Ermakov systems, nonlinear superposition, and solutions of nonlinear equations of motion

Reid, James L.; Ray, John R.

doi:10.1063/1.524625

Cited by 141 publications

(80 citation statements)

References 6 publications

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“…We prove that the theory developed by Lie can easily be adapted to deal with many examples of such SODE systems, and in this way the superposition rules which are generally used for first-order differential equations can also be used for dealing with Ermakov systems. We find room in this way for implicit nonlinear superposition rules in the terminology of [36,41]. The Ermakov-Lewis invariants [29] appear in a natural way as functions defining the foliation associated to the superposition rule.…”

Section: Introductionmentioning

confidence: 99%

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Recent Applications of the Theory of Lie Systems in Ermakov Systems

Cariñena

Lucas

Rañada

2008

SIGMA

112

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Abstract. We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found from this new perspective. We also obtain new results, such as a new superposition rule for the Pinney equation in terms of three solutions of a related Riccati equation.

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Section: Introductionmentioning

confidence: 99%

“…Much work has also been devoted to use Hamiltonian or Lagrangian structures in the study of such a system, see e.g. [30,31], and many generalisations or new insights from the mathematical point of view can be found in [32,33,34,35,36,37,38,39,40].…”

Section: Introductionmentioning

confidence: 99%

Recent Applications of the Theory of Lie Systems in Ermakov Systems

Cariñena

Lucas

Rañada

2008

SIGMA

112

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“…The coupled pair of nonlinear equations (2.7) and (2.8) constitute a Ermakov-Ray-Reid system [29,30,36,37,47]. On introduction of new dependent and independent variables α, β and ζ according to…”

Section: Integrable Ermakov-ray-reid Reductionmentioning

confidence: 99%

“…Nonlinear coupled systems of Ermakov-Ray-Reid type have their roots in the classical work of Ermakov [18] and were introduced by Ray and Reid [29,30]. Subsequently, 2+1-dimensional Ermakov-Ray-Reid systems were constructed in Ref.…”

Section: Introductionmentioning

confidence: 99%

Ermakov-Modulated Nonlinear Schrödinger Models. Integrable Reduction

Rogers¹,

Saccomandi²,

Vergori³

2021

JNMP

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Nonlinear Schrödinger equations with spatial modulation associated with integrable Hamiltonian systems of Ermakov-Ray-Reid type are introduced. An algorithmic procedure is presented which exploits invariants of motion to construct exact wave packet representations with potential applications in a wide range of physical contexts such as, 'inter alia', the analysis of Bloch wave and matter wave solitonic propagation and pulse transmission in Airy modulated NLS models. A particular Ermakov reduction for Mooney-Rivlin materials is set in the broader context of transverse wave propagation in a class of higher-order hyperelastic models of incompressible solids.

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“…Con este formalismo se ha predicho la reflectividad aumentada debido a las discontinuidades en las derivadas en el índice de refracción (aunque n sea constante) [7]. Esta expresión corresponde a ondas contrapropagantes y establece la forma del principio de superposición no lineal [8][9][10][11] para la ecuación del vector de onda.…”

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Generalized Snell’s refraction angle relationship

Fernández¹

2012

Opt. Pura Apl.

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RESUMEN:La representación de amplitud y fase permite obtener ecuaciones diferenciales para la amplitud y el vector de onda. Éstas ecuaciones pueden desacoplarse utilizando el invariante de Ermakov. Para medios estratificados transparentes, se muestra que la componente del vector de onda en la dirección de estratificación , cumple con la relación tan constante, donde es el ángulo de inclinación de propagación. Dicha componente satisface la ecuación diferencial no lineal 2 0, donde es el índice de refracción, es una constante y la magnitud del vector de onda en el vacío. Para variaciones suaves del índice de refracción comparadas con la longitud de onda, la relación anterior deviene en la relación de Snell generalizada sin . En el caso de interfase abrupta entre dos medios homogéneos, se recupera la relación usual de Snell sin sin . Palabras clave: Propagación en Medios Inhomogéneos, Medios Estratificados, Invariante de CamposComplementarios. ABSTRACT:The representation of waves in amplitude and phase variables can be decoupled using the Ermakov invariant. The wave vector component in the direction of stratification kz, satisfies the relationship tan constant, where is the angle of propagation. This component must fulfill the nonlinear differential equation 2 0, where is the refractive index, is a constant and the wave vector magnitude in vacuum. For soft variations of the refractive index compared with the wavelength, this relationship becomes the so called generalized Snell relationship sin . For an abrupt interface, the usual Snell equation sin sin is recovered.

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Ermakov systems, nonlinear superposition, and solutions of nonlinear equations of motion

Cited by 141 publications

References 6 publications

Recent Applications of the Theory of Lie Systems in Ermakov Systems

Recent Applications of the Theory of Lie Systems in Ermakov Systems

Ermakov-Modulated Nonlinear Schrödinger Models. Integrable Reduction

Generalized Snell’s refraction angle relationship

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