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Ferreira González, Chelo

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https://academica-e.unavarra.es/handle/2454/49026

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Ferreira González

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Chelo

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Matemática e Informática

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2476

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2018 - 201912020 - 20212
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    PublicationOpen Access
    The asymptotic expansion of the swallowtail integral in the highly oscillatory region
    (Elsevier, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
    The mathematical models of many short wavelength phenomena, specially wave propagation and optical diffraction, contain, as a basic ingredient, oscillatory integrals with several nearly coincident stationary phase or saddle points. The uniform approximation of those integrals can be expressed in terms of certain canonical integrals and their derivatives [2,16]. The importance of these canonical diffraction integrals is stressed in [14] by means of the following sentence: The role played by these canonical diffraction integrals in the analysis of caustic wave fields is analogous to that played by complex exponentials in plane wave theory. Apart from their mathematical importance in the uniform asymptotic approximation of oscillatory integrals [12], the canonical diffraction integrals have physical applications in the description of surface gravity waves [11], [17], bifurcation sets, optics, quantum mechanics, chemical physics [4] and acoustics (see [1], Section 36.14 and references there in). To our knowledge, the first application of this family of integrals traces back to the description of the disturbances on a water surface produced, for example, by a traveling ship. These disturbances form a familiar pattern of bow and stern waves which was first explained mathematically by Lord Kelvin [10] using these integrals.
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    PublicationOpen Access
    The swallowtail integral in the highly oscillatory region II
    (Kent State University, 2020) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
    We analyze the asymptotic behavior of the swallowtail integral R ∞ −∞ e i(t 5+xt3+yt2+zt)dt for large values of |y| and bounded values of |x| and |z|. We use the simpli ed saddle point method introduced in [López et al., 2009]. With this method, the analysis is more straightforward than with the standard saddle point method and it is possible to derive complete asymptotic expansions of the integral for large |y| and xed x and z. There are four Stokes lines in the sector (−π, π] that divide the complex y−plane in four sectors in which the swallowtail integral behaves di erently when |y| is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of x, y and z. One of them is of Poincaré type and is given in terms of inverse powers of y 1/2 . The other one is given in terms of an asymptotic sequence of the order O(y −n/9 ) when |y| → ∞, and it is multiplied by an exponential factor that behaves di erently in the four mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.
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    PublicationOpen Access
    The swallowtail integral in the highly oscillatory region III
    (Taylor & Francis, 2021) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
    We consider the swallowtail integral Ψ(x,y,z):=∫∞−∞ei(t5+xt3+yt2+zt)dt for large values of |z| and bounded values of |x| and |y|. The integrand of the swallowtail integral oscillates wildly in this region and the asymptotic analysis is subtle. The standard saddle point method is complicated and then we use the modified saddle point method introduced in López et al., A systematization of the saddle point method application to the Airy and Hankel functions. J Math Anal Appl. 2009;354:347–359. The analysis is more straightforward with this method and it is possible to derive complete asymptotic expansions of Ψ(x,y,z) for large |z| and fixed x and y. The asymptotic analysis requires the study of three different regions for argz separated by three Stokes lines in the sector −π