Spectral Transform for Nonlinear Evolution Equations with N Space Dimensions

Abstract

We study a special case of a method we visited recently⁽¹⁾ and which generalizes well-known methods for constructing solutions of non linear evolution equations. The input is an integral equation which is restricted here to the form

$${\psi _j}\left( {k,x,t} \right) = {I_j} + \int {\frac{{d\sigma \left( {\lambda ,\Lambda } \right)}}{{ - k + \Lambda }}T\left( {\lambda ,\Lambda ,x,t} \right)} {\psi _j}\left( {\lambda ,x,t} \right)$$

(1.1)

where k,λ,Λ, ∈ ℂ, ψ_j, I_j ∈ ℂ^N, x ∈ ℂ^N, t ∈ ℝ, T is a linear mapping of ℂ^N into itself, dσ is a measure in ℂ². For j = 1,2…,N, I_j are known column vectors, defined as (I_j)_k = δ_jk