Symmetry Operators of the Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Equation with a Quadratic Operator

Levchenko, E.A.; Trifonov, A. Yu.; Шаповалов, А. В.

doi:10.1007/s11182-014-0194-x

Cited by 2 publications

(9 citation statements)

References 9 publications

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“…We call Equations (30)-(32) the Einstein-Ehrenfest system for (29), following [23,29]. Applying initial condition (14) to Equations (25)- (27) yields…”

Section: Semiclassical Solution Of the Non-local Fisher-kpp Equationmentioning

confidence: 99%

“…Let us substitute expansion (37) and the moment σ (0) (t) determined by Equations (30) and (33) into Equation (29) and take into account estimates (20)- (22) and the fact that both u (0,0) (x, t) and u (0,1) (x, t) have the form of (18). Collecting the lower order terms in √ D 1 , we get the following equation for u (0,0) (x, t):…”

Section: Semiclassical Solution Of the Non-local Fisher-kpp Equationmentioning

confidence: 99%

“…However, for some special types of equations, this is possible. For example, families of symmetry operators are constructed for the nonlocal generalization of the Gross-Pitaevskii equation in [28], and for the multidimensional nonlocal Fisher-KPP equation in [29] in the framework of the semiclassical approximation method.…”

Section: Symmetry Operatorsmentioning

confidence: 99%

“…Here, we briefly describe the construction of a family of symmetry operators and a set of corresponding solutions for the semiclassically reduced Equation (39) according to [29].…”

Section: Symmetry Operatorsmentioning

confidence: 99%

“…In the framework of the formalism of semiclassical asymptotics, we discuss the applicability of the method for constructing symmetry operators developed in [28,29] to the system of model equations under consideration to extend a set of approximate analytical solutions.…”

Section: Introductionmentioning

confidence: 99%

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Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

Шаповалов

Trifonov

2019

Symmetry

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We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

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“…We call Equations (30)-(32) the Einstein-Ehrenfest system for (29), following [23,29]. Applying initial condition (14) to Equations (25)- (27) yields…”

Section: Semiclassical Solution Of the Non-local Fisher-kpp Equationmentioning

confidence: 99%

Section: Semiclassical Solution Of the Non-local Fisher-kpp Equationmentioning

confidence: 99%

Section: Symmetry Operatorsmentioning

confidence: 99%

“…Here, we briefly describe the construction of a family of symmetry operators and a set of corresponding solutions for the semiclassically reduced Equation (39) according to [29].…”

Section: Symmetry Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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