Third order semilinear dispersive equations related to deep water waves

TL;DR

This paper establishes local existence results for third order semilinear dispersive PDEs in two dimensions, which model deep water gravity waves and require advanced pseudodifferential operator techniques beyond classical methods.

Contribution

It introduces a novel approach using pseudodifferential operators with nonsmooth coefficients to prove local existence for complex dispersive equations related to deep water waves.

Findings

Proved local existence of solutions for the equations

Developed methods handling nonsmooth coefficients

Extended analysis beyond classical energy methods

Abstract

We present local existence theorem of the initial value problem for third order semilinear dispersive partial differential equations in two space dimensions. This type of equations arises in the study of gravity wave of deep water, and cannot be solved by the classical energy method. To solve the initial value problem, we make full use of pseudodifferential operators with nonsmooth coefficients.