Variational principle for Zakharov-Shabat equations in two-dimensions. (English) Zbl 1284.35400

Helal, M. A.; Seadawy, A. R.; Ibrahim, R. S.

Appl. Math. Comput. 219, No. 10, 5635-5648 (2013).

Summary: 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 160402060 formulae 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.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35A15	Variational methods applied to PDEs
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51	Soliton equations
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

partial differential equations; calculus of variations; canonical formalism; Lagrangians and variational principles