AI 中文总结
研究二维非线性薛定谔方程中多峰波形自聚焦与坍塌，发现其从立方非线性临界阈值分岔，通过分析力平衡确定稳态，分析谱稳定性得出多峰状态更不稳定，揭示对称破缺等动力学，刻画了特征值分组。
AI 中文摘要
在本工作中，我们研究了具有一般幂律非线性的非线性薛定谔方程中涉及多个脉冲的二维波形的自聚焦和由此产生的坍塌。我们发现，广泛的多峰状态从立方非线性的临界阈值处发生分岔，即随着非线性指数超过该阈值，相关脉冲在临界极限下从无限远处开始并靠近。我们将由此产生的“相互作用粒子系统”确定为指数相互作用尾部（由合适的幂律调制）和线性相位诱导力之间的力平衡。发现由此力平衡产生的平衡与偏微分方程确定的稳态非常吻合。分析了多峰构型的谱稳定性，得出所有相关状态都比早期分析其稳定性的单峰坍塌解更不稳定的结论。事实上，我们揭示了对称破缺以及运动诱导的不稳定动力学，其中前者占主导并最终导致单个主导坍塌点。此外，我们系统地刻画了多峰构型的实部和虚部特征值，将它们按不同大小分组，由解的爆破率G的幂描述。
英文摘要
In the present work, we explore the self-focusing and resulting collapse of two-dimensional waveforms involving multiple pulses in a nonlinear Schroedinger equation with a general power-law nonlinearity. We find that a wide range of multi-peaked states bifurcate from the critical threshold of the cubic nonlinearity, thus representing ``bifurcations from infinity'', i.e., the relevant pulses start at infinite distance in the critical limit and draw nearer, as the nonlinearity exponent increases past that threshold. We identify the resulting ``interacting particle system'' as amounting to a force balance between the exponentially interacting tails (modulated by a suitable power law) and a linear phase-induced force. The equilibria emerging from this force balance are found to be in excellent agreement with the identified steady states of the partial differential equation. The spectral stability of multi-peaked configurations is analyzed, leading to the conclusion that all the relevant states are less stable than the single-peak collapsing solution whose stability was analyzed earlier. Indeed, we reveal both symmetry-breaking, as well as motion-inducing destabilizing dynamics, with the former ones among them being dominant and ultimately leading to a single dominant collapse spot. Moreover, we characterize systematically both the real and imaginary eigenvalues of multi-peaked configurations, partitioning them in groups of different sizes, described by powers of the solution's blowup rate G.
