Asymptotic Perturbation Theory of Waves - download pdf or read online
By Lev Ostrovsky
This e-book is an creation to the perturbation idea for linear and nonlinear waves in dispersive and dissipative media. the main target is at the direct asymptotic approach that's in response to the asymptotic growth of the answer in sequence of 1 or extra small parameters and important finiteness of the perturbations; this leads to sluggish edition of the main-order resolution. the tactic, which doesn't rely on integrability of simple equations, is utilized to quasi-harmonic and non-harmonic periodic waves, in addition to to localized waves resembling solitons, kinks, and autowaves. the fundamental theoretical principles are illustrated through 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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It is assumed that if Re λ i = 0, then Im λ i ≠ 0, ±1, ±2... The latter condition is known in the theory of nonlinear oscillations as the absence of internal resonances . When this condition is met, the corresponding N – m vectors Yi are out of resonance with the fundamental periodic solution. c. c. denotes complex conjugation. 12) 44 Asymptotic Perturbation Theory of Waves where y *jk are periodic functions of θ corresponding to the adjoint matrix Y*. 10), λ1 = .... = λm +1 = 0 ; in particular, y1 = Uθ .
C. 7) so that ∂k ∂ω + =0. , the dispersion equation. 8), we have ∂ω ∂ω ∂ω ∂ω +c g = 0, c g = , so that cgs = ∂t ∂r ∂k ∂ks ( s = x, y , z ) . 10) Here cg is the group velocity vector. 5). In the next order we have A ∂u1 ∂u ∂a iθ ∂a + B 1 + Cu1 = f − H ( 0 ) , H ( 0 ) = A +B e . , n = 1, 2,... 12) n Here fn (T , X) are slowly varying coefficients. 13) n where u1n (T , X) are slowly varying Fourier coefficients. 11) yields ∂a ∂a f1 − A +B f2 ∂X ∂T u1 = , u2 = ... ). 4), D = 0. First we suppose that D2 and similar determinants for higher harmonics are non-zero for all T and X; in the next section we shall consider the resonant interaction of wave harmonics for which the latter condition is violated.
46) 16 Asymptotic Perturbation Theory of Waves where β = (cg2 − c02 ) / (2kc02 ) and γ = 3q / (2kc02 ) ; in the small term with ∂A ∂A ≈ − cg which is true up to the next order. 46), we obtain ∂A ∂2 A 2 + iβ + iγ A A = 0. 47) is widely used in contemporary theory of nonlinear waves. It is called the Nonlinear Schrödinger equation (NSE or NLS). This equation has been thoroughly discussed in the literature, both mathematically and in applications to nonlinear optics, water waves, etc. , [2, 7–9]).
Asymptotic Perturbation Theory of Waves by Lev Ostrovsky