Darboux system: Liouville reduction and an explicit solution (Q2280445)

Problem of constructing orthogonal curvilinear coordinate systems in \(\mathbb R^3\), where it is reduced to six equations for the Christoffel symbols corresponding to a diagonal metric \[ \frac{\partial}{\partial t_i}\Gamma_{jk}= \frac{\partial}{\partial t_k}\Gamma_{ji}=\Gamma_{jk}\Gamma_{ki}+\Gamma_{ji}\Gamma_{ik}-\Gamma_{jk}\Gamma_{ji},\tag{1} \] where \(t=(t_1,t_2,t_3)\), \(\{i,j,k\}\) is a permutation of \(\{1,2,3\}\) and, since \(\Gamma^i_{jk}=0\) for all \(i\ne j\ne k\ne i\), then \(\Gamma^i_{ij}\equiv \Gamma_{ij}\) is considered. Theorem 1 (Section 3). The general solution to the Darboux system (1) under factorization condition (Section 2) \(\Gamma_{jk, t_j}= \Gamma_{kj, t_k} = \Gamma_{jk} \Gamma_{kj}\), where \(\{i, j, k\}=\text{ perm}\{1, 2, 3\}\) is given by the equalities \(\Gamma_{ij}(t) = -\partial_{t_j}\log c_k(t)\), \(\Gamma_{ji}(t) = -\partial_{t_i}\log c_k(t)\), where the functions \(c_i(t)\) are parameterized by six functions of one variable \(a_{ij}(t_j)\), \(i\ne j\), \(i, j=1,2,3\), via the equalities \[ c_1(t) =\frac{a_{12}}{G^{(3)}}(a_{31} + a_{32}-1) + \frac{a_{13}}{G^{(2)}}(a_{21} + a_{23} - 1)-1, \] \[ c_2(t) =\frac{a_{23}}{G^{(1)}}(a_{12} + a_{13}- 1) +\frac{a_{21}}{G^{(3)}}(a_{32} + a_{31} - 1)-1, \] \[ c_3(t) =\frac{a_{31}}{G^{(2)}}(a_{23} + a_{21} - 1)+\frac{a_{32}}{G^{(1)}}(a_{13} + a_{12} - 1)-1, \] where \(G^{(i)}(t) = a_{ik}a_{kj} + a_{ij}a_{jk} - a_{jk}a_{kj}\), \(\{i, j, k\}= \text{perm}\{1, 2, 3\}\), and the functions \(a_{ij}(t_j)\) satisfy the condition \(a_{ij}(0)=1\). Let functions \(v^{(i)}(t)\), \(i=1,2,3\), be related to the Christoffel symbols by the equalities \(\Gamma_{ij}(t)=v^{(i)}_{t_j}(t)/(v^{(j)}(t)-v^{(i)}(t))\), \(i\ne j\), where \(v^{(i)}_{t_j}(t)\equiv\partial v^{(i)}(t)/\partial t_j\). If we make this substitution to the Darboux system (1), we obtain three second-order equations for the functions \(v^{(i)}(t)\). In Theorem 2 of Section 4, the general solution of these three second-order equations under the reduction \((v^{(i)}-v^{(i)}) v^{(i)}_{t_it_j}-v^{(i)}_{t_i} v^{(i)}_{t_j}=0\), \(i\ne j\) is defined by functions of one variable. Explicit formulas for the Lamé coefficients and solutions to the associated linear problem are also constructed in Section 5. In Section 6, it is shown that the previously known reduction to a weakly nonlinear system is a particular case of the approach proposed.