Geodesic
Asymptotic Solutions of the One-Dimensional Linearized Korteweg--de Vries Equation with Localized Initial Data
Matematičeskie zametki, Tome 102 (2017) no. 3, pp. 445-461.

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The Cauchy problem with localized initial data for the linearized Korteweg–de Vries equation is considered. In the case of constant coefficients, exact solutions for the initial function in the form of the Gaussian exponential are constructed. For a fairly arbitrary localized initial function, an asymptotic (with respect to the small localization parameter) solution is constructed as the combination of the Airy function and its derivative. In the limit as the parameter tends to zero, this solution becomes the exact Green function for the Cauchy problem. Such an asymptotics is also applicable to the case of a discontinuous initial function. For an equation with variable coefficients, the asymptotic solution in a neighborhood of focal points is expressed using special functions. The leading front of the wave and its asymptotics are constructed.
Keywords: linearized Korteweg–de Vries equation, Cauchy problem, asymptotic solution, Maslov canonical operator, Green function, Airy function. 
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     author = {S. A. Sergeev},
     title = {Asymptotic {Solutions} of the {One-Dimensional} {Linearized} {Korteweg--de} {Vries} {Equation} with {Localized} {Initial} {Data}},
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S. A. Sergeev. Asymptotic Solutions of the One-Dimensional Linearized Korteweg--de Vries Equation with Localized Initial Data. Matematičeskie zametki, Tome 102 (2017) no. 3, pp. 445-461. http://geodesic.mathdoc.fr/item/MZM_2017_102_3_a9/

[1] V. I. Karpman, Nelineinye volny v dispergiruyuschikh sredakh, Nauka, M., 1973 | MR

[2] Dzh. Uizem, Lineinye i nelineinye volny, Mir, M., 1977 | MR | Zbl

[3] R. Haberman, Applied Partial Differential Equations, Pearson Education, Upper Saddle River, NJ, 2013 | MR | Zbl

[4] R. Bullaf, F. Kodri, Solitony, Mir, M., 1983 | Zbl

[5] R. G. Littlejohn, “The Van Vleck formula, Maslov theory, and phase space geometry”, J. Statist. Phys., 68:1-2 (1992), 7–50 | DOI | MR | Zbl

[6] S. Yu. Dobrokhotov, G. Makrakis, V. E. Nazaikinskii, “Kanonicheskii operator Maslova, odna formula Khermandera i lokalizatsiya resheniya Berri–Balazha v teorii volnovykh puchkov”, TMF, 180:2 (2014), 162–188 | DOI | Zbl

[7] C. Yu. Dobrokhotov, B. Tirotstsi, A. I. Shafarevich, “Predstavleniya bystroubyvayuschikh funktsii kanonicheskim operatorom Maslova”, Matem. zametki, 82:5 (2007), 792–796 | DOI | MR | Zbl

[8] V. P. Maslov, M. V. Fedoryuk, Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR | Zbl

[9] F. Olver, Asimptotika i spetsialnye funktsii, Nauka, M., 1990 | MR | Zbl

[10] S. Yu. Dobrokhotov, G. N. Makrakis, V. E. Nazaikinskii, T. Ya. Tudorovskii, “Novye formuly dlya kanonicheskogo operatora Maslova v okrestnosti fokalnykh tochek i kaustik v dvumernykh kvaziklassicheskikh asimptotikakh”, TMF, 177:3 (2013), 355–386 | DOI | Zbl

[11] C. Yu. Dobrokhotov, V. E. Nazaikinskii, “Prokolotye lagranzhevy mnogoobraziya i asimptoticheskie resheniya lineinykh uravnenii voln na vode s lokalizovannymi nachalnymi usloviyami”, Matem. zametki, 101:6 (2017), 936–943 | DOI

[12] L. Hörmander, “Fourier integral operators. I”, Acta Math., 127:1-2 (1971), 79–183 | DOI | MR | Zbl