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.
Shallow water waves Nonlinear equations Invariants of KdV2 equation Adiabatic invariants
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The authors thank Prof. Eryk Infeld and Prof. George Rowlands for inspiring discussions.
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