A Direct Construction of Solitary Waves for a Fractional Korteweg-de Vries Equation With an Inhomogeneous Symbol

TL;DR

This paper constructs solitary wave solutions for a broad class of fractional KdV equations with inhomogeneous symbols, including highest and intermediate waves, using a novel parameterization and convergence approach.

Contribution

It provides the first simultaneous construction of small, intermediate, and highest solitary waves for the entire family of fractional KdV equations with negative-order dispersion.

Findings

Waves exhibit exponential decay at infinity.

The approach generalizes previous work and applies to a wide range of fractional orders.

The highest wave behavior varies with the fractional parameter s.

Abstract

We construct solitary waves for the fractional Korteweg-De Vries type equation $u_t + (\Lambda^{-s}u + u^2)_x = 0$, where $\Lambda^{-s}$ denotes the Bessel potential operator $(1 + |D|^2)^{-\frac{s}{2}}$ for $s \in (0,\infty)$. The approach is to parameterise the known periodic solution curves through the relative wave height. Using a priori estimates, we show that the periodic waves locally uniformly converge to waves with negative tails, which are transformed to the desired branch of solutions. The obtained branch reaches a highest wave, the behavior of which varies with $s$. The work is a generalisation of recent work by Ehrnstr\"om-Nik-Walker, and is as far as we know the first simultaneous construction of small, intermediate and highest solitary waves for the complete family of (inhomogeneous) fractional KdV equations with negative-order dispersive operators. The obtained waves display exponential decay rate as $|x| \to \infty$.