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

We construct solitary waves for the fractional Korteweg-De Vries type equation uₜ + (Λ⁻ˢu + u²)ₓ = 0, where Λ⁻ˢ denotes the Bessel potential operator (1 + |D|²)-s/2 for s ∈ (0,∞). 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| → ∞.