Representation of high frequency wave fields in the form of integrals over Gaussian beams. Gaussian beams are solutions of corresponding hyperbolic equations or system of equations, which have the ray type asymptotic expansion with complex phase and concentrated in a neighbourhood of a geodesic. The integral representations over Gaussian beams give uniform asymptotic expansions of high frequency wave fields on any compact set. The representations are convenient for calculation both in regulare zone of the geodesic fields and in singular points of the fields. Solution of inverse problem of determination of parameters of different media by boundary measurements. Method is based upon construction of a geometric model which is equivalent to initial physical model and uses Gaussian beams.
Biography
Math.-mech. department of the Leningrad State University — 1970. PhD (candidate of science) — 1978. Doctor of science — 1997.
Main publications:
Katchalov A., Kurylev Y. Multidimensional inverse problems with incomplete boundary spectral data // Comm. in PDE, 1998, 23, no. 1–2, 55–95.
Katchalov A., Kurylev Y., Lassas M. Inverse boundary spectral problems. 2001, CRC Press, 290 p.
A. P. Kachalov, “The ray type solution for the wave of finite deformation in the physically linear nonlinear inhomogeneous elastic medium”, Zap. Nauchn. Sem. POMI, 438 (2015), 118–132; J. Math. Sci. (N. Y.), 224:1 (2017), 79–89
2014
2.
A. P. Kachalov, “The ray type solution for the finite deformation waves in a physically linear nonlinear inhomogeneous medium”, Zap. Nauchn. Sem. POMI, 422 (2014), 47–61; J. Math. Sci. (N. Y.), 206:3 (2015), 260–269
2012
3.
A. P. Kachalov, S. A. Kachalov, “Computations of Rayleigh waves in anisotropic elastic media and impedance”, Zap. Nauchn. Sem. POMI, 409 (2012), 40–48; J. Math. Sci. (N. Y.), 194:1 (2013), 21–25
2011
4.
A. P. Kachalov, “Rayleigh waves in an anisotropic elastic medium and impedance”, Zap. Nauchn. Sem. POMI, 393 (2011), 125–143; J. Math. Sci. (N. Y.), 185:4 (2012), 581–590
A. P. Katchalov, “Quasijets in anisotropic media, Finsler geometry, and Fermi coordinates”, Zap. Nauchn. Sem. POMI, 332 (2006), 48–69; J. Math. Sci. (N. Y.), 142:6 (2007), 2546–2558
A. P. Katchalov, “Gaussian beams, the Hamilton–Jacobi equations and Finsler geometry”, Zap. Nauchn. Sem. POMI, 297 (2003), 66–92; J. Math. Sci. (N. Y.), 127:6 (2005), 2374–2388
A. P. Katchalov, “Gaussian beams for the Maxwell equations on a mainfold”, Zap. Nauchn. Sem. POMI, 285 (2002), 58–87; J. Math. Sci. (N. Y.), 122:5 (2004), 3485–3501
A. P. Katchalov, “Nonstationary electromagnetic Gaussian beams in nonhomogeneous anysotropic media”, Zap. Nauchn. Sem. POMI, 264 (2000), 83–100; J. Math. Sci. (New York), 111:4 (2002), 3667–3677
A. P. Kachalov, Ya. V. Kurylev, “The multidimensional inverse Gel'fand problem with incomplete boundary spectral data”, Dokl. Akad. Nauk, 346:5 (1996), 587–589
1994
10.
M. I. Belishev, A. P. Kachalov, “Operator integral in multidimensional spectral Inverse Problem”, Zap. Nauchn. Sem. POMI, 215 (1994), 9–37; J. Math. Sci. (New York), 85:1 (1997), 1559–1577
M. I. Belishev, A. P. Katchalov, “Boundary controls and quasiphotons in a Riemannian manifold reconstruction problem via dynamical data”, Zap. Nauchn. Sem. POMI, 203 (1992), 21–50; J. Math. Sci., 79:4 (1996), 1172–1190
A. P. Katchalov, “Space-time Gaussian beams of the electromagnetic waves”, Zap. Nauchn. Sem. LOMI, 186 (1990), 115–121; J. Math. Sci., 73:3 (1995), 370–374
A. P. Katchalov, Ya. V. Kurylev, “Transformation operator method for inverse scattering problem”, Zap. Nauchn. Sem. LOMI, 179 (1989), 73–87; J. Soviet Math., 57:3 (1991), 3111–3122
M. I. Belishev, A. P. Katchalov, “Application of boundary control theory methods to spectral inverse problem for inhomogeneous string”, Zap. Nauchn. Sem. LOMI, 179 (1989), 14–22; J. Soviet Math., 57:3 (1991), 3072–3077
A. P. Katchalov, Ya. V. Kurylev, “Asymptotics of the Jost-function for the two-dimensional Schrödinger operator”, Zap. Nauchn. Sem. LOMI, 173 (1988), 96–103; J. Soviet Math., 55:3 (1991), 1712–1717
16.
A. P. Katchalov, “Two-parameter asymptotic formulas for space-time Gaussian beams in an elastic media”, Zap. Nauchn. Sem. LOMI, 173 (1988), 87–95; J. Soviet Math., 55:3 (1991), 1705–1712
A. P. Kachalov, M. M. Popov, “Application of Gaussian beam method to the computations of theoretical seisinograms”, Zap. Nauchn. Sem. LOMI, 156 (1986), 73–97
1985
18.
A. P. Kachalov, “Application of “quaaiphotons” to the computations of wave fields in elasticity media”, Zap. Nauchn. Sem. LOMI, 148 (1985), 89–103
1984
19.
A. P. Katchalov, “A coordinate system for describing the “quasiphoton””, Zap. Nauchn. Sem. LOMI, 140 (1984), 73–76; J. Soviet Math., 32:2 (1986), 151–153
A. P. Kachalov, “Space-time ray method for waves of small deformation in a nonlinear elastic medium”, Zap. Nauchn. Sem. LOMI, 140 (1984), 61–72; J. Soviet Math., 32:2 (1986), 143–150
A. P. Katchalov, M. M. Popov, I. Pshenchik, “On validity of Gaussian beams summation method in problems with corner points on boundaries”, Zap. Nauchn. Sem. LOMI, 128 (1983), 65–71
1981
22.
A. P. Kachalov, M. M. Popov, “Application of the Gaussian beam summation method for the computation of wave fields in the high-frequency approximation”, Dokl. Akad. Nauk SSSR, 258:5 (1981), 1097–1100
A. P. Katchalov, “The weak inhomogeneous optical fiber”, Zap. Nauchn. Sem. LOMI, 117 (1981), 134–146
1978
24.
A. P. Katchalov, “Behavior of the roots of the equation $w'_1(z)+\sigma w_1(z)=0$”, Zap. Nauchn. Sem. LOMI, 78 (1978), 90–94; J. Soviet Math., 22:1 (1983), 1056–1059
1976
25.
A. P. Katchalov, “Some equations for the problem of diffraction by a convex shell”, Zap. Nauchn. Sem. LOMI, 62 (1976), 124–125; J. Soviet Math., 11:5 (1979), 743–744
26.
A. P. Katchalov, “Ray method for flexural vibrations of a shell immersed in a liquid”, Zap. Nauchn. Sem. LOMI, 62 (1976), 111–123; J. Soviet Math., 11:5 (1979), 733–742
1974
27.
A. P. Katchalov, “Elastic Wave Propagation in Piezodielectric Medium”, Zap. Nauchn. Sem. LOMI, 42 (1974), 155–161
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