex nonlinear dispersive system with derivatives up to second order, extending analysis techniques to higher-order equations.

Findings

Initial value problem is locally well-posed for high-regularity Sobolev data.

Uses energy method combined with gauge transformation and Bona-Smith approximation.

Applicable to systems arising in nonlinear science and geometric analysis.

Abstract

This paper investigates the initial value problem for a system of one-dimensional fourth-order dispersive partial differential-integral equations with nonlinearity involving derivatives up to second order. Examples of the system arise in relation with nonlinear science and geometric analysis. Applying the energy method based on the idea of a gauge transformation and Bona-Smith approximation technique, we prove that the initial value problem is time-locally well-posed on the real line for initial data in a Sobolev space with high regularity.