Algebro-Geometric Finite Gap Solutions to the Korteweg--de Vries\n Equation as Primitive Solutions

2019/07/22 by Patrik V. Nabelek, Nabelek, Patrik V.
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Waves and Solitons #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1907.09667

openalex publication_date 2019/07/22 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

In this paper we show that all algebro-geometric finite gap solutions to the\nKorteweg--de Vries equation can be realized as a limit of N-soliton solutions\nas N diverges to infinity (see remark 1 for the precise meaning of this\nstatement). This is done using the the primitive solution framework initiated\nby [5,28,31]. One implication of this result is that the N-soliton solutions\ncan approximate any bounded periodic solution to the Korteweg--de Vries\nequation arbitrarily well in the limit as N diverges to infinity. We also study\nprimitive solutions numerically that have the same spectral properties as the\nalgebro-geometric finite gap solutions but are not algebro-geometric solutions.\n

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