Search Results (12)

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order ($\alpha$, where $0<\alpha \le 1$). This approach is based on the semiclassical approximation and the principles of the Maslov method. The fractal time derivative is introduced in the framework of $F^{\alpha }$ calculus. The Fisher–KPP equation is decomposed into a system of nonlinear equations that describe the dynamics of interacting quasiparticles within classes of trajectory-concentrated functions. A key element in constructing approximate quasiparticle solutions is the interplay between the dynamical system of quasiparticle moments and an auxiliary linear system of equations, which is coupled with the nonlinear system. General constructions are illustrated through examples that examine the effect of the fractal parameter ($\alpha$) on quasiparticle behavior. Full article

We review constructions of three-dimensional ‘quantum’ black holes. Such spacetimes arise via holographic braneworlds and are exact solutions to an induced higher-derivative theory of gravity consistently coupled to a large-c quantum field theory with an ultraviolet cutoff, accounting for all orders of semi-classical backreaction. Notably, such quantum-corrected black holes are much larger than the Planck length. We describe the geometry and horizon thermodynamics of a host of asymptotically (anti-) de Sitter and flat quantum black holes. A summary of higher-dimensional extensions is given. We survey multiple applications of quantum black holes and braneworld holography. Full article

The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example. Full article

We apply the original semiclassical approach to the kinetic ionization equation with the nonlocal cubic nonlinearity in order to construct the family of its asymptotic solutions. The approach proposed relies on an auxiliary dynamical system of moments of the desired solution to the kinetic equation and the associated linear partial differential equation. The family of asymptotic solutions to the kinetic equation is constructed using the symmetry operators acting on functions concentrated in a neighborhood of a point determined by the dynamical system. Based on these solutions, we introduce the nonlinear superposition principle for the nonlinear kinetic equation. Our formalism based on the Maslov germ method is applied to the Cauchy problem for the specific two-dimensional kinetic equation. The evolution of the ion distribution in the kinetically enhanced metal vapor active medium is obtained as the nonlinear superposition using the numerical–analytical calculations. Full article

We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights. Full article

A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given. Full article

The asymptotic phase is a fundamental quantity for the analysis of deterministic limit-cycle oscillators, and generalized definitions of the asymptotic phase for stochastic oscillators have also been proposed. In this article, we show that the asymptotic phase and also amplitude can be defined for classical and semiclassical stochastic oscillators in a natural and unified manner by using the eigenfunctions of the Koopman operator of the system. We show that the proposed definition gives appropriate values of the phase and amplitude for strongly stochastic limit-cycle oscillators, excitable systems undergoing noise-induced oscillations, and also for quantum limit-cycle oscillators in the semiclassical regime. Full article

We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonlocal Gross–Pitaevskii equation. We solve the eigenvalue problem for the nonlocal stationary Gross–Pitaevskii equation basing on the semiclassical asymptotics found for the Cauchy problem of the parametric family of linear equations associated with the time-dependent Gross–Pitaevskii equation in the space of extended dimension. The approach proposed uses symmetries of equations in the space of extended dimension. Full article

We apply the semi-classical limit of the generalized $SO\left(3\right)$ map for representation of variable-spin systems in a four-dimensional symplectic manifold and approximate their evolution terms of effective classical dynamics on $T^{*}{\mathcal{S}}_2$. Using the asymptotic form of the star-product, we manage to “quantize” one of the classical dynamic variables and introduce a discretized version of the Truncated Wigner Approximation (TWA). Two emblematic examples of quantum dynamics (rotor in an external field and two coupled spins) are analyzed, and the results of exact, continuous, and discretized versions of TWA are compared. Full article

We study Gaussian wave beam and wave packet types of solutions to the linearized cold plasma system in a toroidal domain (tokamak). Such solutions are constructed with help of Maslov’s complex germ theory (short-wave or semi-classical asymptotics with complex phases). The term “semi-classical” asymptotics is understood in a broad sense: asymptotic solutions of evolutionary and stationary partial differential equations from wave or quantum mechanics are expressed through solutions of the corresponding equations of classical mechanics. This, in particular, allows one to use useful geometric considerations. The small parameter of the expansion is $h=\lambda /2\pi L$ where $\lambda$ is the wavelength and L the dimension of the system. In order to apply the asymptotic algorithm, we need this parameter to be small, so we deal only with high-frequency waves, which are in the range of lower hybrid waves used to heat the plasma. The asymptotic solution appears to be a Gaussian wave packet divided by the square root of the determinant of an appropriate Jacobi matrix (“complex divergence”). When this determinant is zero, focal points appear. Our approach allows one to write out asymptotics near focal points. We also claim that this approach is very practical and leads to formulas that can be used for numerical simulations in software like Wolfram Mathematica, Maple, etc. For the particular case of high-frequency beams, we present a recipe for constructing beams and packets and show the results of their numerical implementation. We also propose ideas to treat the more difficult general case of arbitrary frequency. We also explain the main ideas of asymptotic theory used to obtain such formulas. Full article

Traditional first order JWKB method ( $=:{\left(JWKB\right)}_1$ ) is a conventional semiclassical approximation method mainly used in quantum mechanical systems for accurate solutions. ${\left(JWKB\right)}_1$ general solution of the Time Independent Schrodinger’s Equation (TISE) involves application of the conventional asymptotic matching rules to give the accurate wavefunction in the Classically Inaccessible Region (CIR) of the related quantum mechanical system. In this work, Bessel Differential Equation of the first order ( $=:{\left(BDE\right)}_1$ ) is chosen as a mathematical model and its ${\left(JWKB\right)}_1$ solution is obtained by first transforming into the normal form via the change of independent variable. The ${\left(JWKB\right)}_1$ general solution for appropriately chosen initial values in both normal and standard form representations is analyzed via the generalized ${\left(JWKB\right)}_1$ asymptotic matching rules regarding the ${\tilde{S}}_{ij}$ matrix elements given in the literature. Instead of applying the common ${\left(JWKB\right)}_1$ asymptotic matching rules relying on the physical nature of the quantum mechanical system, i.e., a physically acceptable (normalizable) wavefunction, a pure semiclassical analysis is studied via the ${\left(BDE\right)}_1$ model mathematically. Finally, an application to a specific case of the exponential potential decorated quantum mechanical bound state problem is presented. Full article