Adiabatic invariants of the extended KdV equation

TL;DR

This paper investigates adiabatic invariants in the extended KdV equation, showing they can be directly calculated without transformations and are nearly conserved, extending understanding of invariants beyond the classical KdV.

Contribution

It introduces a direct method for calculating adiabatic invariants of the extended KdV equation without using near-identity transformations.

Findings

Adiabatic invariants are nearly conserved in numerical tests.

The direct calculation method simplifies analysis of higher-order KdV equations.

Exact invariants are recovered when expansion parameters are zero.

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.