On a Coupled Nonlinear Schrödinger System: A Ermakov Connection

Abstract

Dedicated to the memory of Sergei Manakov]Ermakov-type invariants are isolated for a subsystem of an N -component coupled nonlinear Schrödinger system. An algorithmic procedure is presented which reduces this Ermakov-Ray-Reid system to quadrature. The method is illustrated in the single component case by application to a nonlinear system descriptive of the propagation of transverse waves in incompressible hyperelastic media subject to rotation. An extended Ermakov-Ray-Reid system is presented which, if it has underlying Hamiltonian structure, is also amenable to the algorithm.

On the mathematical and geometrical structure of the determining equations for shear waves in nonlinear isotropic incompressible elastodynamics

“…This format has been fundamental to unify all of the Carroll's results [9,10,11] and to generalize them. Moreover, this format has been used by Rogers [36] to show a relationship between (1.1) and the Ermakov-Ray-Reid system. For example, let us consider a special class of such similarity solutions: the Carroll's finite amplitude circularly-polarized harmonic progressive waves [9].…”

On Dirichlet two-point boundary value problems for the Ermakov–Painlevé IV equation

“…16) where K I and K II are constants of integration and it is required that R 2 I > 2β(E + β), R 2 II > 2β(E + β). The relations (A.15), (A.16) are subject to the constraint U + V = 1 and the latter relation may be used to calculate V in terms of U as given by (A.15) without recourse to (A.14).…”

“…Moreover, while solitonic and their associated Painlevé reductions generically admit linear representations, Ermakov-Ray-Reid systems likewise have been shown to have associated linear structure [15]. Despite these important commonalities investigations of Painlevé and Ermakov-Ray-Reid type systems have preceded independently until recently in [16] wherein hybrid Ermakov-Painlevé II symmetry reductions have been derived for N+1-dimensional resonant nonlinear Schrödinger systems. A range of boundary value problems for the Painlevé II equation has been investigated in [17][18][19][20][21], notably in the context of multi-ion electrodiffusion.…”

A Novel Ermakov–Painlevé II System: ‐Dimensional Coupled NLS and Elastodynamic Reductions

“…This kind of procedure has been recently adopted in other nonlinear elastodynamics contexts in [49,50].…”