Da-jun Zhang ; Shi-min Liu ; Xiao Deng - The solutions of classical and nonlocal nonlinear Schrödinger equations with nonzero backgrounds: Bilinearisation and reduction approach

ocnmp:10036 - Open Communications in Nonlinear Mathematical Physics, February 6, 2023, Volume 3 - https://doi.org/10.46298/ocnmp.10036
The solutions of classical and nonlocal nonlinear Schrödinger equations with nonzero backgrounds: Bilinearisation and reduction approachArticle

Authors: Da-jun Zhang ; Shi-min Liu ; Xiao Deng

    In this paper we develop a bilinearisation-reduction approach to derive solutions to the classical and nonlocal nonlinear Schrödinger (NLS) equations with nonzero backgrounds. We start from the second order Ablowitz-Kaup-Newell-Segur coupled equations as an unreduced system. With a pair of solutions $(q_0,r_0)$ we bilinearize the unreduced system and obtain solutions in terms of quasi double Wronskians. Then we implement reductions by introducing constraints on the column vectors of the Wronskians and finally obtain solutions to the reduced equations, including the classical NLS equation and the nonlocal NLS equations with reverse-space, reverse-time and reverse-space-time, respectively. With a set of plane wave solution $(q_0,r_0)$ as a background solution, we present explicit formulae for these column vectors. As examples, we analyze and illustrate solutions to the focusing NLS equation and the reverse-space nonlocal NLS equation. In particular, we present formulae for the rouge waves of arbitrary order for the focusing NLS equation.


    Volume: Volume 3
    Published on: February 6, 2023
    Accepted on: January 31, 2023
    Submitted on: September 13, 2022
    Keywords: Nonlinear Sciences - Exactly Solvable and Integrable Systems
    Funding:
      Source : OpenAIRE Graph
    • Basalt fibre reinfored HDPE for wave energy converters; Code: 132958
    • Agronomic Big Data Analytics for improved crop management; Funder: UK Research and Innovation; Code: 132850

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