Total masses of solutions to general Toda systems with singular sources (Q6540263)

The authors consider the following Toda system on \(\mathbb{R}^2\)\N\[\N\left\{\begin{array}{ll} \Delta u_i+4\sum_{j=1}^na_{ij}e^{u_j}=4\pi \gamma_i\delta_0, \ \ &\gamma_i>-1,\\\N\displaystyle{\int_{\mathbb{R}^2}e^{u_i}dx}<\infty, &1\leq i\leq n, \end{array}\right.\tag{1}\N\]\Nwhere \((a_{ij})\) is the Cartan matrix of rank \(n\) associated to a complex simple Lie algebra \(\mathfrak{g}\), and \(\delta_0\) is the Dirac measure at \(0\). A solution of problem \((1)\) is a function \(u=(u_1,\dots,u_n)\in C^2(\mathbb{R}^2\setminus \{0\},\mathbb{R}^n)\) satisfying the equation \(\Delta u_i+4\sum_{j=1}^na_{ij}e^{u_j}=0\) in \(\mathbb{R}^2\setminus \{0\}\), and \(u_i(x)=2\gamma_i\log|x|+O(1)\) near \(0\), for \(i=1,\dots,n\).\N\NThe main result of the paper concerns the asymptotic behavior of the solutions. The authors prove that, if \(u=(u_1,\dots,u_n)\) is a solution to system \((1)\), then, setting\N\[\NU_i=\sum_{j=1}^na^{ij}u_j, \ \ \ \text{and} \ \ \ \gamma^i=\sum_{j=1}^na^{ij}\gamma_j, \ \ \ i=1,\dots,n,\N\]\Nwhere \((a^{ij})\) is the inverse of \((a_{ij})\), one has\N\N\[\NU_i(z)=2(\gamma^i-\langle\omega_i-\kappa \omega_i,\omega_0\rangle)\log|z|+O(1) \ \ \ \text{as} \ \ z\rightarrow \infty,\N\]\Nand\N\[\N\sigma_i(u)=2\langle\omega_i-\kappa \omega_i,\omega_0\rangle.\N\]\NHere,\N\[\N\sigma_i(u)=\frac{4}{2\pi}\int_{\mathbb{R}^2}e^{u_i}dx, \ \ \ i=1,\dots,n,\N\]\Nis the total mass of \(u_i\), \(\kappa\) is the longest element of the Weyl group of the Lie algebra \(\mathfrak{g}\), \(\langle \cdot,\cdot \rangle\) is the pairing between the real Cartan subalgebra \(\mathfrak{h}_0\) and its dual \(\mathfrak{h}_0'\), \(\omega_i\in \mathfrak{h}_0'\) is the \(i\)-th fundamental weight, and \(\omega_0\in \mathfrak{h}_0\). The total mass \(\sigma_i(u)\) turns out to be an even integer if \(\gamma_i\in \mathbb{Z}_{\geq 0}\).\N\NThe above theorem generalizes previous results obtained in the case of Toda systems of types \(A\), \(G_2\) and \(B\), \(C\).