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Adiabatic Invariants of Second Order Korteweg-de Vries Type Equation

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Nonlinear Systems, Vol. 1

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In this chapter we analyze the existence and forms of invariants of the extended Korteweg-de Vries equation (KdV2). This equation appears when the Euler equations for shallow water are extended to the second order, beyond Korteweg-de Vries (KdV). We show that contrary to KdV for which there is an infinite number of invariants, for KdV2 there exists only one, connected to mass (volume) conservation of the fluid. For KdV2 we found only so-called adiabatic invariants, that is, functions of the solutions which are constants neglecting terms of higher order than the order of the equation. In this chapter we present two methods for construction of such invariants. The first method, a direct one, consists in using constructions of higher KdV invariants and eliminating non-integrable terms in an approximate way. The second method introduces a near-identity transformation (NIT) which transforms KdV2 into equation (asymptotically equivalent) which is integrable. For the equation obtained by NIT, exact invariants exist, but they become approximate (adiabatic) when the inverse NIT transformation is applied and original variables are restored. Numerical tests of the exactness of adiabatic invariants for KdV2 in several cases of initial conditions are presented. These tests confirm that relative changes in these approximate invariants are small indeed. The relations of KdV invariants and KdV2 adiabatic invariants to conservation laws are discussed, as well.

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Acknowledgements

The authors thank Prof. Eryk Infeld and Prof. George Rowlands for inspiring discussions.

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Authors and Affiliations

  1. Faculty of Physics and Astronomy, Institute of Physics, University of Zielona Góra, Szafrana 4a, 65-246, Zielona Góra, Poland

    Piotr Rozmej

  2. Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-246, Zielona Góra, Poland

    Anna Karczewska

Authors
  1. Piotr Rozmej
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  2. Anna Karczewska
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Corresponding author

Correspondence to Piotr Rozmej .

Editor information

Editors and Affiliations

  1. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Victoriano Carmona

  2. Departamento de Física Aplicada I, Universidad de Sevilla, Sevilla, Spain

    Jesús Cuevas-Maraver

  3. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Fernando Fernández-Sánchez

  4. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Elisabeth García- Medina

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Rozmej, P., Karczewska, A. (2018). Adiabatic Invariants of Second Order Korteweg-de Vries Type Equation. In: Carmona, V., Cuevas-Maraver, J., Fernández-Sánchez, F., García- Medina, E. (eds) Nonlinear Systems, Vol. 1. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-66766-9_6

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