Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations

Abstract

Fractional integrals and derivatives of Airy functions (Riesz potentials) are considered. For half integrals D −1/2 Ai(x) and D −1/2 Gi(x) explicit representations are found in terms of the products of Airy functions. Here Ai(x) and Gi(x) are the Airy function of the first kind and the Scorer function, respectively. Based on that representations are obtained for all semiinteger derivatives of Ai(x) and Gi(x). Applications to Korteweg-de Vries type equations are provided. (2000). 35C15, 33E20, 35Q53. Mathematics Subject Classification

Asymptotic expansions for Riesz potentials of Airy functions and their products

“…where J ν (x) is the Bessel function of order ν. Riesz potentials (sometimes also called Riesz fractional derivatives) of fundamental solutions are of great importance in studying global solvability, properties and the long-time behavior of the corresponding Cauchy problems (see [2,3,4,5] and the references therein). In the current paper we are concerned with obtaining asymptotic expansions as x → ±∞ of the Riesz potentials of the Airy function Ai(x) and the Scorer function Gi(x) = −HAi(x), where H is the Hilbert transform (see (5) below).…”

“…Riesz fractional derivatives of these functions of order α = 1/2 stand out as the highest Riesz potentials that are still uniformly bounded on the whole real axis (see [2,3]). Moreover, all semi-integer derivatives of Ai(x) and Gi(x) can be expressed in terms of the products of Airy functions (see [5]). We also provide formulas that allow one to obtain asymptotic expansions of the products of Airy functions Ai(x)Bi(x), Ai 2 (x) and…”

Riesz potentials for Korteweg-de Vries solitons

“…More recently, Riesz potentials have been used for studying structural properties of solutions of KdV (see [22][23][24]). Notice that solutions of KdV do not have a zero mean property with respect to the spatial coordinate.…”

“…At first relations (1.1) were established only for α = 1/2 (see [22]). It was done on the basis of the special function representations for the half derivatives of Airy functions.…”

“…[4]), u 0 (x, t) = 2sech 2 (X) with X = x − 4t, is chosen as such an example. Using the same approach as in [22][23][24] we study the Riesz potentials for u 0 (x, t) in parallel with their Hilbert transforms, i.e. as conjugate pairs.…”