TY - JOUR

T1 - Numerical direct scattering transform for breathers

AU - Mullyadzhanov, I. I.

AU - Gudko, A. S.

AU - Mullyadzhanov, R. I.

AU - Gelash, A. A.

PY - 2024/2/3

Y1 - 2024/2/3

N2 - We consider the model of the focusing one-dimensional nonlinear Schrödinger equation (fNLSE) in the presence of an unstable constant background, which exhibits coherent solitary wave structures—breathers. Within the inverse scattering transform (IST) method, we study the problem of the scattering data numerical computation for a broad class of breathers localized in space. Such a direct scattering transform (DST) procedure requires a numerical solution of the auxiliary Zakharov–Shabat system with boundary conditions corresponding to the background. To find the solution, we compute the transfer matrix using the second-order Boffetta–Osborne approach and recently developed high-order numerical schemes based on the Magnus expansion. To recover the scattering data of breathers, we derive analytical relations between the scattering coefficients and the transfer matrix elements. Then we construct localized single- and multi-breather solutions and verify the developed numerical approach by recovering the complete set of scattering data with the built-in accuracy providing the information about the amplitude, velocity, phase and position of each breather. To combine the conventional IST approach with the efficient dressing method for multi-breather solutions, we derive the exact relation between the parameters of breathers in these two frameworks.

AB - We consider the model of the focusing one-dimensional nonlinear Schrödinger equation (fNLSE) in the presence of an unstable constant background, which exhibits coherent solitary wave structures—breathers. Within the inverse scattering transform (IST) method, we study the problem of the scattering data numerical computation for a broad class of breathers localized in space. Such a direct scattering transform (DST) procedure requires a numerical solution of the auxiliary Zakharov–Shabat system with boundary conditions corresponding to the background. To find the solution, we compute the transfer matrix using the second-order Boffetta–Osborne approach and recently developed high-order numerical schemes based on the Magnus expansion. To recover the scattering data of breathers, we derive analytical relations between the scattering coefficients and the transfer matrix elements. Then we construct localized single- and multi-breather solutions and verify the developed numerical approach by recovering the complete set of scattering data with the built-in accuracy providing the information about the amplitude, velocity, phase and position of each breather. To combine the conventional IST approach with the efficient dressing method for multi-breather solutions, we derive the exact relation between the parameters of breathers in these two frameworks.

KW - breathers

KW - modulation instability

KW - solitons

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85184007324&origin=inward&txGid=7194ccb359e694390ce915e16f49c003

UR - https://www.webofscience.com/wos/woscc/full-record/WOS:001146884300003

UR - https://www.mendeley.com/catalogue/fa8d897d-19bc-3779-9698-8aa09a184b58/

U2 - 10.1098/rspa.2023.0529

DO - 10.1098/rspa.2023.0529

M3 - Article

VL - 480

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 1364-5021

IS - 2282

M1 - 20230529

ER -