On the behavior of some explicit solutions of the harmonic maps equation
Let \(M\) be a \(n + 1\)-dimensional Minkowski space \(M(n,1)\), and \(N\) be an \(m\)-dimensional Riemannian manifold \(N\). The authors consider harmonic maps \(u : M \to N\) of a special type, called null-solutions, which satisfy the following equation: \[ {\partial^2 u^i\over \partial t^2} - \sum^n_{p = 1} {\partial^2 u^i \over \partial x^{p^2}} = {\partial u^j\over \partial t} \;{\partial u^k \over \partial t} - \sum^n_{p = 1} {\partial u^j \over \partial x^p} {\partial u^k \over \partial x^p} = 0 \] and are of the type \(u(t,x) = f(x + t) + g(x - t)\) for some positive functions \(f\) and \(g\) with finite supports and finite number of critical points. For such solutions they find some representations. For example, if \(n = 1\) for such a solution \(u^i = F^i(f^1, g^1)\), \(1 \leq i \leq n\), where \(F\) is some function of two variables.
scientific article; zbMATH DE number 796918