New family of asymptotic solutions of Helmholtz equation

Maltsev, Nick

doi:10.1063/1.530596

Cited by 4 publications

(8 citation statements)

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“…For example, algorithms that extend the Wentzel, Kramers and Brilluen (WKB) technique have recently been developed to find power series solutions to Eq. (12) [23]. For related work, see also Ref.…”

Section: The Ermakov-pinney Equation In Cosmologymentioning

confidence: 99%

Ermakov-Pinney equation in scalar field cosmologies

2002

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It is shown that the dynamics of cosmologies sourced by a mixture of perfect fluids and self-interacting scalar fields are described by the non-linear, Ermakov-Pinney equation. The general solution of this equation can be expressed in terms of particular solutions to a related, linear differential equation. This characteristic is employed to derive exact cosmologies in the inflationary and quintessential scenarios. The relevance of the Ermakov-Pinney equation to the braneworld scenario is discussed. reviews, see, e.g., [3]). The simplest mechanism for inflation utilises the potential energy associated with the self-interaction of a scalar inflaton field to drive the accelerated expansion. High redshift observations of type Ia supernovae suggest that the universe is experiencing another phase of accelerated expansion at the present epoch [4]. This, combined with the CMB data, presents a picture of the universe dominated by a dark energy component [5,6]. One possible source of this dark energy is a scalar quintessence field that interacts with baryonic and non-baryonic matter in such a way that its potential energy is currently dominating the cosmic dynamics [7]. The observations favour a model of structure formation where 70% of the energy density of the universe is presently in the form of quintessence [6]. The remaining fraction of the energy density is comprised of visible and cold dark matter which collectively act as a pressureless perfect fluid.The ekpyrotic scenario has recently been proposed as an alternative to the standard inflationary cosmology [8]. In this scenario the big bang is interpreted as the collision of two domain walls or branes travelling through a fifth dimension. Before the collision, the effective dynamics on the four dimensional branes is described by Einstein's gravity minimally coupled to a self-interacting scalar field. The field parametrizes the separation between the branes and at early times slowly rolls down a negative potential. This results in an accelerated collapse of the universe and since the field is minimally coupled, its energy density is related to the Hubble parameter by the standard Friedmann equation. Thus, self-interacting scalar fields play a central role in modern cosmology and in view of the above developments, it is important to investigate cosmologies that contain both a scalar field and a perfect fluid.In this paper, we develop an analytical approach to models of this type by expressing the cosmological field equations in terms of an Ermakov system [9,10,11]. In general, an Ermakov system is a pair of coupled, second-order, non-linear ordinary differential equations (ODEs) [11] and such systems often arise in studies of nonlinear optics [12], nonlinear elasticity [13], molecular structures [14] and quantum cosmology [15]. (For further references, see, e.g., Refs. [16]). In the one-dimensional case, the two equations decouple and the system reduces to a single equation known as the Ermakov-Pinney equation [9,17]. This is given by 1

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Section: The Ermakov-pinney Equation In Cosmologymentioning

confidence: 99%

Ermakov-Pinney equation in scalar field cosmologies

2002

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“…1 shows that in the limit x → ∞ the condition (12) holds for the roots 1 and 3, while in the limit x → ∞ it does for the roots 1 and 2 which are the complex conjugates to each other in this region. Thus, κ (1) (x; ) is the root 1 for N = 1. For N = 2 (see Fig.…”

Section: Example: a Linear Potentialmentioning

confidence: 97%

“…What is important, one of two relevant roots κ(x; ) of Eqs. (10) obeys the condition (12) for any point x ∈ [a, b] (hereinafter this root will be denoted as κ (1) (x; )). For other roots this condition breaks when x crosses turning points.…”

Section: Formalismmentioning

confidence: 99%

“…In 1994 year two generalized WKB approaches has been proposed -in the paper [1], for solving the Helmholtz equation, and in the papers [2,3] (see also [4]), for solving the one-dimensional Schrödinger equation (OSE). Both are based on the same ideato use the REDs in order to obtain WKB series regular at the turning points (later, in 2002 year, this idea has been reopened in [5]).…”

Section: Introductionmentioning

confidence: 99%

“…However, despite the common idea, these approaches differ from each other, because they treat the REDs in a different way. For example, in the approach [1] the searched-for solutions are expanded traditionally in integer powers of Plank's constant . While in the formalism [2,3] the role of a small parameter is played by some function ǫ(x; ) of the spatial variable x and Plank's constant.…”

Section: Introductionmentioning

confidence: 99%

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Hybrid Taylor-WKB series

Chuprikov

2009

Preprint

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A generalized WKB approach for constructing WKB series endowed with some properties of Taylor ones is presented. Apart from the Riccati equation itself its formalism involves also the Riccati-equation's derivatives (REDs) obtained by differentiating of the former with respect to a spatial variable. For any smooth potential barrier given in the finite spatial interval to include turning points, the zeroth-order term of presented WKB series is regular everywhere in this interval. Moreover, the more REDs are used, the more exact the zeroth-order solution is.

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The WKB method and differential consequences of the Riccati equation

Chuprikov

2011

Math Notes

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New family of asymptotic solutions of Helmholtz equation

Abstract: A new family of asymptotic solutions of the Helmholtz equation can be derived. It is shown that new solutions are regular at caustics and turning points and do not require any special functions.

Cited by 4 publications

References 3 publications

Ermakov-Pinney equation in scalar field cosmologies

Ermakov-Pinney equation in scalar field cosmologies

Hybrid Taylor-WKB series

The WKB method and differential consequences of the Riccati equation

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