Periodic Waves and Monotonicity of Wave Velocity in the pgKdV Equation with Two Arbitrarily High-Order Nonlinearities

• Tianrui An ^[1] ; Yanfei Dai ^[1]
  1. [1] Zhejiang Normal University

     Zhejiang Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper investigates periodic traveling waves in a class of perturbed generalized Korteweg–de Vries (pgKdV) equations with two higher-order nonlinear terms, (um)t + (un)χ + uχχχ + εuχχ + uχχχχ = 0, where m and n are arbitrary positive integers satisfying n > m. In contrast to previous studies restricted to particular relations between the exponents, we establish the existence of periodic traveling waves under this general condition. By combining geometric singular perturbation theory, bifurcation methods, Abelian integral analysis, the criterion function method and symbolic computation, we establish the existence of periodic traveling waves for sufficiently small perturbations.We also show that the wave velocity is strictly increasing on every compact subinterval of the energy interval, provided that the perturbation parameter is sufficiently small. The explicit upper and lower bounds for the limiting wave velocity are obtained. When both m and n are odd, we further prove the existence of at least two periodic traveling wave solutions.

    These results extend several previous studies and provide a partial answer to an open problem posed in [1]. Numerical simulations are presented to illustrate the theoretical results.

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