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Transformations and Soliton Solutions for a Variable-coefficient Nonlinear Schrödinger Equation in the Dispersion Decreasing Fiber with Symbolic Computation

Zeng Z.-F., Liu J.-G., Jiang Y., Nie B.

Fundamenta Informaticae

2016

Vol. 145, nr 2

207--219

Describing the dispersion decreasing fiber, a variable-coefficient nonlinear Schrödinger equation is hereby under investigation. Three transformations have been obtained from such a equation to the known standard and cylindrical nonlinear Schrödinger equations with the relevant constraints on the variable coefficients presented, which turn out to be more general than those previously published in the literature. Meanwhile, several families of exact dark-soliton-like and bright-soliton-like solutions are constructed. Also, we obtain some similarity solutions, which can be illustrated in terms of the elliptic and the second Painlevé transcendent equations.

Extended Generalized Hyperbolic-function Method and New Exact Solutions of the Generalized Hamiltonian and NNV Equations by the Symbolic Computation

Liu J.-G., Zeng Z.-F.

Fundamenta Informaticae

2014

Vol. 132, nr 4

501--517

In this paper, with the aid of the computerized symbolic computation, we present an extended generalized hyperbolic-function method. Being concise and straightforward, it can be applicable to seek more types of solutions for certain nonlinear evolution equations (NLEES). In illustration, we choose the generalized Hamiltonian equations and the (2 + 1)-dimensional Nizhnik- Novikov-Veselov (NNV) equations to demonstrate the validity and advantages of the method. As a result, abundant new exact solutions are obtained including soliton-like solutions, traveling wave solutions etc. Themethod can be also applied to other nonlinear partial differential equations (NPDEs).