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<p>Classes of moving boundary problems of Stefan-type are here shown to be exactly solvable in terms of classical Airy functions both for the linearised and solitonic Korteweg–de Vries equations. In the latter case, a Miura transformation is applied to a class of Airy-type similarity solutions derived via a Painlevé II reduction of the mKdV equation. Reciprocal transformations are then applied to obtain, in turn, Airy-type solution of associated moving boundary problems for both a nonlinear evolution equation of magma theory and a novel reciprocal Korteweg–de Vries equation which incorporates a source term.</p>

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On Korteweg–de Vries and associated reciprocal moving boundary problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年1月4日

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  • Colin Rogers

摘要

Classes of moving boundary problems of Stefan-type are here shown to be exactly solvable in terms of classical Airy functions both for the linearised and solitonic Korteweg–de Vries equations. In the latter case, a Miura transformation is applied to a class of Airy-type similarity solutions derived via a Painlevé II reduction of the mKdV equation. Reciprocal transformations are then applied to obtain, in turn, Airy-type solution of associated moving boundary problems for both a nonlinear evolution equation of magma theory and a novel reciprocal Korteweg–de Vries equation which incorporates a source term.