We present several families of nonlinear reaction diffusion equations with variable coefficients including generalizations of Fisher-KPP and Burgers type equations. Special exact solutions such as traveling wave, rational, triangular wave and N-wave type solutions are shown. By means of similarity transformations the variable coefficients are conditioned to satisfy Riccati or Ermakov systems of equations. When the Riccati system is used, conditions are established so that finite-time singularities might occur. We explore solution dynamics across multi-parameters. In the suplementary material, we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions.

In this paper, we consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinsky-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability.

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