By Lev Ostrovsky
This publication is an advent to the perturbation conception for linear and nonlinear waves in dispersive and dissipative media. the main target is at the direct asymptotic process that's in response to the asymptotic enlargement of the answer in sequence of 1 or extra small parameters and significant finiteness of the perturbations; this leads to sluggish edition of the main-order answer. the strategy, which doesn't depend upon integrability of simple equations, is utilized to quasi-harmonic and non-harmonic periodic waves, in addition to to localized waves equivalent to solitons, kinks, and autowaves. the fundamental theoretical rules are illustrated by means of many actual examples in the course of the book.
Readership: Graduate scholars and younger researchers in nonlinear technology, physicists and utilized mathematicians.
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Additional resources for Asymptotic Perturbation Theory of Waves
2 illustrates the exponential increase of particle velocity amplitude (and, correspondingly, the vertical displacement amplitude) in a wave propagating upward. 2. 5 km. Let us also mention propagation of an acoustic impulse which is short compared with the characteristic scale of spatial variation of medium parameters. Since the wave frequency and the pulse duration in a stationary non-dispersive medium remain constant, the wave number k is 31 Perturbation Method for Quasi-Harmonic Waves proportional to c0 along the propagation direction l and, since the entire pulse is short, it is possible to neglect variations of c0 and ρ at its scale.
After zeroing the expressions at each power of µ , we obtain, in each order of approximation, a set of linear equations for perturbations: Gu ( n ) = H ( n ) . 3) near the solution U(θ, A), namely, Gu (n) ∂u (1) ∂Q(0) ∂A(0) ∂B (0) ∂U ( n ) = (ω A − kB ) + + ω −k u . 5) depends only on the solutions obtained in previous approximations. In particular, ∂U ∂U (0) + B (0) H (1) = f (0) − A(0) , f = f ( U, T , X ). 5) have variable coefficients which are, by definition, periodic in θ .
32 Asymptotic Perturbation Theory of Waves Strictly speaking, these equations are good for a rarefied dielectric (gas, plasma); otherwise, the electric field E in the last equation should be changed to some average field Eeff acting on a given electron from all other particles in the medium. The latter can be treated in a similar fashion. 2). 42) and write the corresponding vectors and matrices of coefficients: H E u = , A = P S C= 0 0 0 0 µ0 0 0 0 0 ε0 1 0 0 0 0 1 0 0 1 0 , B= 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 , ∂H 0 ∂E0 µ0 ∂T + ∂X 0 0 0 ε ∂E0 + ∂H 0 0 0 0 0 ∂X ∂T f , = .