Exact solutions of the nonlinear Schr\"odinger equation iψₜ+{Δ}{ψ}=a₀{ψ}/B +a₁{ψ}{}{ψ}²+a₂{ψ}{}{ψ}⁴, for which initial conditions can be imposed on a cylinder, are presented. A symmetry group of the equation is used to reduce it to an ordinary differential equation which is then solved with the help of a singularity analysis. Solutions are obtained in terms of elementary functions, Jacobi elliptic functions, and Painlev\'e transcendents.
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Gagnon et al. (1989) studied this question.