Dobrokhotov, S. Yu.; Zhevandrov, P. N.; Kuz’mina, V. M. Asymptotics of the solution of the Cauchy-Poisson problem in a layer of nonconstant thickness. (English. Russian original) Zbl 0810.35084 Math. Notes 53, No. 6, 657-660 (1993); translation from Mat. Zametki 53, No. 6, 141-145 (1993). This article is devoted to the Cauchy-Poisson problem \[ \begin{alignedat}{4} \varepsilon^ 2 \Delta\Phi +\Phi_{yy} &=0 &\quad &(-H(x)< y<0), &\qquad &\Phi_ y+\varepsilon^ 2 \nabla \Phi\nabla H = 0 &\quad &(y=- H(x)),\\ \varepsilon^ 2 \Phi_{tt} +\Phi_ y &= 0 &\quad &(y=0), &\qquad &\Phi= \varphi_ 0,\quad \varepsilon\Phi_ t =\varphi_ 1 &\quad &(y=0,\;t=0).\end{alignedat} \] The main result is asymptotic representations of solutions to this problem for sufficiently small \(t\) in the neighborhood of some set of singularities that is naturally defined with solutions of the Hamiltonian system with Hamiltonian \({\mathcal H}= [| p| th(| p| H(X))]^{1/2}\). Reviewer: P.Zabreiko (Minsk) Cited in 6 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35C20 Asymptotic expansions of solutions to PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation Keywords:Cauchy-Poisson problem; asymptotic representations of solutions; Hamiltonian system PDFBibTeX XMLCite \textit{S. Yu. Dobrokhotov} et al., Math. Notes 53, No. 6, 1 (1993; Zbl 0810.35084); translation from Mat. Zametki 53, No. 6, 141--145 (1993) Full Text: DOI References: [1] V. A. Borovikov, Yu. V. Vladimirov, and M. Ya. Kel’bert, Preprint No. 236, Inst. Applied Math, of the Academy of Sciences of the USSR, Moscow (1984). [2] J. N. Newman in: T. Miloh (ed.), Mathematical Approaches in Hydrodynamics, SIAM, Philadelphia (1991), pp. 153-162. [3] C. Chestei, B. Friedman, and F. Ursell, Proc. Phil. Soc,53, 599-611 (1953). [4] N. Bleinstein, J. Math. Mech.,17, 533-560 (1967). [5] S. Yu. Dobrokhotov and P. N. Zhevandrov, Funkts. Analiz Prilozhen.,19, No. 1, 43-54 (1985). [6] V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1975). [7] A. A. Korobkin and I. V. Sturova, Zh. Prikl. Mat. Teor. Fiz.,3, 54-60 (1990). [8] V. P. Maslov, Usp. Mat. Nauk,38, No. 6, 3-36. [9] S. Yu. Dobrokhotov, P. N. Zhevandrov, V. P. Maslov, and A. I. Shafarevich, Mat. Zametki,49, No. 4, 31-46 (1991). [10] P. C. Sabatier, J. Fluid Mech.,126, 27-58 (1983). · Zbl 0528.76020 [11] S. Yu. Dobrokhotov and P. N. Zhevandrov, in: Oscillations and Waves in a Fluid [in Russian], Gor’kii (1988), pp. 32-41. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.