TY - JOUR

T1 - Magnus Expansion for the Direct Scattering Transform

T2 - High-Order Schemes

AU - Mullyadzhanov, R. I.

AU - Gelash, A. A.

N1 - Publisher Copyright:
© 2021, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/2

Y1 - 2021/2

N2 - We describe in detail the construction of numerical fourth- and sixth-order schemes using the Magnus expansion for solving the Zakharov–Shabat system, which makes it possible to solve accurately the direct scattering problem for the nonlinear Schrödinger equation. To avoid numerical instabilities inherent in the procedure of solving the direct scattering problem, high-precision arithmetic is used. At present, application of the proposed schemes in combination with the highprecision arithmetic is a unique tool that can be used for analysis of complex wave fields, which contain a great number of solitons, and allows one to determine the complete discrete spectrum, including both eigenvalues and normalization constants. In this work, we study the errors in the proposed scheme using an example of the potential in the form of a hyperbolic secant. It is found that the time of calculation of the scattering matrix using the sixth-order algorithm is almost twice as long compared with that for the standard second-order Boffetta–Osborne algorithm, whereas the time gain resulting from the reduction of number of the wave-field discretization points with retaining the desired precision can reach an order of magnitude or more. Exact solution of the direct scattering problem requires that the discretization increment of high-amplitude wave fields should be comparable with the characteristic width of the largest solitons contained in such fields. In this case, the discretization increment can be sufficiently smaller than that required for reconstruction of the full Fourier spectrum of the wave field. Application of the proposed high-order approximation schemes can be of fundamental importance for successful operation with a great number of complex nonlinear wave fields, such as, e.g., that in the process of the statistical study of the scattering data.

AB - We describe in detail the construction of numerical fourth- and sixth-order schemes using the Magnus expansion for solving the Zakharov–Shabat system, which makes it possible to solve accurately the direct scattering problem for the nonlinear Schrödinger equation. To avoid numerical instabilities inherent in the procedure of solving the direct scattering problem, high-precision arithmetic is used. At present, application of the proposed schemes in combination with the highprecision arithmetic is a unique tool that can be used for analysis of complex wave fields, which contain a great number of solitons, and allows one to determine the complete discrete spectrum, including both eigenvalues and normalization constants. In this work, we study the errors in the proposed scheme using an example of the potential in the form of a hyperbolic secant. It is found that the time of calculation of the scattering matrix using the sixth-order algorithm is almost twice as long compared with that for the standard second-order Boffetta–Osborne algorithm, whereas the time gain resulting from the reduction of number of the wave-field discretization points with retaining the desired precision can reach an order of magnitude or more. Exact solution of the direct scattering problem requires that the discretization increment of high-amplitude wave fields should be comparable with the characteristic width of the largest solitons contained in such fields. In this case, the discretization increment can be sufficiently smaller than that required for reconstruction of the full Fourier spectrum of the wave field. Application of the proposed high-order approximation schemes can be of fundamental importance for successful operation with a great number of complex nonlinear wave fields, such as, e.g., that in the process of the statistical study of the scattering data.

UR - http://www.scopus.com/inward/record.url?scp=85111700508&partnerID=8YFLogxK

U2 - 10.1007/s11141-021-10096-6

DO - 10.1007/s11141-021-10096-6

M3 - Article

AN - SCOPUS:85111700508

VL - 63

SP - 786

EP - 803

JO - Radiophysics and Quantum Electronics

JF - Radiophysics and Quantum Electronics

SN - 0033-8443

IS - 9-10

ER -