Semiclassically concentrated solutions for the one-dimensional Fokker–Planck equation with a nonlocal nonlinearity

Bellucci, Stefano; Trifonov, A. Yu.

doi:10.1088/0305-4470/38/7/l01

Cited by 6 publications

(11 citation statements)

References 27 publications

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“…For the linear caseL ′ The nonlocal Gross-Pitaevskii equation (known as the Hartree-type equation in the mathematical literature) [42], the nonlocal Fokker-Planck equation [45], and the nonlocal FKPP equation [43] are examples of nonlinear equations for which semiclassical asymptotics have been constructed.…”

Section: The Consistent System and A Semiclassical Approximationmentioning

confidence: 99%

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Levchenko

Шаповалов

Trifonov

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tThe classical group analysis approach used to study the symmetries of integro-differential equations in a semiclassical approximation is considered for a class of nearly linear integrodifferential equations. In a semiclassical approximation, an original integro-differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis.The approach is illustrated by the calculation of the Lie symmetries in explicit form for a special case of the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov population equation.

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Section: The Consistent System and A Semiclassical Approximationmentioning

confidence: 99%

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Levchenko

Шаповалов

Trifonov

2012

Journal of Mathematical Analysis and Applications

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“…By analogy with [10,15] For the functions belonging to P τ (Y (τ )), the following asymptotic estimates can be immediately obtained…”

Section: Semiclassical Approximationmentioning

confidence: 99%

Nonlinear Fokker-Planck Equation in the Model of Asset Returns

Шаповалов¹

2008

SIGMA

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Abstract. The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a FokkerPlanck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leading term of the Cauchy problem solution asymptotic in a formal small parameter in semiclassical approximation following the complex WKBMaslov method in the class of trajectory concentrated functions.

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“…The total EES for moments of all orders was derived in constructing approximate semiclassical solutions for a one-dimensional FPKE in [11].…”

Section: The Cauchy Problem and The Evolution Operatormentioning

confidence: 99%

“…The group properties of the semiclassical approximation were considered in quantum mechanics and in some models of the quantum field theory [21]. The semiclassical method for solving the Cauchy problem in the class of TCF's has been developed for the Hartree type equation [17,18,19,20] and for the one-dimensional FPKE [11,22,23]. For the Hatree type equation, symmetry operators have been found in the TCF class.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity

Шаповалов¹,

Rezaev²,

Trifonov³

2007

SIGMA

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Semiclassically concentrated solutions for the one-dimensional Fokker–Planck equation with a nonlocal nonlinearity

Cited by 6 publications

References 27 publications

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation

Nonlinear Fokker-Planck Equation in the Model of Asset Returns

Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity

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