Transverse Instability and Bifurcation Analysis of the line soliton for the NLS equation on a Fractured Strip

Abstract

We consider the nonlinear Schrödinger equation on a two-dimensional strip with an attractive $\delta$ interaction and power nonlinearity. We investigate the transverse stability and bifurcation of line solitons as the width of the strip varies. We first establish local well-posedness in $H^1$, conservation of mass and energy, and global existence in the $H^1$-subcritical regime. We then identify a critical width $L_{\ast }$ at which the line soliton undergoes a transverse instability. More precisely, we prove orbital stability for $L<L_{\ast }$ and orbital instability for $L>L_{\ast }$. At the critical width, a simple eigenvalue of the linearized operator crosses zero, and we construct, via the Lyapunov-Schmidt reduction, a branch of positive nontrivial stationary solutions bifurcating from the line soliton. We determine the direction of this bifurcation by computing the second-order variation of the width along the branch. Finally, we investigate the orbital stability of the bifurcating solitons and obtain a stability criterion which can be evaluated in the regime of sufficiently small interaction strength.