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    Area of Science:

    • Nonlinear dynamics
    • Mathematical physics
    • Computational methods

    Background:

    • The Zakharov-Shabat system is crucial for modeling integrable nonlinear phenomena.
    • Solving the direct scattering problem for the nonlinear Schrödinger equation is computationally challenging.
    • Previous methods limited the analysis of complex wave fields with multiple solitons.

    Purpose of the Study:

    • To develop high-order numerical schemes for the Zakharov-Shabat system.
    • To systematically construct numerical solutions for the direct scattering problem.
    • To enable the analysis of wave fields with a large number of solitons.

    Main Methods:

    • Application of the Magnus expansion to the Zakharov-Shabat system.
    • Development and implementation of second-, fourth-, and sixth-order numerical schemes.
    • Numerical simulations of wave fields with up to 128 solitons.

    Main Results:

    • Successful application of the Magnus expansion for constructing high-order schemes.
    • Demonstration of the capability to simulate complex wave fields with numerous solitons.
    • Identification of the necessity for precise numerical techniques to determine eigenvalues and phase coefficients.

    Conclusions:

    • The developed numerical schemes provide a robust foundation for studying integrable nonlinear systems.
    • This approach facilitates the analysis of large optical wave packets and their scattering data.
    • The findings offer fundamental insights into the origin of nonlinear effects in wave propagation.