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 (fKdV) type equation: $$\begin{array}{r}{u}_t+({\mathrm{\Lambda }}^{-s}u+u^2{)}_x=0,\end{array}$$ where ${\mathrm{\Lambda }}^{-s}$ denotes the Bessel potential operator $(1+|D{|}^2{)}^{-s/2}$ for $s\in (0,\mathrm{\infty })$ . The approach is to parameterize 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 generalization 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) fKdV equations with negative-order dispersive operators. The obtained waves display exponential decay rate as $|x|\to \mathrm{\infty }$ .