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In this paper, we study the bifurcation problem from a line soliton for astationary nonlinear Schr\"{o}dinger equation on the product space $\mathbb{R}\times \mathbb{T}$. We extend earlier results to a larger class of thenonlinearity in the equation. The salient point of our analysis relies on alower bound of solution to the ``auxiliary equation'' and then on theapplication of the Crandall-Rabinowitz argument

Pitchfork bifurcation at line solitons for nonlinear Schr\"{o}dinger equations on the product space $\mathbb{R} \times \mathbb{T}$