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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 \(D^sf(x)=\frac{1}{2\pi}\int_{-\infty}^\infty | \xi| ^s \hat{f}(\xi)e^{i \xi x} d\xi,\) where \(\hat{f}(\xi)\) is the Fourier transform of the function \(f(x)\). Some applications to Korteweg-de Vries type equations and for the Ostrovsky equation are given.

35C15 Integral representations of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
33E20 Other functions defined by series and integrals
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