Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

Abstract

A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations of the form pu′′+qu=u∑k=1Nλkrk is obtained. Such equations are known as Sturm-Liouville equations with potentials polynomially dependent on the spectral parameter or Sturm-Liouville pencils (SLP). It allows one to write a general solution of the equation as a power series in terms of the spectral parameter λ. The coefficients of the series are given in terms of recursive integrals involving a particular solution of the equation pu0′′+qu0=0. The convenient form of the solution of SLP provides an efficient numerical method for solving corresponding initial value, boundary value, and spectral problems. A special case of the SLP arises in relation with the Zakharov-Shabat system. We derive a SPPS representation for its general solution and consider other applications as the one-dimensional Dirac system and the equation describing a damped string. Several numerical examples illustrate the efficiency and the accuracy of the numerical method based on the SPPS representations which besides its natural advantages like the simplicity in implementation and accuracy is applicable to the problems admitting complex coefficients, spectral parameter dependent boundary conditions, and complex spectrum.