Highest Cusped Waves for the Fractional KdV Equations

TL;DR

This paper establishes the existence and asymptotic behavior of highest cusped traveling wave solutions for fractional KdV equations across a range of fractional orders, using analytical and computational methods.

Contribution

It proves the existence of highest cusped solutions for fractional KdV equations with all  in (-1,0) and determines their precise asymptotic behavior at zero.

Findings

Existence of highest cusped solutions for fractional KdV equations.

Exact asymptotic behavior of solutions at zero.

Combination of analytical and computer-assisted proof techniques.

Abstract

In this paper we prove the existence of highest, cusped, traveling wave solutions for the fractional KdV equations $f_t + f f_x = |D|^{\alpha} f_x$ for all $\alpha \in (-1,0)$ and give their exact leading asymptotic behavior at zero. The proof combines careful asymptotic analysis and a computer-assisted approach.