一类带一阶贝塞尔正则化的Airy方程的小速度孤立波稳定性与整体适定性
Small-speed solitary-wave stability and global well-posedness for a Bessel-regularized Airy equation
AI总结：
本研究针对带一阶贝塞尔正则化的Airy方程，证明了小速度下孤立波的轨道稳定性，建立了相关动力学框架并验证了稳定性准则的非退化性与斜率条件。
摘要：
我们证明了一类带二次相互作用项且经一阶贝塞尔势滤波的Airy方程，在足够小的正速度下极小化孤立波的轨道稳定性。稳定性假设已得到验证：约束极小化给出一个负方向；小速度KdV极限与剖面预解式的算子范数扰动给出一维平移核；带一致有界导数的可重标分支给出二次不变量的正斜率。由此，一个守恒的Lyapunov泛函证明了在3/2阶导数能量空间中的轨道稳定性，且该扰动还给出小速度下中心极小化子的唯一性。动力学框架由直接的Bourgain空间双线性估计、平方可积类中的局部适定性、1/4阶导数起的无条件唯一性，以及半阶导数能量空间中的整体适定性提供。对每个给定的正速度，约束极小化构造出光滑正剖面，其对称、远离中心处严格单调且指数局域化。在一般正速度下，稳定性准则仍依赖于非退化性与斜率条件。
英文摘要：
We prove orbital stability of minimizing solitary waves at sufficiently small positive speeds for an Airy equation with a quadratic interaction filtered by a Bessel potential of order one. The stability hypotheses are verified: constrained minimization gives one negative direction, the small-speed KdV limit and operator-norm perturbation of the profile resolvent give a one-dimensional translation kernel, and a differentiable rescaled branch with uniformly bounded derivative gives a positive slope of the quadratic invariant. A conserved Lyapunov functional then proves orbital stability in the three-halves-derivative energy space and the perturbation also gives uniqueness of centered minimizers at small speed. The dynamical framework is supplied by a direct Bourgain-space bilinear estimate, local well-posedness in the square-integrable class, unconditional uniqueness from one-quarter derivative onward, and global well-posedness in the half-derivative energy space. For every prescribed positive speed, constrained minimization constructs smooth positive profiles that are symmetric, strictly monotone away from their center, and exponentially localized. At general positive speeds the stability criterion remains conditional on nondegeneracy and the slope condition.
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