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

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