On multi-component Ermakov systems in a two-layer fluid: a variational approach

Abstract

By introducing Madelung-type transformations, a two-layer fluid model with a circular paraboloidal bottom topography is reduced to coupled nonlinear Schrödinger equations incorporating harmonic traps and de Broglie–Bohm quantum potentials. A kind of multi-component Ermakov system is obtained via a variational approach and a multi-parameter Gaussian ansatz. In particular, three typical reductions to generalized Ermakov systems are discussed. Notable integrals of motion and additional Hamiltonian structures are employed to derive analytical solutions in terms of elliptic integrals for the multi-component Ermakov systems.

On Multi‐component Ermakov Systems in a Two‐Layer Fluid: Integrable Hamiltonian Structures and Exact Vortex Solutions

“…Because the multicomponent Ermakov systems were proposed in 1996 [35], there has been limited work done [13,[35][36][37]. In this paper, by introducing a general elliptic vortex ansatz, we have successfully constructed multicomponent Ermakov systems in the two-layer fluid.…”

“…In [10], Becker and Bercovici investigated the interactions of water waves with a deformable sea-bed while gravity currents and related phenomena were studied by Simpson [11]. Important contributions were also made by Rogers et al [12] and An et al [13] with regard to analytical solutions and reductions to particular Ermakov systems.…”

“…withḢ N (t) = 0. Indeed, by employing the Madelung transformation and variational approach, An and her coauthors recently constructed multicomponent Ermakov systems for the two-layer fluid without Coriolis force [13]. Yet, due to the limitations inherent in the variational approach, it loses its power in certain physically meaningful cases, such as when Coriolis force is present and when other kinds of geometry are considered.…”

“…To the best of our knowledge, to date, except for Refs. [13,[35][36][37], little work has been done on multicomponent Ermakov systems. Therefore, it seems worthy of investigating whether alternative exact methods exist that lead to multicomponent Ermakov systems.…”