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

2024/07/02 by Yadav, Swati, Xue, Jun
#35C07 #76B03 #76B25 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.02717

Abstract

We construct solitary waves for the fractional Korteweg-De Vries type equation u_t + (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 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| → ∞.

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