AI 中文总结
针对断裂带区域上带吸引δ相互作用与幂次非线性的二维NLS方程，确定线孤子横向不稳定性的临界宽度，构造分岔解并给出其稳定性准则。
AI 中文摘要
我们研究具有吸引δ相互作用和幂次非线性的二维带形区域上的非线性薛定谔（NLS）方程。我们探讨带形区域宽度变化时线孤子的横向稳定性与分岔。首先，我们建立了该方程在H¹空间中的局部适定性、质量与能量守恒性，以及在H¹次临界区域中的整体存在性。接着，我们确定了线孤子发生横向不稳定性的临界宽度L₊，具体而言，证明了当宽度L<L₊时线孤子具有轨道稳定性，当L>L₊时则具有轨道不稳定性。在临界宽度处，线性化算子的一个简单特征值穿过零点，我们通过Lyapunov-Schmidt约化构造出从线孤子分岔出的一支正的非平凡定态解。我们通过计算沿该分支宽度的二阶变分确定分岔方向。最后，我们研究了分岔出的孤子的轨道稳定性，并得到了可在相互作用强度足够小的区域中进行评估的稳定性准则。
英文摘要
We consider the nonlinear Schrödinger equation on a two-dimensional strip with an attractive $δ$ 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 $L_*$ at which the line soliton undergoes a transverse instability. More precisely, we prove orbital stability for $L<L_*$ and orbital instability for $L>L_*$. 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.
