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<p>By introducing a change of variables, Grimshaw et al. (Stud.Appl.Math.129:414-436, 2012) and Hakkaev et al. (Stud.Appl.Math.139:405-433, 2017) successfully transformed the short pulse (SP) model into a standard Schrödinger equation, based on which the properties of the periodic travelling wave solutions of SP equation were investigated. However, the transformation is no longer applicable to the modified short pulse (mSP) equation. In this paper, we refine the previously introduced transformation and propose two novel changes of variables, thereby establishing a direct connection between the mSP equation and the simple harmonic motion equation. Using these transformations, we explicitly construct the periodic travelling wave solutions of the mSP equation for both the focusing and defocusing cases. Furthermore, by employing the bifurcation theory of dynamical systems, we analyze the phase portraits of the traveling wave system corresponding to the mSP equation.</p>

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Periodic Travelling Wave Solutions of the Modified Short Pulse Equation

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月5日

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  • Yujing Chen,
  • Aiyong Chen,
  • Xiaokai He

摘要

By introducing a change of variables, Grimshaw et al. (Stud.Appl.Math.129:414-436, 2012) and Hakkaev et al. (Stud.Appl.Math.139:405-433, 2017) successfully transformed the short pulse (SP) model into a standard Schrödinger equation, based on which the properties of the periodic travelling wave solutions of SP equation were investigated. However, the transformation is no longer applicable to the modified short pulse (mSP) equation. In this paper, we refine the previously introduced transformation and propose two novel changes of variables, thereby establishing a direct connection between the mSP equation and the simple harmonic motion equation. Using these transformations, we explicitly construct the periodic travelling wave solutions of the mSP equation for both the focusing and defocusing cases. Furthermore, by employing the bifurcation theory of dynamical systems, we analyze the phase portraits of the traveling wave system corresponding to the mSP equation.