Strongly interacting multi-solitons for generalized Benjamin-Ono equations

TL;DR

This paper constructs and analyzes strongly interacting multi-soliton solutions for the generalized Benjamin-Ono equation, revealing their asymptotic behavior and uniqueness in certain cases.

Contribution

It introduces a method to construct multi-soliton solutions with specific interaction patterns and proves their uniqueness for two-soliton cases with supercritical power.

Findings

Multi-soliton solutions exhibit interactions with separation growing as √t.

Constructed solutions are unique for two-solitons with p>3.

The asymptotic behavior of soliton positions is characterized.

Abstract

We consider the generalized Benjamin-Ono equation: $$\partial_tu+\partial_x(-|D|u+|u|^{p-1}u)=0,$$ with $L^2$-supercritical power $p>3$ or $L^2$-subcritical power $2<p<3$. We will construct strongly interacting multi-solitary wave of the form: $\sum_{i=1}^nQ(\cdot-t-x_i(t))$, where $n\geq 2$, and the parameters $x_i(t)$ satisfying $x_{i}(t)-x_{i+1}(t)\sim \alpha_k \sqrt{t}$ as $t\rightarrow +\infty$, for some universal positive constants $\alpha_k$. We will also prove the uniqueness of such solutions in the case of $n=2$ and $p>3$.