Variational principle for Zakharov–Shabat equations in two-dimensions

Abstract

We study the corresponding scattering problem for Zakharov and Shabat compatible differential equations in two-dimensions, the representation for a solution of the nonlinear Schrödinger equation is formulated as a variational problem in two-dimensions. We extend the derivation to the variational principle for the Zakharov and Shabat equations in one-dimension. We also developed an approximate analytical technique for finding discrete eigenvalues of the complex spectral parameters in Zakharov and Shabat equations for a given pulse-shaped potential, which is equivalent to the physically important problem of finding the soliton content of the given initial pulse. Using a trial function in a rectangular box we find the functional integral. The general case for the two box potential can be obtained on the basis of a different ansatz where we approximate the Jost function by polynomials of order n instead of a piecewise linear function. We also demonstrated that the simplest version of the variational approximation, based on trial functions with one, two and n-free parameters respectively, and treated analytically.