Phase approach in the method of parameter evolution

New variant of the Method of Parameter Evolution based on using phase variables (PhV-MPE) is suggested. Operator of Evolution for Sturm–Liouville eigenvalue problems is derived. PhV-MPE is generalized for solving non-self-conjugated boundary eigenvalue problems with nonlinear containing spectral parameter in the equation coefficients and boundary conditions. Evolution schemes, which permit to use embedded grids effectively, are found for computing high accuracy solution.