Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation

TL;DR

This paper establishes global well-posedness for a higher-order Benjamin-Ono equation in the energy space and analyzes the solution's limit behavior as a small parameter approaches zero, connecting it to the classical Benjamin-Ono equation.

Contribution

It proves global well-posedness for a higher-order Benjamin-Ono equation and demonstrates convergence to the classical Benjamin-Ono solution as a parameter tends to zero.

Findings

Global well-posedness in $H^1(R)$

Convergence of solutions as $oldsymbol{oldsymbol{ extit{ extepsilon}} o 0}$

Conditions on coefficients for limit behavior

Abstract

In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation $$ \partial_tv-b\mathcal{H}\partial^2_xv- a\epsilon \partial_x^3v=cv\partial_xv-d\epsilon \partial_x(v\mathcal{H}\partial_xv+\mathcal{H}(v\partial_xv)), $$ is globally well-posed in the energy space $H^1(\mathbb R)$. Moreover, we study the limit behavior when the small positive parameter $\epsilon$ tends to zero and show that, under a condition on the coefficients $a$, $b$, $c$ and $d$, the solution $v_{\epsilon}$ to this equation converges to the corresponding solution of the Benjamin-Ono equation.