By Robert C. Gunning, Hugo Rossi
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Extra resources for Analytic Functions of Several Complex Variables
The analogy between these techniques has been already noted. Here we show that, in the limit of “brutal” linearization corresponding to the neglect of all nonlinear contributions, just as the nonlinear evolution equation becomes a linear equation, so the technique of solution via the spectral transform goes over into the technique of solution via the Fourier transform. Indeed all the elements leading to this conclusion have been provided in the preceding sections. All we need here is to collect the relevant formulae, which are reported below even if this entails some repetition.
In the following we shall take this point of view, although we keep some of the terminology (such as “reflection” and “transmission” coefficients; see below) that clearly has originated from the quantal scattering problem. ,N . 1 17 Direct spectral problem solution of (1) characterized by the asymptotic boundary conditions (3a) (3b) +( x , k ) T( k)exp( -i kx), x+ - 00 , +( x , k ) exp( -i kx ) +R ( k ) exp(i kx ) , + -+ x+ + co This asymptotic behaviour is clearly consistent with (2), and it identifies uniquely the eigenfunction +(x, k), as well as the transmission coefficient T( k ) and the reflection coefficient R( k).
For such solutions, the standard analysis, based on the Fourier transform technique of solution, becomes applicable. s. -l), namely the form of the particular nonlinear evolution equation under consideration. s. -1). But if the spectral transform of u(x,t) has no discrete eigenvalues, namely if u( x, t ) contains no solitons, there is no mechanism available to maintain the solution u ( x , t ) (or any part of it) localized in the long-time limit. Thus, in the asymptotic long-time limit, the solution becomes locally small everywhere.
Analytic Functions of Several Complex Variables by Robert C. Gunning, Hugo Rossi