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

The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative examples of the one-dimensional symmetry operators are presented.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semiclassical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained

Symmetry operators of the two-component Gross–Pitaevskii equation with a Manakov-type nonlocal nonlinearity

We consider an integro-differential 2-component multidimensional Gross-Pitaevskii equation with a Manakov-type cubic nonlocal nonlinearity. In the framework of the WKB-Maslov semiclassical formalism, we obtain a semiclassically reduced 2-component nonlocal Gross- Pitaevskii equation determining the leading term of the semiclassical asymptotic solution. For the reduced Gross-Pitaevskii equation we construct symmetry operators which transform arbitrary solution of the equation into another solution. Constructing the symmetry operator is based on the Cauchy problem solution technique and uses an intertwining operator which connects two solutions of the reduced Gross-Pitaevskii equation. General structure of the symmetry operator is illustrated with a 1D case for which a family of symmetry operators is found explicitly and a set of exact solutions is generated

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

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

Adomian decomposition method for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovsky-Piskunov equation

The Adomian decomposition method is applied to construct an approximate solution of the generalized onedimensional Fisher Kolmogorov–Petrovsky–Piskunov equation describing the population dynamics with nonlocal competitive losses. An approximate solution is constructed in the class of decreasing functions. The diffusion operator is taken as a reversible linear operator. The inverse operator is presented in terms of the diffusion propagator. An example of the approximate solution of the Cauchy problem for the function of competitive losses and for the initial function of the Gaussian type is considered

Simultaneous asynchronous detection-discrimination of signals at the output of multibeam channels with fading

The paper is devoted to the problem of simultaneous detection and discrimination of M-signals with unknown time position at the output of a multibeam channel with common fading. The signal search is assumed to carry out at a large a priori interval containing many elements of resolution in terms of delays. Based on the theory of outliers of random processes, the characteristics of simultaneous detection and distinguishing of signals are found and investigated

Adomyan decomposition method for a two-component nonlocal reaction-diffusion model of the Fisher-Kolmogorov-Petrovsky-Piskunov type

We consider an approach to constructing approximate analytical solutions for the one-dimensional twocomponent reaction diffusion model describing the dynamics of population interacting with the active substance surrounding the population. The system of model equations includes the nonlocal generalized Fisher–Kolmogorov–Petrovsky–Piskunov equation for the population density and the diffusion equation for the density of the active substance. Both equations contain additional terms describing the mutual influence of the population and the active substance. To find approximate solutions of the system of model equations, we first use the perturbation method with respect to the small parameter of interaction between the population and the active substance. Then we apply the well-known iterative method developed by G. Adomian to solve equations for terms of perturbation series. In the method proposed, the solution is presented as a series whose terms are determined by the corresponding iterative procedure. In this work, the diffusion operator is taken as the operator for which the inverse operator is expressed in terms of the diffusion propagator. This allows one to find the approximate solutions in the class of functions decreasing at infinity. As an illustration, we consider an example of solving the Cauchy problem for the initial functions of a Gaussian form

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher-KPP equation

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure

Amplitude estimation of signal with unknown duration

Quasilikelihood and maximum likelihood algorithms for estimating the amplitude of arbitrary waveform signal with unknown duration have been synthesized. Characteristics of the synthesized algorithms have been also found