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<p>A family of nonsingular rational solutions of the Kadomtsev–Petviashvili (KP) I equation is investigated. These solutions have multiple peaks whose heights are time-dependent and the peak trajectories in the <i>xy</i>-plane are altered after collision. The anomalous scattering occurs due to a non-trivial internal dynamics among the peaks in a slow time scale. This phenomenon is explained by relating the peak locations to the roots of complex heat polynomials. It follows from the long-time asymptotics of the solutions that the peak trajectories separate as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42286_2025_119_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(\sqrt{|t|})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mo stretchy="false">|</mo> <mi>t</mi> <mo stretchy="false">|</mo> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42286_2025_119_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">\(|t| \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>t</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. The formation of 1-lump solution from a broad range of initial conditions is investigated numerically. The numerical results show that the 1-lump solution is stable under both weak and strong perturbations.</p>

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Analytical and Numerical Studies of KPI Lumps

  • 期刊论文

  • 发表日期：

    2025年5月13日

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  • Sarbarish Chakravarty,
  • Michael Zowada

摘要

A family of nonsingular rational solutions of the Kadomtsev–Petviashvili (KP) I equation is investigated. These solutions have multiple peaks whose heights are time-dependent and the peak trajectories in the xy-plane are altered after collision. The anomalous scattering occurs due to a non-trivial internal dynamics among the peaks in a slow time scale. This phenomenon is explained by relating the peak locations to the roots of complex heat polynomials. It follows from the long-time asymptotics of the solutions that the peak trajectories separate as \(O(\sqrt{|t|})\) O ( | t | ) as \(|t| \rightarrow \infty \) | t | → ∞ . The formation of 1-lump solution from a broad range of initial conditions is investigated numerically. The numerical results show that the 1-lump solution is stable under both weak and strong perturbations.