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Inverse problem for N × N hyperbolic systems on the plane and the N‐wave interactions
Author(s) -
Sung L. Y.,
Fokas A. S.
Publication year - 1991
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160440503
Subject(s) - mathematics , inverse , norm (philosophy) , hermitian matrix , inverse problem , diagonal , matrix (chemical analysis) , combinatorics , mathematical analysis , pure mathematics , geometry , materials science , political science , law , composite material
We study regiorously the solvability of the direct and inverse problems associated with Ψ x – J Ψ y = Q Ψ,( x , y ) ∈ ℝ 2 , where (i) Ψ is an N × N ‐matrix‐valued function on ℝ 2 ( N ≦ 2), (ii) J is a constant, real, diagonal N × N matrix with entries, J 1 > J 2 > … > J N and (iii) Q is off‐diagonal with rapidly decreasing (Schwartz) component functions. In particular we show that the direct problem is always solvable and give a small norm condition for the solvability of the inverse problem. In the particular case that Q is skew Hermitian the inverse problem is solvable without the small norm assumption. Furthermore we show how these results can be used to solve certain Cauchy problems for the associated nonlinear evolution equations. For concreteness we consider the N‐wave interactions and show that if a certain norm of Q ( x , y , 0) is smallor if Q ( x , y , 0) is skew Hermitian the N‐wave interations equation has a unique global solution.