非线性散射问题

The Scattering Problem in the Nonlinear Medium

三维介质中的谐波在遇到障碍物后的散射问题,数学上可表示为Helmholtz方程的边值问题,其中无穷远点满足Sommerfeld散射条件.在非线性介质中,波动方程可表示为u_(tt)-c^2△u=F(x,u),当F(x,u)满足适当条件时,代入入射波的表达式U(x,t)=e^(-iωt)u(x),即得到在有界区域内散射波满足的方程△u+k_1~2u=f(x,u).对非线性介质在小跳跃度和小扰动下散射问题的解的存在性进行讨论,同时对一类非线性函数f(x,u)在大跳跃度情况下给出散射问题解的存在性.

This paper consideres the scattering problem of time harmonic wave after its interaction with the obstacle in three dimensions.The mathematical formulation of this problem involves Helmholtz equations with boundary value conditions at the interface and Sommerfeld radiation condition at infinity.The wave equation can be expressed by U（tt）- c^2△U=F（x,U）in the nonlinear medium.Under some assumptions on F（x,U）,a kind of solution of time harmonic wave U（x,t）=e^（-iωt）u（x）is considered.Then the wave equation yields△u＋k1^2u=f（x,u） which the incidental wave satisfies in a bounded domain. Here, the existence of the solution of the scatU＇ring problem with small jump and small perturbation is studied and also for a kind of nonlinear function f（x, u） some results of existence with big jump are given.