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<p>This work aims to investigate the asymptotic behavior analysis of solutions to the Cauchy problem of new coupled integrable dispersionless equations. Utilizing the gauge transformations, spectral analysis and inverse scattering method, we show that the solutions of new coupled integrable dispersionless equations can be expressed in terms of the solutions of two matrix Riemann–Hilbert problems formulated in the complex <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11040_2025_9507_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-plane. Applying the nonlinear steepest descent method to the two associated matrix-valued Riemann–Hilbert problems, we obtain precise leading-order asymptotic formulas and uniform error estimates for the solutions of new coupled integrable dispersionless equations.</p>

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Gauge Transformations and Long-Time Asymptotics for the New Coupled Integrable Dispersionless Equations

  • 期刊论文

  • 发表日期：

    2025年5月2日

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  • Xumeng Zhou,
  • Xianguo Geng,
  • Minxin Jia,
  • Yunyun Zhai

摘要

This work aims to investigate the asymptotic behavior analysis of solutions to the Cauchy problem of new coupled integrable dispersionless equations. Utilizing the gauge transformations, spectral analysis and inverse scattering method, we show that the solutions of new coupled integrable dispersionless equations can be expressed in terms of the solutions of two matrix Riemann–Hilbert problems formulated in the complex \(\lambda \) λ -plane. Applying the nonlinear steepest descent method to the two associated matrix-valued Riemann–Hilbert problems, we obtain precise leading-order asymptotic formulas and uniform error estimates for the solutions of new coupled integrable dispersionless equations.