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

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 |