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The Super N-Periodic Wave Solutions and Their Dynamical Behaviors in the N = 1 Supersymmetric KdV-Type Equation

  • Xia Li [1] ; Zhonglong Zhao [1] ; Zhaohua Li [1] ; Xianzhong Yao [2]
    1. [1] North University of China

      North University of China

      China

    2. [2] Shanxi University of Finance and Economics

      Shanxi University of Finance and Economics

      China

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 6, 2025
  • Idioma: inglés
  • Enlaces
  • Resumen

    • The quasi-periodic wave solutions and dynamical behaviors in the N = 1 supersymmetric KdV-type equation are investigated by employing super Riemann-theta functions and Hirota’s bilinear method. In particular, the 3-periodic, 4-periodic and N-periodic wave solutions are constructed, along with their asymptotic behaviors.

      For N ≥ 3, the determination of quasi-periodic wave solutions is reformulated as an over-determined system of equations due to fewer unknown parameters than constraint equations. We formulate the over-determined system as a least squares optimization problem that can be solved via the Gauss-Newton method. Therefore, the numerical quasi-periodic wave solutions are obtained by reformulating the constraint conditions.

      At the same time, the asymptotic properties between the super N-periodic wave solutions and the super N-soliton solutions are presented. Furthermore, because of the presence of the Grassmann variable, the quasi-periodic wave solutions emerge an “influencing band" in the central region and the bandwidth for the “influencing band" becomes bigger with the growth of N. The dynamic behaviors including peak locations and trough locations, inter-peak distances and periodicity for the quasi-periodic waves are displayed systematically. The method presented in this work can be further generalized to other supersymmetric KdV-type equations to explore more complex structures of nonlinear waves.

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