Improper singular points of corank 1 of systems of differential equations not solved for the derivatives
Consider a system of differential equations not solved for the derivatives where \(\mathbf{x}=(x^{1},x^{2},\dots, x^{n})\), \(\mathbf{p}=(p^{1},p^{2},\dots ,p^{n})\) and \(\mathbf{F}=(F^{1},F^{2},\dots, F^{n})\) are \(n\)-dimensional vectors. It is assumed that the function \(\mathbf{F}(t,\mathbf{x},\mathbf{p})\) is jointly \(C^{s+1}\)-smooth, \(s\geq 1\), with respect to the arguments. The tangent space of \(\mathbf{F}( t,\mathbf{x},\mathbf{p}) =0\) is given by the system of linear equations \[ \frac{\partial F^{i}}{\partial t} \,dt+\sum_{j=1}^{n}\frac{\partial F^{i}}{\partial x^{j}} \,dx^{j}+\sum_{j=1}^{n}\frac{\partial F^{i}}{\partial p^{j}} dp^{j}=0, \quad i=1,2,\dots, n. \] By substituting the relations \(dx^{i}=p^{i}dt\) into the previous equation, we obtain the system of linear equations \[ G^{i}dt+\sum_{j=1}^{n}\frac{\partial F^{i}}{\partial p^{j}} \,dp^{j}=0, \quad i=1,2,\dots ,n, \] where \(G^{i}=\frac{\partial F^{i}}{\partial t} +\sum_{j=1}^{n}\frac{\partial F^{i}}{\partial x^{j}}p^{j}\) for the unknowns \(dt,dp^{1},\dots ,dp^{n}\). We denote the matrix of the last system by \(A\). A point \(\mathbf{T}_{0}=(t_{0},\mathbf{x}_{0},\mathbf{p}_{0})\) is said to be admissible for \(\mathbf{F}(t,\mathbf{x},\mathbf{p})=0\) if \(\mathbf{F}(t_{0},\mathbf{x}_{0},\mathbf{p}_{0})=0.\) An admissible point \(\mathbf{T}_{0}=(t_{0},\mathbf{x}_{0},\mathbf{p}_{0})\) is said to be singular if \( \det(\partial F/\partial p)=0\) at that point. A singular point \(\mathbf{T}_{0}\) is said to be improper if \(\text{rank}A<n\). An improper singular point \(\mathbf{T}_{0}\) is referred to as a point of corank \(c>0\) if \(\text{rank}A=n-c\) at that point. The author deals with the analysis of \(\mathbf{F}(t,\mathbf{x},\mathbf{p})=0\) in a neighborhood of improper singular points of corank 1.
scientific article; zbMATH DE number 1916148