Existence Of Constrained State Solution For Second Harmonic Generation Systems

Nonlinear partial differential equations can describe the important natural phenomena well and have been widely concerned by a large number of researchers.In nonlinear optic theory,the propagation of soliton in quadratic nonlinear fiber couplers can be described by second harmonic generation systems.This paper studies existence of solutions for second harmonic generation systems and second harmonic generation systems with perturbations.It provides a theoretical basis for people to understand the two coupling optical effects.This article includes two chapters.The Chapter 1 considers the existence of solutions to the second harmonic generation equations under the condition?RN u2 = a,?RN v2 = b.Here N = 1,2,?,a,b>0 are constant,?1 and ?2 are unknown.We use the critical point theory and function rearrangement method to obtain the existence of constrained state solutions of the above systems.In Chapter 2,we mainly consider the existence of constrained state solutions to second harmonic generation systems with perturbations.under the condition?RNu2 = a,?RN v2 = b.Here ?,?>0,N = 1,2,3,?1 and ?2 are unknown.For any a,b>0,any minimizing sequence for the second harmonic generation systems with perturbations is,up to translation,precompact in Lp(RN)×Lp(RN)for p ?(2,2*),where 2*= 6,N= 3;2*=?,N =1,2.Furthermore,we obtain existence of constrained state solutions for second harmonic generation systems with perturbations.