Adiabatic invariants of the extended KdV equation

Abstract

When the Euler equations for shallow water are taken to the next order, beyond KdV, momentum and energy are no longer exact invariants. (The only one is mass.) However, adiabatic invariants (AI) can be found. When the KdV expansion parameters are zero, exact invariants are recovered. Existence of adiabatic invariants results from general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. Here we present the direct method of calculations of adiabatic invariants. It does not need a transformation to a moving reference frame nor performing a near-identity transformation. Numerical tests show that deviations of AI from almost constant values are indeed small.

6 pages, 10 figures. Abstract and introduction extended, several new references added. 4 new figures added. Appendix describing an application of near-identity transformation to obtain adiabatic invariants of KdV2 equation included

References in corpus (2)

• Energy invariant for shallow water waves and the Korteweg -- de Vries equation. Is energy always an invariant?
• Symmetries and conservation laws for the Karczewska--Rozmej--Rutkowski--Infeld equation

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• New exact superposition solutions to KdV2 equation
• Martingale solution to stochastic extended Korteweg - de Vries equation
• Remarks on existence/nonexistence of analytic solutions to higher order KdV equations
• Existence of mild solution to stochastic extended Korteweg - de Vries equation
• Signatures of chaotic dynamics in wave motion according to the extended KdV equation