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<p>This paper presents a Riemann–Hilbert (RH) approach to the inverse scattering transform, which is employed to solve the initial value problem for a higher-order short pulse (SP) equation on the line with zero boundary conditions. The RH problem is formulated for two types of scattering data: one with <i>N</i> simple poles and the other with <i>N</i> higher-order poles. Using the asymptotic behavior of the spectral variable at <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the solution of the higher-order SP equation is recovered from the solution of the RH problem. In the reflectionless case, we generalize the residue theorem, which allows us to derive multiple higher-order pole solutions for the higher-order SP equation by solving a linear algebraic system. Finally, general formulae are derived for the solutions of the higher-order SP equation, which cover cases of <i>N</i> simple poles and <i>N</i> higher-order poles. As applications, explicit simple-pole and double-pole solutions are displayed in both analytical and graphical forms. Depending on the parameter choices, these correspond to singular-loop solitons and two distinct types of breathers.</p>

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Riemann–Hilbert approach for a higher-order short pulse equation on the line

  • 期刊论文

  • 发表日期：

    2026年3月21日

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  • Cong Lv,
  • Q. P. Liu,
  • Shoufeng Shen

摘要

This paper presents a Riemann–Hilbert (RH) approach to the inverse scattering transform, which is employed to solve the initial value problem for a higher-order short pulse (SP) equation on the line with zero boundary conditions. The RH problem is formulated for two types of scattering data: one with N simple poles and the other with N higher-order poles. Using the asymptotic behavior of the spectral variable at \(k=0\) k = 0 , the solution of the higher-order SP equation is recovered from the solution of the RH problem. In the reflectionless case, we generalize the residue theorem, which allows us to derive multiple higher-order pole solutions for the higher-order SP equation by solving a linear algebraic system. Finally, general formulae are derived for the solutions of the higher-order SP equation, which cover cases of N simple poles and N higher-order poles. As applications, explicit simple-pole and double-pole solutions are displayed in both analytical and graphical forms. Depending on the parameter choices, these correspond to singular-loop solitons and two distinct types of breathers.