Nonlinear Wave Equations by Schmidt H.-J.
By Schmidt H.-J.
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Extra resources for Nonlinear Wave Equations
Sample text
For example, the linear free 1-dimensional Schr¨odinger equation is obtained 1 2 p if = 1. The common technique to solve this kind of wave equation with ω(p) = 2m is the Fourier transform, comprised by the following equations 1 q(x, t) = √ b(k, t)eikx dk, 2π R 1 b(k, t)ω(k)eikx dk, ω(−i∂x )q = √ 2π R ibt = ω(k)b, b(k, t) = e−iω(k)t b(k, 0). (194) (195) (196) (197) If the initial value q(x, 0) of the wave function is given, its Fourier transform b(k, 0) has to be multiplied by e−iω(k)t and Fourier re-transformed to give the desired solution q(x, t).
Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP34 (1972) 62-69 [9] A. Galindo and P. '#t' ParametricPlot#Evaluate#M#t', Z#t' ss. nb 2 ParametricPlot#Evaluate#M#t', Z#t' ss. 5 -1 h Graphics h The swept phase space area I(E) as a function of time listI Table#2 S +n 1/ S, NIntegrate#Evaluate#Z#t' ^ 2 ss. nb 5 Damped motion in phase space ParametricPlot#Evaluate#M#t', Z#t' ss. nb 6 Dissipation of energy Plot$Evaluate$k +1 Cos#M#t''/ 1 cccc Z#t' ^ 2 ss.
The second part consists of data of the bound states φn (x) of the potential q(x) with eigenvalues −κ2n , n = 1, . . , N. Since the potential is localized they exponentially decay for large values of x: φn (x) ∼ Cn e−κn x for x → ∞. (203) The reflection coefficient ρ(k) together with the Cn , κn define the “scattering data”, in symbols S(q) = {ρ(k), (Cn , κn )n=1,... ,N }. (204) S(q) can also be considered as the “scattering transform” of q, in analogy to the Fourier transform considered above. The scattering transform is designed to solve non-linear wave equations analogously to the solution of linear wave transformations by the Fourier transform.