正则化 sine–Gordon 方程中行波的谱稳定性与快慢结构
Spectral stability and slow--fast structure of traveling waves in a regularized sine--Gordon equation
AI总结：
本文研究含两种四阶正则化机制的耗散 sine–Gordon 方程行波解的谱稳定性，推导 Melnikov 型速度选择条件，结合 Evans 函数技术证明扭结与反扭结解无不稳定特征值，相关结果经数值模拟验证。
AI中文摘要：
我们研究含两种不同四阶正则化机制（混合时空惯性项与纯空间椭圆项）的耗散 sine–Gordon 方程中扭结与反扭结行波解的动力学及谱稳定性。该模型包含阻尼、偏置驱动与高阶耗散效应，其研究动机源于长约瑟夫森结中 fluxon 动力学的精细描述。利用集体坐标约化，我们推导了用于速度选择的 Melnikov 型条件，得到渐近传播速度的显式预测，并通过对完整偏微分方程的直接数值模拟验证了该预测。采用适配正则化诱导的奇异快慢结构的 Evans 函数技术分析谱稳定性：通过在一致分裂域上构建线性化问题，证明无额外点谱从原点分岔；在本质谱边缘附近，采用平方根变换解决分支奇异性并建立提升谱变量的解析性。在 λ=0 及本质谱边缘附近的数值 Evans 函数计算证实了分析结果，表明扭结与反扭结解均不存在不稳定特征值。
英文摘要：
We investigate the dynamics and spectral stability of traveling kink and antikink solutions in a dissipative sine--Gordon equation with two distinct fourth--order regularization mechanisms: a mixed space--time (inertial) term and a purely spatial (elliptic) term. The model includes damping, bias forcing, and higher--order dissipative effects, and is motivated by refined descriptions of fluxon dynamics in long Josephson junctions. Using a collective--coordinate reduction, we derive a Melnikov--type condition for speed selection, yielding explicit predictions for asymptotic propagation speeds, which are validated by direct numerical simulations of the full partial differential equation. Spectral stability is analyzed using Evans function techniques adapted to the singular slow--fast structure induced by the regularization. By formulating the linearized problem on a consistent--splitting domain, we show that no additional point spectrum bifurcates from the origin. Near the edges of the essential spectrum, a square--root transformation is used to resolve branch singularities and establish analyticity in a lifted spectral variable. Numerical Evans function computations near $λ=0$ and near the essential spectrum edges confirm the analytical results, indicating absence of unstable eigenvalues for both kink and antikink solutions.
