On persistence properties in weighted spaces for solutions of the fractional Korteweg–de Vries equation

Abstract

Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these models do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with the dispersive part of the equation. Altogether, this analysis is complemented by unique continuation principles that determine optimal spatial decay. This work is intended to establish the above questions for a weakly dispersive perturbation of the inviscid Burgers equation. More precisely, we consider the fractional Korteweg–de Vries equation, which comprises the Burgers–Hilbert equation and dispersive effects weaker than those of the Benjamin–Ono equation.

Periodic Hölder waves in a class of negative-order dispersive equations

“…The resulting singularities occur in at least two ways: wave breaking [18], in which the spatial derivative of a bounded solution of (1) blows up, or sharp crests in travelling-wave solutions -reminiscent of the highest Stokes' wave [30] -which is the subject of this paper. We refer also to [12,21,22,24,28] for other results concerning singularities, well-posedness, persistence, and existence time of solutions.…”

Highest Cusped Waves for the Burgers-Hilbert equation

“…For other results see e.g. [19] for existence time, [49] for well-posedness and [39,40] for some results for related equations.…”

Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions

“…In general, the fKdV equation in dimention d = 1 has been used in a variety of wave phenomena, and the lower dimensional case 0 < α < 1 has been investigated as a model to measure the influence of dispersive effects on the dynamics of Burger's equation. For other studies on the 1d fKdV equation, for example, see [7,45,64,74,65,77], for the higher dimensional fKdV, see [76,38,79,75], and reference therein.…”

“…Later, for α = 1 and d = 2 , the local well-posedness was improved by Schippa [79], who also extended the local well-posedness to 1 ≤ α < 2 in H s (R d ) with s > d+3 2 − α. We also mention that for d ≥ 1 and 0 < α < 2, the well-posedness of the fKdV equation has been studied in weighted spaces H s (R d ) ∩ L 2 (ω(x) dx) with certain weights ω(x), e.g., (1 + |x|) a , see [41,31,30,76,74]. Now, that we reviewed the existence of solutions, we point a very specific solution, a solitary wave, traveling in time preserving its shape.…”