Physical vs mathematical origin of the extended KdV and mKdV equations

Saleh Baqer; Theodoros P. Horikis; Dimitrios J. Frantzeskakis; Saleh Baqer; Theodoros P. Horikis; Dimitrios J. Frantzeskakis

doi:10.3934/math.2025427

AIMS Mathematics

2025, Volume 10, Issue 4: 9295-9309. doi: 10.3934/math.2025427

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Physical vs mathematical origin of the extended KdV and mKdV equations

1.
Department of Mathematics, Kuwait University, Kuwait City 13060
2.
Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
3.
Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece

Received: 13 February 2025 Revised: 02 April 2025 Accepted: 16 April 2025 Published: 23 April 2025
MSC : 35B20, 35B40, 35C08, 35Q51, 35Q53, 35Q55, 76X05

The higher-order Korteweg-de Vries (KdV) and modified KdV (mKdV) equations are derived from a physical model describing a three-component plasma composed of cold fluid ions and two species of Boltzmann electrons at different temperatures. While the higher-order KdV equation is well established, the corresponding mKdV equation is typically derived using the system's integrability properties. In this work, we present the extended mKdV equation, derived directly from the physical system, offering a fundamentally different form from its integrable counterpart. We explore the connections between the two equations via Miura transformations and analyze their solutions within the framework of asymptotic integrability.
Citation: Saleh Baqer, Theodoros P. Horikis, Dimitrios J. Frantzeskakis. Physical vs mathematical origin of the extended KdV and mKdV equations[J]. AIMS Mathematics, 2025, 10(4): 9295-9309. doi: 10.3934/math.2025427

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Abstract

The higher-order Korteweg-de Vries (KdV) and modified KdV (mKdV) equations are derived from a physical model describing a three-component plasma composed of cold fluid ions and two species of Boltzmann electrons at different temperatures. While the higher-order KdV equation is well established, the corresponding mKdV equation is typically derived using the system's integrability properties. In this work, we present the extended mKdV equation, derived directly from the physical system, offering a fundamentally different form from its integrable counterpart. We explore the connections between the two equations via Miura transformations and analyze their solutions within the framework of asymptotic integrability.

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