Higher-order stationary dispersive equations on bounded intervals: a relation between the order of an equation and the growth of its convective term

TL;DR

This paper investigates boundary value problems for higher-order stationary nonlinear dispersive equations, establishing existence, uniqueness, and the relationship between the equation's order and the convective term's growth.

Contribution

It introduces a relation between the order of the dispersive equation and the critical growth of the convective term for solutions to exist and be unique.

Findings

Existence and uniqueness of solutions were proved.

A critical relation between the order of the equation and the convective term was established.

Continuous dependence of solutions on data was demonstrated.

Abstract

A boundary value problem for a stationary nonlinear dispersive equation of order $2l+1\;\; l\in \mathbb{N}$ with a convective term in the form $u^ku_x\;\; k\in \mathbb{N}$ was considered on an interval $(0,L)$. The existence, uniqueness and continuous dependence of a regular solution as well as a relation between $l$ and critical values of $k$ have been established.