$N$-soliton strings in four-dimensional space–time

We investigate infinite relativistic strings in the Minkowski space $E_{1,3}$ set theoretically. We show that the set of such strings is uniquely parameterized by elements of the Poincaré group $\mathcal P$, of the group $\mathcal D$ of scaling transformations of Minkowski space, and of a certain subgroup $\mathcal W_0$ of the group of Weyl transformations of the two-metric and also by elements of the set of scattering data for a pair of first-order spectral problems that are characteristic of the theory of the nonlinear Schrödinger equation. The coefficients of the spectral problems are related to the second quadratic forms of the worldsheet. In this context, we define $N$-soliton strings. We discuss a hierarchy of surfaces that occurs in this analysis and corresponds to the known hierarchy associated with the nonlinear Schrödinger equation.