Small-Time Existence and Two Classes of Solutions for the n-Dimensional Coupled Yukawa Equations

We discuss the Yukawa equations, a system of nonlinear partial differential equations which has applications to meson-nucleon interactions. First, we determine small-time local existence of solutions under fairly general initial data. In particular, we show that small-time solutions are analytic, and by the higher order Cauchy–Kowalevski theorem, unique. Secondly, we obtain a class of stationary solutions. For the 1 + 1 model, we find that space-periodic solutions may be obtained through an application of multiple scales analysis. In this case, the coupled Yukawa equations result in a sort of non-local Gross-Pitaevskii equation. We outline the method for stationary solutions to the n + 1 problem as well. Finally, we consider a separate class of solutions, namely travelling waves. The wave solutions we obtain here are distinct from those discussed previously in the literature. For these solutions, we are able to determine the asymptotic behavior of the solutions.