Designable Integrability of the Variable Coefficient Nonlinear Schrödinger Equations

JINGSONG HE* †, YISHEN LI‡† DepartmentofMathematics, NingboUniversity,Ningbo,Zhejiang315211, P.R.China‡ DepartmentofMathematics, USTC,Hefei,Anhui230026 ,P.R.ChinaAbstract. Thedesignableintegrability(DI)[40]ofthe variablecoefficientnonlinearSchr¨odingerequation(VCNLSE) isfirst introducedby constructionofan explicit transformationwhichmapsVCNLSE to the usual nonlinear Schr¨odinger equation(NLSE). One novel feature of VCNLSEwith DI is that its coefficients can be designed artificially and analytically by using transforma-tion. A special example between nonautonomous NLSE and NLSE is given here. Further, theoptical super-lattice potentials (or periodic potentials) and multi-well potentials are designed,which are two kinds of important potential in Bose-Einstein condensation(BEC) and nonlinearoptical systems. There are two interesting features of the soliton of the VCNLSE indicated bythe analytic and exact formula. Specifically, its the profile is variable and its trajectory is nota straight line when it evolves with time t.

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