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Designable Integrability of the Variable Coefficient Nonlinear Schrödinger Equations


Address for correspondence: J. He, Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, PR China; e-mail:


The designable integrability (DI) [51] of the variable coefficient nonlinear Schrödinger equations (VCNLSEs) is first introduced by construction of an explicit transformation, which maps VCNLSE to the usual nonlinear Schrödinger equations (NLSEs). One novel feature of VCNLSE with DI is that its coefficients can be designed artificially and analytically by using transformation. A special example between nonautonomous NLSEs and NLSEs is given here. Further, the optical super-lattice potentials (or periodic potentials) and multiwell potentials are designed, which are two kinds of important potential in Bose–Einstein condensation and nonlinear optical systems. There are two interesting features of the soliton of the VCNLSEs indicated by the analytic and exact formula. Specifically, its profile is variable and its trajectory is not a straight line when it evolves with time t.