Integral representations for products of Airy functions

The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related to the Korteweg-de Vries (KdV) and the KadomtsevPetviashvili models. KdV can be obtained from the equation in question when the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation. (2000). 35B40, 35C15, 35C20, 35Q53. Mathematics Subject Classification

“…At first glance the structure of (2.2) seems similar to that of the integrals representing the products of Airy 960 V. Varlamov ZAMP functions (see [16]), e.g.,…”

Section: Statement Of the Problemmentioning

confidence: 92%

Oscillatory integrals generated by the Ostrovsky equation

Варламов¹

2005

“…The integral representations obtained in this way involve the modified Bessel functions of order 1 6 and they are thus somewhat more complicated than the results obtained in [1].…”

Section: Introductionmentioning

confidence: 92%

“…On the positive real axis (which is a Stokes line) we simply take the average of these two expressions (as discussed in [1]) to obtain…”

Section: The Products Ai(z)bi 2 (Z) and Bi 3 (Z)mentioning

confidence: 99%

“…In a previous paper [1] it was shown how integral representations for the product of two Airy functions could be derived directly from the differential equation which they satisfy, and in this paper we wish to apply that technique to the product of three Airy functions. The integral representations obtained in this way involve the modified Bessel functions of order 1 6 and they are thus somewhat more complicated than the results obtained in [1].…”

Section: Introductionmentioning

confidence: 99%

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Integral representations for products of Airy functions ¶Part 2. Cubic products

Reid

1997

Self Cite

Integral representations are obtained for some cubic products of the Airy functions Ai(z) and Bi(z). These integral representations are of the Laplace contour type but they involve the modified Bessel functions of order 1 6 . From these results it is then possible to evaluate a number of definite integrals involving such cubic products.Mathematics Subject Classification (1991). 33C10, 34A20, 34E05, 44A15.

“…They are widely used in the theory of differential equations and asymptotic expansions (see [5,6,7]). Being special functions in their own right, the products of Airy functions also deserve serious attention and were studied in the papers [3,4,17,2,16,20,21,22].…”

Section: Introductionmentioning

confidence: 99%

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

Варламов¹

2007

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