# zbMATH — the first resource for mathematics

The effect of a small background inhomogeneity on the asymptotic properties of linear perturbations. (English. Russian original) Zbl 1272.35020
J. Appl. Math. Mech. 74, No. 2, 127-134 (2010); translation from Prikl. Mat. Mekh. 74, No. 2, 179-190 (2010).
Summary: General regularities in the evolution of one-dimensional unstable linear perturbations on a weakly inhomogeneous background are studied when, at the initial instant, the perturbations are concentrated in the {$$\delta$$}-neighbourhood of a certain point. Times are considered when these perturbations do not fall outside the limits of a certain domain of size $$l$$ such that $$\delta \ll l \ll L$$, where $$L$$ is the large characteristic size of the background inhomogeneity. With contain assumptions, the effect of the background inhomogeneity on the asymptotic behaviour of the perturbations at long times is taken into account in a general form. The first corrections to the well known asymptotic relation for the evolution of perturbations on a homogeneous background, that arise because of background inhomogeneity, are obtained using Hamilton’s method. An example of the use of the proposed approximate method is considered and the error in the approximation is estimated.
##### MSC:
 35B20 Perturbations in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
Full Text:
##### References:
 [1] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001 [2] Lifshitz, E.M.; Pitaevskii, L.P., Physical kinetics, (1981), Pergamon Oxford [3] Fedoryuk, M.F., The method of steepest descent, (1977), Nauka Moscow [4] Maslov, V.P., Operator methods, (1973), Nauka Moscow · Zbl 0288.47042 [5] Iordanskii, S.V., The stability of inhomogeneous states and continual integrals, Zh exper teor fiz, 94, 7, 180-189, (1988) [6] Belov, V.V.; Dobrokhotov OYu; Dobrokhotov, SYu., Isotropic tori, complex germ and Maslov index, the norm of a form and quasimodes of multidimensional problems, Mat zamet, 69, 4, 483-514, (2001) [7] Maslov, V.P., Asymptotic methods for solving pseudodifferential equations, (1987), Nauka Moscow · Zbl 0625.35001 [8] Maslov, V.P., Perturbation theory and asymptotic methods, (1972), Dunod Paris · Zbl 0653.35002 [9] Wienstein A. The Maslov Gerbe. arXiv:math/0312274v1 [math.SG] 13 Dec. 2003. [10] Hunt, R.E.; Crighton, D.G., Instability of flows in spatially developing media, Proc. roy. soc. London ser. A., 435, 1893, 109-128, (1991) · Zbl 0731.76032 [11] Kulikovskii, A.G.; Lozovskii, A.V.; Pashchenko, N.T., The evolution of perturbations on a weakly inhomogeneous background, Prikl mat mekh, 71, 5, 761-774, (2007) · Zbl 1164.74445 [12] Sivashinsky, G.I., Nonlinear analysis of hydrodynamic instability in laminr flames. I. derivation of basic equations, Acta astronaut., 4, 11/12, 1177-1206, (1997) · Zbl 0427.76047 [13] Landau, L.D., On the theory of slow combustion, Zh eksper teor fiz, 14, 6, 240-245, (1944) [14] Markshtein, G.H., Experimental and theoretical studies of flame front stability., J. aeronaut. sci., 18, 3, 199-209, (1951)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.