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<p>In this paper, we introduce a pair of 2+1-dimensional non-isospectral vector fields whose commutation gives rise to a new integrable system which can reduces to the celebrated Pavlov equation. We refer to the new integrable system as the modified Pavlov (mP) system: (a) We study the inverse scattering problem for the multidimensional vector field and Cauchy Problem, and analyze the longtime behaviour of the formal solutions of mP system by introducing a nonlinear Riemann-Hilbert (RH) problem. Meanwhile, a class of particular solutions with arbitrary differential functions is characterized; (b) The complexification of the independent variables <i>x</i>, <i>y</i>, <i>t</i> of the 2+1-dimensional mP system yields the 4+2 dimensions, then reducing the systems from 4+2 to the 3+1 and 3+2 dimensions, and the Lax pairs of the reduced systems are constructed; (c) The Cauchy initial value problems (IVPs) of the high-dimensional mP systems, i.e., 3+1, 3+2 and 4+2 dimensions, are solved by the inverse scattering transform (IST) method; (d) In analogy to 2+1 dimensions, the asymptotics behaviour of the solutions for the 3+1-dimensional mP system is investigated.</p>

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Inverse Scattering and Riemann-Hilbert Problems for a Class of High-Dimensional Modified Pavlov Systems

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • Linlin Gui,
  • Yufeng Zhang

摘要

In this paper, we introduce a pair of 2+1-dimensional non-isospectral vector fields whose commutation gives rise to a new integrable system which can reduces to the celebrated Pavlov equation. We refer to the new integrable system as the modified Pavlov (mP) system: (a) We study the inverse scattering problem for the multidimensional vector field and Cauchy Problem, and analyze the longtime behaviour of the formal solutions of mP system by introducing a nonlinear Riemann-Hilbert (RH) problem. Meanwhile, a class of particular solutions with arbitrary differential functions is characterized; (b) The complexification of the independent variables x, y, t of the 2+1-dimensional mP system yields the 4+2 dimensions, then reducing the systems from 4+2 to the 3+1 and 3+2 dimensions, and the Lax pairs of the reduced systems are constructed; (c) The Cauchy initial value problems (IVPs) of the high-dimensional mP systems, i.e., 3+1, 3+2 and 4+2 dimensions, are solved by the inverse scattering transform (IST) method; (d) In analogy to 2+1 dimensions, the asymptotics behaviour of the solutions for the 3+1-dimensional mP system is investigated.