Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Yarema A Prykarpatskyy

doi:10.3390/sym13061077

Abstract

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

Highlights

Dubrovin’s Integrability SchemeWe begin by recalling some very interesting works by B
As the right-hand side of the flow (10) defines a vector field ut = K [u] on a suitably chosen smooth functional manifold M ⊂ C ∞ (R; R) (being here locally diffeomorphic to the jet-manifold J ∞ (R; R), we checked the existence of a suitable infinite hierarchy of conservation laws for the flow (10) and related Hamiltonian structures on M via the gradient-holonomic integrability scheme [4,5]
The derived above modified Krichever-Novikov type Equation (28) is an integrable bi-Hamiltonian flow on the functional manifold M for arbitrary c2 ∈ R: vt = v xxx − 3 where the Poisson operator v x v xx v3

Summary

Introduction

We begin by recalling some very interesting works by B. As the right-hand side of the flow (10) defines a vector field ut = K [u] on a suitably chosen smooth functional manifold M ⊂ C ∞ (R; R) (being here locally diffeomorphic to the jet-manifold J ∞ (R; R), we checked the existence of a suitable infinite hierarchy of conservation laws for the flow (10) and related Hamiltonian structures on M via the gradient-holonomic integrability scheme [4,5] It means that this hierarchy is suitably ordered and satisfies the well known Noether-Lax equation dφ/dt + K 0∗ [u] · φ = 0 on M, where φ := φ[u; λ] = grad ξ [u; λ] ∈ T ∗ ( M )—the functional gradient of a functional ξ (λ) on M, depending on the constant parameter λ ∈ C as |λ| → ∞, and chosen to be a generating function of conservation laws to the vector field K : M → T ( M).

Reduction Integrability Properties

Poissonian Structure on M3

Lax-Type Integrability Analysis

Concluding Remarks