Searcharxiv⌕ Search

arXiv subjects

Sergei K. Suslov

Publications and source records attributed to Sergei K. Suslov.

At least 37 records · Page 2Linked to original sources
arXiv2012

Wave Packets and Coherent Structures for Nonlinear Schroedinger equations in Variable Nonuniform Media

We determine conditions under which a generic gauge invariant nonautonomous and inhomogeneous nonlinear partial differential equation in the two-dimensional space-time continuum can be transform into standard autonomous forms. In addition to the nonlinear Schroedinger equation, important examples include the derivative nonlinear Schroedinger equation, the quintic complex Ginzburg-Landau equation, and the Gerdjikov-Ivanov equation. This approach provides a mathematical description of nonstationary media supporting unidimensional signal propagation and/or total field trapping. In particular, we study self-similar and nonspreading wave packets for Schroedinger equations. Some other important coherent structures are also analyzed and applications to nonlinear waves in inhomogeneous media such as atmospheric plasma, fiber optics, hydrodynamics, and Bose-Einstein condensation are discussed.

math-ph↗
arXiv2012

Relativistic Coulomb Integrals and Zeilberger's Holonomic Systems Approach. I

With the help of computer algebra we study the diagonal matrix elements , where O are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem. Using Zeilberger's extension of Gosper's algorithm and a variant to it, three-term recurrence relations for each of these expectation values are derived together with some transformation formulas for the corresponding generalized hypergeometric series. In addition, the virial recurrence relations for these integrals are also found and proved algorithmically.

quant-ph↗
arXiv2012

An eigenvalue problem for the associated Askey-Wilson polynomials

To derive an eigenvalue problem for the associated Askey-Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey-Wilson polynomials introduced by Ismail and Rahman. The Askey-Wilson operator, applied in each variable separately, maps this function to the ordinary Askey-Wilson polynomials with different sets of parameters. A third Askey-Wilson operator is found with the help of a computer algebra program which links the two, and an eigenvalue problem is stated.

math.CA↗
arXiv2012

The Berry Phase for Simple Harmonic Oscillators

We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system.

quant-ph↗
arXiv2012

On a Hidden Symmetry of Quantum Harmonic Oscillators

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.

quant-ph↗
arXiv2011

On The Harmonic Oscillator Group

We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a view point of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schroedinger group of the free particle.

math-ph↗
arXiv2011

Exact Wave Functions for Generalized Harmonic Oscillators

We transform the time-dependent Schroedinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time-evolution of exact wave functions of generalized harmonic oscillators is determined in terms of solutions of certain Ermakov and Riccati-type systems. In addition, we show that the classical Arnold transformation is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.

math-ph↗
arXiv2011

On Integrability of Nonautonomous Nonlinear Schroedinger Equations

We show, in general, how to transform the nonautonomous nonlinear Schroedinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is outlined. A few simple integrable systems are discussed.

math-ph↗
arXiv2011

The Riccati System and a Diffusion-Type Equation

We discuss a method of constructing solution of the initial value problem for duffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered.

math-ph↗
arXiv2010

Soliton-like Solutions for Nonlinear Schroedinger Equation with Variable Quadratic Hamiltonians

We construct one soliton solutions for the nonlinear Schroedinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of a complete (super) integrability of generalized harmonic oscillators. The soliton wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schroedinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painleve transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons, and Jacobi elliptic and second Painleve transcendental solutions for several variable Hamiltonians that are important for current research in nonlinear optics and Bose-Einstein condensation. The Feshbach resonance matter wave soliton management is briefly discussed from this new perspective.

math-ph↗
arXiv2010

The Degenerate Parametric Oscillator and Ince's Equation

We construct Green's function for the quantum degenerate parametric oscillator in terms of standard solutions of Ince's equation in a framework of a general approach to harmonic oscillators. Exact time-dependent wave functions and their connections with dynamical invariants and SU(1,1) group are also discussed.

math-ph↗
arXiv2010

Quantum Integrals of Motion for Variable Quadratic Hamiltonians

We construct the integrals of motion for several models of the quantum damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the time-dependent Schroedinger equation with variable quadratic Hamiltonians. An extension of Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application.

math-ph↗
arXiv2010

Dynamical Invariants for Variable Quadratic Hamiltonians

We consider linear and quadratic integrals of motion for general variable quadratic Hamiltonians. Fundamental relations between the eigenvalue problem for linear dynamical invariants and solutions of the corresponding Cauchy initial value problems for the time-dependent Schroedinger equation are emphasized. An eigenfunction expansionof the solution of the initial value problem is also found. A nonlinear superposition principle for the generalized Ermakov systems is established as a result of decomposition of the general quadratic invariant in terms of the linear ones.

math-ph↗
arXiv2009

Mathematical Structure of Relativistic Coulomb Integrals

We show that the diagonal matrix elements $< Or^{p} >,$ where $O$ $={1,β,i\mathbf{αn}β}$ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, may be considered as difference analogs of the radial wave functions. Such structure provides an independent way of obtaining closed forms of these matrix elements by elementary methods of the theory of difference equations without explicit evaluation of the integrals. Three-term recurrence relations for each of these expectation values are derived as a by-product. Transformation formulas for the corresponding generalized hypergeometric series are discussed.

math-ph↗
arXiv2009

Relativistic Kramers-Pasternack Recurrence Relations

Recently we have evaluated the matrix elements $ $,$ where $O$ $={1,β, i\mathbf{αn}β} $ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, in terms of generalized hypergeometric functions $_{3}F_{2}(1) $ for all suitable powers and established two sets of Pasternack-type matrix identities for these integrals. The corresponding Kramers--Pasternack three-term vector recurrence relations are derived here.

math-ph↗