反自对偶杨-米尔斯方程拟格兰姆行列式与拟朗斯基行列式\(N\)孤子解之间的渐近等价性
Asymptotic Equivalence Between Quasi-Grammian and Quasi-Wronskian $N$-Soliton Solutions of the Anti-Self-Dual Yang-Mills Equation
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中文总结 AI 辅助
研究反自对偶杨-米尔斯方程\(N\)孤子解，通过建立拟格兰姆行列式与拟朗斯基行列式表示间的渐近等价性，计算WZW₄模型作用密度可视化孤子行为，得到相移因子，两种表示描述同一类解，暗示与高维佐藤理论联系并给出\(N\leq4\)时精确解。
中文摘要 AI 辅助
在反自对偶杨-米尔斯（ASDYM）方程的\(J\)矩阵形式中，建立了拟格兰姆行列式与拟朗斯基行列式表示的\(N\)孤子解之间的渐近等价性（相差一个常数矩阵因子）。此形式即杨方程，是四维韦斯-祖米诺-维滕（WZW₄）模型的运动方程且与ASDYM方程等价。通过对\(G = U(2)\)计算WZW₄模型的作用密度以可视化孤子行为，表明两种表示具有相同渐近孤子轮廓，并明确得到\(N\)孤子碰撞的相移因子。这两种表示描述了同一类ASDYM \(N\)孤子解，可视为流体动力学中KP/KdV型多孤子的四维类似物，暗示了ASDYM方程与高维佐藤理论之间的可能联系。还给出了\(N\leq4\)时的精确拟格兰姆行列式\(N\)孤子解。
英文摘要
Asymptotic equivalence between the quasi-Grammian and quasi-Wronskian representations of $N$-soliton solutions in the $J$-matrix formulation of the anti-self-dual Yang-Mills (ASDYM) equation is established up to a constant matrix factor. This formulation, known as the Yang equation, serves as the equation of motion of the four-dimensional Wess-Zumino-Witten (WZW$_4$) model and is equivalent to the ASDYM equation. To visualize the solitonic behavior, the action density of the WZW$_4$ model is evaluated for $\mathrm{G}=\mathrm{U}(2)$, demonstrating that the quasi-Grammian and quasi-Wronskian representations exhibit the same asymptotic soliton profiles, while the phase shift factors associated with $N$-soliton collisions are obtained explicitly. Hence, by virtue of the particle-like nature of solitons, the two representations describe the same class of ASDYM $N$-soliton solutions. These solitons can be regarded as a four-dimensional analogue of KP/KdV-type multi-solitons in fluid dynamics, suggesting a possible connection between the ASDYM equation and higher-dimensional Sato theory. Exact quasi-Grammian $N$-soliton solutions are also presented for $N\leq4$.
