Journal
APPLIED MATHEMATICS AND COMPUTATION
Volume 219, Issue 10, Pages 5635-5648Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.amc.2012.10.079
Keywords
Partial differential equations; Calculus of variations; Canonical formalism; Lagrangians and variational principles
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We study the corresponding scattering problem for Zakharov and Shabat compatible differential equations in two-dimensions, the representation for a solution of the nonlinear Schrodinger 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. (C) 2012 Elsevier Inc. All rights reserved.
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