In this paper, we study the blow-up analysis of an affine Toda system corresponding to minimal surfaces into ${\mathbb{S}}^4$. This system is an integrable system which is a natural generalization of sinh-Gordon equation. By exploring a refined blow-up analysis in the bubble domain, we prove that the blow-up values are multiple of 8π, which generalizes the previous results proved in [39], [37], [27], [22] for the sinh-Gordon equation. More precisely, let $(u_k^1,u_k^2,u_k^3)$ be a sequence of solutions of$-\mathrm{\Delta }{u}^1=e^{u^1}-e^{u^3},-\mathrm{\Delta }{u}^2=e^{u^2}-e^{u^3},-\mathrm{\Delta }{u}^3=-\frac{1}{2}{e}^{u^1}-\frac{1}{2}{e}^{u^2}+e^{u^3},u^1+u^2+2u^3=0,$ in $B_1(0)$, which has a uniformly bounded energy in $B_1(0)$, a uniformly bounded oscillation on $\partial B_1(0)$ and blows up at an isolated blow-up point {0}, then the local masses $({\sigma }_1,{\sigma }_2,{\sigma }_3)\ne 0$ satisfy$$\begin{gathered} \begin{array}{ccc}\hfill {\sigma }_1& \hfill =\hfill & m_1(m_1+3)+m_2(m_2-1)\hfill \\ \hfill {\sigma }_2& \hfill =\hfill & m_1(m_1-1)+m_2(m_2+3)\hfill \\ \hfill {\sigma }_3& \hfill =\hfill & m_1(m_1-1)+m_2(m_2-1)\hfill \end{array} \\ \text{for some}\begin{array}{c}(m_1,m_2)\in \mathbb{Z}\text{ either}\hfill \\ m_1,m_2=0,1\text{ mod }4,\text{ or }\hfill \\ m_1,m_2=2,3\text{ mod }4.\hfill \end{array} \end{gathered}$$ Here ${\sigma }_i:=\frac{1}{2\pi }{\lim }_{\delta \to 0}{\lim }_{k\to \infty }{\int }_{B_{\delta }(0)}{e}^{u_k^i}dx$.

在本文中，我们研究了对应于最小曲面的仿射 Toda 系统的爆破分析 ${\mathbb{秒}}^4$. 该系统是可积系统，是sinh-Gordon方程的自然推广。通过探索气泡域中精细的爆破分析，我们证明了爆破值是 8 π 的倍数，这概括了之前在 [39]、[37]、[27]、[22] 中证明的结果：辛格-戈登方程。更准确地说，让$({你}_{克}^1,{你}_{克}^2,{你}_{克}^3)$ 是一系列的解决方案$-\mathrm{\Delta }{你}^1={\mathrm{电子}}^{{你}^1}-{\mathrm{电子}}^{{你}^3},-\mathrm{\Delta }{你}^2={\mathrm{电子}}^{{你}^2}-{\mathrm{电子}}^{{你}^3},-\mathrm{\Delta }{你}^3=-\frac{1}{2}{\mathrm{电子}}^{{你}^1}-\frac{1}{2}{\mathrm{电子}}^{{你}^2}+{\mathrm{电子}}^{{你}^3},{你}^1+{你}^2+2{你}^3=0,$ 在 ${乙}_1(0)$，其具有均匀有界的能量 ${乙}_1(0)$，均匀有界振荡 $\partial {乙}_1(0)$ 并在一个孤立的爆炸点 {0} 爆炸，然后当地群众 $({\sigma }_1,{\sigma }_2,{\sigma }_3)\ne 0$ 满足$$\begin{gathered} \begin{array}{ccc}\hfill {\sigma }_1& \hfill =\hfill & {米}_1({米}_1+3)+{米}_2({米}_2-1)\hfill \\ \hfill {\sigma }_2& \hfill =\hfill & {米}_1({米}_1-1)+{米}_2({米}_2+3)\hfill \\ \hfill {\sigma }_3& \hfill =\hfill & {米}_1({米}_1-1)+{米}_2({米}_2-1)\hfill \end{array} \\ \text{对于一些}\begin{array}{c}({米}_1,{米}_2)\in \mathbb{Z}\text{ 任何一个}\hfill \\ {米}_1,{米}_2=0,1\text{ 模组 }4,\text{ 要么 }\hfill \\ {米}_1,{米}_2=2,3\text{ 模组 }4.\hfill \end{array} \end{gathered}$$ 这里 ${\sigma }_{\mathrm{一世}}：=\frac{1}{2\pi }{\mathrm{林}}_{\delta \to 0}{\mathrm{林}}_{克\to \infty }{\int }_{{乙}_{\delta }(0)}{\mathrm{电子}}^{{你}_{克}^{\mathrm{一世}}}dX$.