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

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 Ehrnström-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 $|x|\to \infty$.

21 pages