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<p>We derive full asymptotics of the modified KdV equation (mKdV) with a higher-order perturbative term. We extend the perturbative theory of infinite-dimensional integrable systems developed by Deift and Zhou (Acta Math 188(2):163–262, 2002), and some new and simpler proofs of certain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> bounds and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> a priori estimates developed recently in Chen et al. (Lett Math Phys 115(5):94, 2025). We show that the perturbed equation exhibits the same long-time behavior as the completely integrable mKdV. This is the first extension of the methodology of Deift and Zhou (2002) to members of the AKNS hierarchy other than the NLS on the real line.</p>

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Long-Time Asymptotics of A Perturbed Modified KdV Equation

  • 期刊论文

  • 发表日期：

    2026年4月13日

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  • Gong Chen,
  • Jiaqi Liu,
  • Yuanhong Tian

摘要

We derive full asymptotics of the modified KdV equation (mKdV) with a higher-order perturbative term. We extend the perturbative theory of infinite-dimensional integrable systems developed by Deift and Zhou (Acta Math 188(2):163–262, 2002), and some new and simpler proofs of certain \(L^\infty \) L ∞ bounds and \(L^p\) L p a priori estimates developed recently in Chen et al. (Lett Math Phys 115(5):94, 2025). We show that the perturbed equation exhibits the same long-time behavior as the completely integrable mKdV. This is the first extension of the methodology of Deift and Zhou (2002) to members of the AKNS hierarchy other than the NLS on the real line.