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<p>We propose a general approach to quasi-deform the Korteweg–De Vries (KdV) equation by deforming its Hamiltonian. The standard abelianization process based on the inherent <i>sl</i>(2) loop algebra leads to an infinite number of anomalous conservation laws, that yield conserved charges for definite space-time parity of the solution. Judicious choice of the deformed Hamiltonian yields an integrable system with scaled parameters as well as a hierarchy of deformed systems, some of which possibly are quasi-integrable. One such system maps to the known quasi-deformed nonlinear Schrödinger (NLS) soliton in the already known weak-coupling limit, whereas a generic scaling of the KdV amplitude <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41598_2025_86381_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(u\rightarrow u^{1+\varepsilon }\)</EquationSource> </InlineEquation> also suggests quasi-integrability under an order-by-order expansion. In general, these deformed KdV solutions need to be parity-even for quasi-conservation that agrees with our analytical results. Following the recent demonstration of quasi-integrability in regularized long wave (RLW) and modified regularized long wave (mRLW) systems by ter Braak et al. (Nucl Phys B 939:49–94, 2019), that are particular cases of the present approach, general soliton solutions should numerically be accessible.</p>

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On quasi-integrable deformation scheme of the KdV system

  • 期刊论文

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  • 发表日期：

    2025年1月18日

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  • Kumar Abhinav,
  • Partha Guha

摘要

We propose a general approach to quasi-deform the Korteweg–De Vries (KdV) equation by deforming its Hamiltonian. The standard abelianization process based on the inherent sl(2) loop algebra leads to an infinite number of anomalous conservation laws, that yield conserved charges for definite space-time parity of the solution. Judicious choice of the deformed Hamiltonian yields an integrable system with scaled parameters as well as a hierarchy of deformed systems, some of which possibly are quasi-integrable. One such system maps to the known quasi-deformed nonlinear Schrödinger (NLS) soliton in the already known weak-coupling limit, whereas a generic scaling of the KdV amplitude \(u\rightarrow u^{1+\varepsilon }\) also suggests quasi-integrability under an order-by-order expansion. In general, these deformed KdV solutions need to be parity-even for quasi-conservation that agrees with our analytical results. Following the recent demonstration of quasi-integrability in regularized long wave (RLW) and modified regularized long wave (mRLW) systems by ter Braak et al. (Nucl Phys B 939:49–94, 2019), that are particular cases of the present approach, general soliton solutions should numerically be accessible.