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 $u_t+({\mathrm{\Lambda }}^{-s}u+u^2{)}_x=0$, where ${\mathrm{\Lambda }}^{-s}$ denotes the Bessel potential operator $(1+\mathrm{\mid }D{\mathrm{\mid }}^2{)}^{-\frac{s}{2}}$ for $s\in (0,\mathrm{\infty })$. 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 $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$.