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<p>In mathematical physics, the Kadomtsev–Petviashvili I (KPI) equation is a fundamental model for studying (2+1)-dimensional nonlinear wave propagation. The aim of this paper is to apply the Riemann–Hilbert approach to solve the KPI equation. Firstly, we decompose the KPI equation into two soliton equations, which results in three spectral matrices. Secondly, we perform a spectral analysis of these three spectral matrices and derive the Jost solutions, scattering matrix, as well as their analytic and symmetric properties. By using a standard dressing procedure in the Riemann–Hilbert problem, we establish the associated matrix Riemann–Hilbert problem. Then, by solving the Riemann–Hilbert problem for non-regular cases and introducing a special matrix, we obtain the compact <i>N</i>-soliton solution formula expressed in terms of determinants. Finally, as applications of the <i>N</i>-soliton formula, we give the specific one-soliton and two-soliton solutions of the KPI equation under suitable reductions. In addition, combined with the presented graphs, we discuss the influence of spectral parameters on the propagation direction of solitary waves.</p>

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N-soliton solutions for the Kadomtsev–Petviashvili I equation via Riemann–Hilbert approach

  • 期刊论文

  • 发表日期：

    2026年5月20日

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  • Jian Chang,
  • Zhaqilao

摘要

In mathematical physics, the Kadomtsev–Petviashvili I (KPI) equation is a fundamental model for studying (2+1)-dimensional nonlinear wave propagation. The aim of this paper is to apply the Riemann–Hilbert approach to solve the KPI equation. Firstly, we decompose the KPI equation into two soliton equations, which results in three spectral matrices. Secondly, we perform a spectral analysis of these three spectral matrices and derive the Jost solutions, scattering matrix, as well as their analytic and symmetric properties. By using a standard dressing procedure in the Riemann–Hilbert problem, we establish the associated matrix Riemann–Hilbert problem. Then, by solving the Riemann–Hilbert problem for non-regular cases and introducing a special matrix, we obtain the compact N-soliton solution formula expressed in terms of determinants. Finally, as applications of the N-soliton formula, we give the specific one-soliton and two-soliton solutions of the KPI equation under suitable reductions. In addition, combined with the presented graphs, we discuss the influence of spectral parameters on the propagation direction of solitary waves.