A Unified Method for Solving Linear and Nonlinear Evolution Equations and an Application to Integrable Surfaces

Abstract

We call a nonlinear equation integrable iff it admits a Lax pair formulation [1], i.e. iff it can be written as the compatibility condition of two linear equations. One of these equations involves derivatives with respect to space variables only. This equation can be regarded as a linear eigenvalue equation; its spectral analysis gives rise to a nonlinear Fourier transform. The other linear equation defining the Lax pair, simply determines the evolution of the nonlinear Fourier data. The method of using a nonlinear Fourier transform to solve the Cauchy problem for decaying initial data is called the inverse spectral method [2].