The stochastic Liouville equation and Padé approximants

The applicability of Padé approximant techniques to solving the stochastic Liouville equation is discussed. The special case of an axially symmetric spin system undergoing isotropic Brownian motion is studied. Two types of expansions are explored which yield efficient algorithms for spectral simulations.     

A new program based on stochastic Liouville equation for the analysis of superhyperfine interaction in CW-ESR spectroscopy

In the slow-motion region, ESR spectra cannot be expressed as a sum of simple Lorentzian lines. Studies of Freed and co-workers, on nitroxides in liquids gained information on the microscopic models of rotational dynamics, relying much on computer programs for simulation of ESR spectra based on the stochastic Liouville equation (SLE). However, application of Freed's method to copper system of biological interest has been for a long time precluded by lack of a full program able to simulate ESR spectra containing more than one hyperfine interaction. Direct extension of the Freed's approach in order to include superhyperfine interaction is not difficult from a theoretical point of view but the resulting algorithm is problematical because it leads to substantial increase in the dimensions of the matrix related to the spin-hamiltonian operator. In this paper preliminary results of a new program, written in C, which includes the superhyperfine interactions are presented. This preliminary version of the program does not take into account a restoring potential, so it can be used only in isotropic diffusion conditions. A comparison with an approximate method previously developed in our laboratory, based on a post-convolution approach, is discussed.     

The non-Markovian stochastic Liouville equation and anomalous quantum relaxation kinetics

We analyze within the continuous-time random walk approach, the kinetics of phase and population relaxation in quantum systems induced by noise with the anomalously slowly decaying correlation function P(t) ∝ (wt)^-α, where 0 < α < 1. The relaxation kinetics is shown to be anomalously slow. Moreover, for α < 1, in the limit of a short characteristic time of fluctuations w^-1, the kinetics is independent of w. As α → 1, the relaxation regime changes from the static limit to narrowing of fluctuation. Simple analytical expressions are obtained that describe the specific features of the kinetics.     

A comment on the generalized Liouville equation

The generalized Liouville equation is studied in a new light using the Lie derivative of a differential form with respect to a vector field.     

Integrable systems as nonlinear realizations of infinite-dimensional symmetries: The Liouville equation example

The Liouville equation is shown to have a natural interpretation in terms of the nonlinear realization of an infinite parameter conformal group in 1+1-dimensions. The relevant zero-curvature representation and Bäcklund transformations get a simple treatment in this approach. The proposed construction can hopefully be generalized to other integrable systems.     

Liouville equation and schottky problem

An Ansatz for the Poincaré metric on compact Riemann surfaces is proposed. This implies that the Liouville equation reduces to an equation resembling a nonchiral analogous of the higher genus relationships (KP equation) arising within the framework of Schottky's problem solution. This approach connects uniformization (Fuchsian groups) and moduli space theories with KP hierarchy. Besides its mathematical interest, the Ansatz has some applications within the framework of quantum Riemann surfaces arising in 2D gravity.Partly supported by a SERC fellowship and by the European Community Research Programme Gauge Theories, applied supersymmetry and quantum gravity, contract SC1-CT92-0789.     

Prolongation theory. A new nonlinear Schrödinger equation

We discuss a new kind of nonlinear Schrödinger equation from the viewpoint of prolongation theory. It is shown that the equation possess a Lax pair with a 3 × 3 matrix structure. It is further demonstrated that by a multiple scale perturbation of Zakharovet al. it can be reduced to the usual KdV equation.     

A new application of Riccati equation to some nonlinear evolution equations

By means of symbolic computation, a new application of Riccati equation is presented to obtain novel exact solutions of some nonlinear evolution equations, such as nonlinear Klein-Gordon equation, generalized Pochhammer-Chree equation and nonlinear Schrödinger equation. Comparing with the existing tanh methods and the proposed modifications, we obtain the exact solutions in the form as a non-integer power polynomial of tanh (or tan) functions by using this method, and the availability of symbolic computation is demonstrated.     

Comment on the Liouville equation and multisolitons

It has been asserted in the literature that a particular solution of the Liouville equation, ?_xtσ = e^σ, can be interpreted as an N-soliton. It is shown here that this solution id in fact familiar one-soliton written in a misleading form.     

Topology of solutions to the Liouville equation

Suggestions concerning the generalization of geometric quantization to the case of non-linear field theories are given. Preliminary results for the Liouville field theory are presented.     

A new auxiliary equation and exact travelling wave solutions of nonlinear equations

A new auxiliary ordinary differential equation and its solutions are used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the auxiliary equation which has more new exact solutions. More new exact travelling wave solutions are obtained for the quadratic nonlinear Klein–Gordon equation, the combined KdV and mKdV equation, the sine-Gordon equation and the Whitham–Broer–Kaup equations.     

Quantum equations of motion and the Liouville equation

Equations of motion for explicitly time-dependent operators in the Heisenberg and Schrödinger pictures, respectively, are reviewed. A simple transformation reduces the related equation in the Heisenberg picture to closed form. An algorithm is introduced for the classical limit which, in either picture, returns the classical equation of motion for dynamical functions. Applications of this algorithm to the equation of motion for the density matrix reduces it to the classical Liouville equation. A property related to this algorithm is established for the commutator of any two analytic functions of finite polynomials of conjugate variables. Thus, if \(\hat F\) and ? are two such functions and \(\hat F^\prime \) and ?′ contain an arbitrary permutation of variables, then $$[\hat F,\hat G] = [\hat F^\prime ,\hat G^\prime ] + \hbar ^2 \hat{A}$$ where  is a remainder commutator. In the classical limit [ \(\hat F\) , ?] is invariant to such permutation.     

Simulation of slow-motion CW EPR spectrum using stochastic Liouville equation for an electron spin coupled to two nuclei with arbitrary spins: matrix elements of the Liouville superoperator

An algorithm is developed that extends the well known nitroxide slow-motional continuous wave electron paramagnetic resonance (EPR) simulation technique developed originally by Meirovitch et al. [E. Meirovitch, D. Inger, E. Inger, G. Moro, J.H. Freed, J. Chem. Phys. 77 (1982) 3915-3938], and implemented by Schneider and Freed [D.J. Schneider, J.H. Freed, Calculating slow motional magnetic resonance spectra: a user's guide, in: Biological Magnetic Resonance, vol. 6, Plenum Publishing Corporation, 1989]. This paper deals with the more general case of coupling of one electron spin to two nuclear spins. A complete listing of the matrix elements of the Liouville superoperator for this extension has been included. This advance has been successfully tested by reproducing the observed spectral lineshapes of a solution of the novel radical Mes(*)(CH(3))P-PMes(*) [Mes(*)=2,4,6 (tBu)(3)C(2)H(2)] in tetrahydrofuran (THF), in which the radical is undergoing slow tumbling, with the coupling of one electron spin to two physically and magnetically inequivalent phosphorus ((31)P) nuclei.     

A new approximate analytical approach for dispersion relation of the nonlinear Klein-Gordon equation

A novel approach is presented for obtaining approximate analytical expressions for the dispersion relation of periodic wavetrains in the nonlinear Klein-Gordon equation with even potential function. By coupling linearization of the governing equation with the method of harmonic balance, we establish two general analytical approximate formulas for the dispersion relation, which depends on the amplitude of the periodic wavetrain. These formulas are valid for small as well as large amplitude of the wavetrain. They are also applicable to the large amplitude regime, which the conventional perturbation method fails to provide any solution, of the nonlinear system under study. Three examples are demonstrated to illustrate the excellent approximate solutions of the proposed formulas with respect to the exact solutions of the dispersion relation. (c) 2001 American Institute of Physics.     

A new method to obtain approximate symmetry of nonlinear evolution equation from perturbations

A novel method for obtaining the approximate symmetry of a partial differential equation with a small parameter is introduced. By expanding the independent variable and the dependent variable in the small parameter series, we obtain more affluent approximate symmetries. The method is applied to two perturbed nonlinear partial differential equations and new approximate solutions are derived.     

A new spiky soliton and an explosive mode of the nonlinear driftwave equation

A new type of nonlinear wave modes which occurs in the electrostatic drift waves in an inhomogeneous magnetized plasma is presented. The author predicts the existence of a new type of spiky solitary wave and an explosive mode with a negative potential as stationary solutions of this equation. These solutions are a consequence of a density gradient and not connected with a temperature gradient. These new nonlinear wave solutions appear to make a step forward in the general scheme of nonlinear normal modes for plasma waves. Using these nonlinear wave modes, the author can explain the solitary structure and the explosive event concerning nonlinear drift waves propagating in space     

A new model equation for nonlinear Rossby waves and some of its solutions

In this paper, we investigate the $(2+1)$ dimensional nonlinear Rossby waves under non-traditional approximation. Using the asymptotic methods of multiple scales and weak nonlinear perturbation expansions, we derive a new modified Zakharov–Kuznetsov equation from the barotropic potential vorticity equation with the complete Coriolis parameter, the topography and the dissipation. Based on the new auxiliary equation method, new exact solutions of the new mZK equation are obtained when the dissipation is absent. However, the new auxiliary equation method fails to solve the new mZK equation with the dissipative term. Therefore, the weak nonlinear method and the homotopy perturbation method are developed to solve the obtained new mZK equation. Through numerical simulations, the results show the effects of different parameters on Rossby waves.