Inverse Scattering Problem on the Half-plane for a First Order Hyperbolic System
Mansur I. Ismailov, Gebze Institute of Technology, Turkey mismailov@gyte.edu.tr
We consider the following system of first-order PDE on the half-plane { 𝑥,𝑡 :𝑥≥ 0, 𝑡 ∈ℝ}
𝜕!𝜓−𝜎𝜕!𝜓=[𝜎,𝑄(𝑥,𝑡)]𝜓, (1) where 𝜎= 𝑑𝑖𝑎𝑔{𝜉!,𝜉!,𝜉!,𝜉!} constant diagonal matrix with 𝜉! ≥𝜉! >0> 𝜉! ≥ 𝜉!,
𝐴,𝐵 = 𝐴𝐵−𝐵𝐴 and [𝜎,𝑄(𝑥,𝑡)] is an 4×4 matrix function with measurable complex-valued rapidly decreasing (Schwartz) entries. The matrix function [𝜎,𝑄(𝑥,𝑡)] is called the potential.
Consider the system (1) under the boundary condition
𝜓!(0,𝑡)=𝐻𝜓!(0,𝑡),𝑑𝑒𝑡𝐻≠ 0, (2) where 𝜓(𝑥,𝑡) = 𝜓!(𝑥,𝑡)
𝜓!(𝑥,𝑡) and 𝜓!(𝑥,𝑡), 𝜓!(𝑥,𝑡) are 2 dimensional vector functions, and asymptotic condition
𝜓! 𝑥,𝑡 =ℱ!𝑎 𝑡 +𝑜 1 ,𝑥→+∞ (3) in space ℒ!(ℝ!,ℂ!), where ℱ!𝑎 𝑡 = 𝑎!(𝑡+𝜉!𝑥)
𝑎!(𝑡+𝜉!𝑥) , 𝑎 𝑡 denotes the profile of incident waves.
For arbitrary 𝑎 𝑡 ∈ℒ!(ℝ,ℂ!), the system (1) with the conditions (2) and (3) have a unique solution in space ℒ!(ℝ!,ℂ!) and the asymptotic relation
𝜓! 𝑥,𝑡 = 𝒯!𝑏 𝑡 +𝑜 1 ,𝑥→+∞
holds in space ℒ!(ℝ!,ℂ!), where 𝒯!𝑏 𝑡 = 𝑏!(𝑡+𝜉!𝑥)
𝑏!(𝑡+𝜉!𝑥) , 𝑏 𝑡 denotes the profile of scattered waves.
An operator 𝑆! transforming the incident waves into the scattering waves is called the scattering operator on the half-plane for the system (1) with the boundary condition (2):
𝑆!:𝑎(𝑡) →𝑏(𝑡).
In this talk, we establish the uniqueness and non-uniqueness of the solution of the inverse problem of determining the potential from the scattering operator 𝑆!.
