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Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations. (English) Zbl 1155.35008
The author introduces functions w - (x)=Ai(x)Bi(x)-Ai 2 (x) and w + (x)=Ai(x)Bi(x)+Ai 2 (x), where Ai(x),/,Bi(x) are Airy functions of the first and second kind, respectively. He proves that half integral of the Airy functions D -1/2 Ai(x) and D -1/2 Gi(x) (Gi(x) is the Scorer function) can be expressed in the terms of w - (x) and w + (x)· Based on that the author establishes general formulas for computing semi-integral derivatives of Ai(x) and Gi(x). Here the functional derivatives of order s>-1 of the function f(x) defined by D s f(x)=1 2π - |ξ| s f ^(ξ)e iξx dξ, where f ^(ξ) is the Fourier transform of the function f(x). Some applications to Korteweg-de Vries type equations and for the Ostrovsky equation are given.
MSC:
35C15Integral representations of solutions of PDE
35Q53KdV-like (Korteweg-de Vries) equations
33E20Functions defined by series and integrals